Evolution-20: How selection advantage arises in evolution

(Please see here for previous posts in this series.)

In the mathematics of evolutionary change, the selection advantage is a key mathematical quantity that determines the rate at which a favorable mutation spreads through the population. The selection advantage is a quantification of the net result of advantages that a variety of a species gains by virtue of its fertility and fecundity and longevity. As we saw before, even a small selection advantage can lead to rapid spread of the mutation.
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Evolution-19: The Boeing 747 in the junkyard

(Please see here for previous posts in this series.)

As I have emphasized repeatedly in this series, the hardest thing to appreciate about evolution is how a cumulative sequence of very tiny changes can lead to big changes. The problem is that our senses can only detect gross differences between organisms and our minds can only comprehend short time scales and to appreciate evolution requires us to overcome those limitations. This is why skeptics need to actually study the details and convince themselves that it works.

I have the same problems when it comes to teaching modern physics topics like quantum mechanics or special relativity. Our senses and intuition have evolved to enable us to deal with objects that are on a human (or ‘classical’) size scale and traveling at speeds that are not too great. But the effects of quantum mechanics only become manifest when describing the very small, subatomic level of particles that we cannot see, and there our intuition completely breaks down. Similarly, the effects of special relativity become manifest only for objects traveling close to the speed of light, which we do not encounter in everyday life and again our intuition is incapable of dealing with it. So when physicists talk about a single electron simultaneously traveling by many different paths from a single initial starting point to a final point, or twins aging at different rates depending on their speed of travel, these ideas initially seem preposterous.

When teaching these subjects, I warn my students that their intuition is quite likely to lead them astray, that what their gut feelings tell them is reasonable or unreasonable is undependable, and that they have to constantly check those intuitive reactions by doing calculations to convince themselves that these counter-intuitive results drop out naturally from a coherent theory

The same thing is true for evolution. Mutations are too small to be visible and time scales are too long to comprehend, so one should not depend upon what seems reasonable to make judgments. Steven Pinker (How the Mind Works 1997, p. 163) points out that: “A hypothetical mouse subjected to a selection pressure so weak that it cannot be measured could nonetheless evolve to the size of an elephant in only twelve thousand generations.” This is quite an amazing result. It is not at all intuitive and is hard to convince oneself that this could be possible unless one does the calculations, or trusts those who do the calculations.

But people who want to throw doubt on evolution exploit this breakdown of intuition by making statements of broad generality. For example, one often hears that the evolution of life as described by natural selection is as likely as a tornado sweeping through a junkyard and spontaneously assembling a Boeing 747 airplane. This analogy was initially proposed by astrophysicist Fred Hoyle in his 1983 book The Intelligent Universe. Hoyle and his co-worker Chandra Wickremasinghe used this example to support their alternative theory of panspermia, that life originated elsewhere in the universe and arrived on Earth from outer space via meteors.

Neither Hoyle nor Wickremasinghe are creationists and have their own reasons to want to discredit natural selection, but intelligent design creationists seized on this vivid image of the 747 in the junkyard and exploit it heavily in their anti-Darwin crusade, and Wickremasinghe has even appeared as a witness for them at some court trials.

To counter this analogy, one needs to look at exactly what natural selection says and compare it with what its opponents portray it as. Jerry Coyne (a professor in the Department of Ecology and Evolution at the University of Chicago) in a devastating review of intelligent design creationist Michael Behe’s new book gives a nice example using the familiar example of throwing dice.

Take for example, some adaptation of a gene that, starting from the original organism, requires twenty mutations at twenty different locations for the desirable new feature of the organism to appear, with the mutations occurring in a specific order so that each mutation confers a slight selection advantage to the organism. Suppose that the random mutations are represented by the throw of a die and the required mutation at a particular site occurs when you throw a six. This means that it will take an average of six throws for the first mutation to occur. Recall that evolution is a step-by-step process that builds on past successes and I have already described how even a slight selection advantage is sufficient for a single mutation to become universal in the population, so this mutation will be stable. It will then take another six throws for the second advantageous mutation to occur, and so on, so that it will take an average of 120 throws for all twenty mutations to occur. If the dice is thrown at the rate of one a second, that means it will take about two minutes for all twenty mutation to have gone into effect.

What the Boeing 747 analogy does is to assume that you have twenty dice and throw them all at once and that all twenty must come up six simultaneously for the new feature to appear. The odds against this are astronomically high. At the same rate of one toss per second, this would take more than one hundred million years. As Coyne says, “This sequential way of getting twenty sixes is infinitely faster than Behe’s method. And this is the way natural selection and mutation really work, not by the ludicrous scenario presented by Behe.”

Arguing by analogy and example is often necessary when trying to explain esoteric points, but is also tricky and has to be done with care. No analogy is a perfect replica of the actual process and you have to make sure that the analogy you select corresponds accurately to the phenomenon being analogized as far as the crucial elements are concerned. In the case of evolution, the key point to bear in mind is that a sequential series of changes, each of which is beneficial and stable, takes much less time (i.e. is far more likely) to occur than for them to occur simultaneously. This is why intelligent design creationists try to desperately find examples of systems that (they argue) could not have occurred by sequential changes. But they have failed.

POST SCRIPT: If FDR had been like George Bush. . .

Jacob Sager Weinstein says that he “got tired of right-wingers saying, “If the media had been as hard on FDR as they are on Bush, we’d have lost World War II.” So I started wondering. . . What if FDR had run his war like GWB?”

Here is the video that resulted from his musings.

Evolution-18: Missing links

(Please see here for previous posts in this series.)

About ten years ago, a group of engineering students came into my office. They were taking part in a scavenger hunt during Engineers Week and the one item that was very hard for them to find was a ‘slide rule’. They had little idea of what it was and no idea how it worked or what one even looked like but they knew it was old technology and they figured that I was old enough to possibly own one.

They were partly right. I had once owned a slide rule as a physics undergraduate in Sri Lanka but unfortunately did not have mine anymore.

For those not familiar with slide rules, the standard type looks like a ruler with another sliding ruler attached, and you use it to do complicated calculations. It was the precursor to the handheld calculator but with the arrival of cheap electronic versions of the latter, the slide rule went extinct. I actually owned a more unusual type of slide rule that was cylindrical rather than linear and was like a collapsible telescope. It had the advantage that it was small enough to carry around in your pocket, and being able to whip out a slide rule when the occasion demanded defined the nerds of that time.

The difficulty that the engineering students had in getting hold of a slide rule is due to the fact that when a new technology comes along, the old technology is superceded and devices built using it become extinct because no more units get produced and the old units get thrown away and end up in landfills. As another example of this phenomenon, at Case we have a state of the art Freedman Center where you can take information stored in any form (say phonograph records of any vintage, old 8mm home movies, etc.) and convert to digital formats. The one old format they cannot currently convert is videotapes made using the Betamax format and this is because they simply cannot get hold of any working Betamax players anywhere anymore. It has gone extinct. (If anyone has an unused but working Betamax player they would like to donate to the Freedman Center, they would be happy to take it off your hands.)

Like slide rules and Betamaxes, you would be hard pressed now to find many things that were mass-produced even just a decade or two ago. Apart from those in the collections of museums and idiosyncratic collectors, most other artifacts have disappeared forever.

Imagine an extraterrestrial archeologist visiting the Earth and seeing an electronic calculator. Even if he looks around fairly carefully, he would likely not find slide rules or indeed any earlier versions of such devices and conclude, erroneously, that Earthlings are brilliant designers who went straight from nothing to a fairly sophisticated calculator in one jump.

The analogy with missing links in evolution is obvious. This difficulty of finding well-preserved specimens of old things is even worse with the fossil record in evolution. As an organism evolves to a newer form, the old versions rapidly disappear and become extinct. As Steve Jones points out (Almost Like a Whale, 1999, p. 161), “When we see any structure highly perfected for any particular habit, as the wings and birds for flight, we should bear in mind that animals displaying early transitional grades of the structure will seldom continue to exist to the present day, for they will have been supplanted by the very process of natural selection.”

Even when it comes to fossils, since fossilization occurs only under very special conditions of organism and soil and climate, most of these dead organisms disappear forever and thus it is hard to find a continuous record of evolution. Even if a fossil is formed, the Earth itself is a dynamic system, with its crusts moving over each other, erosion, sedimentation, glacier movement, and even the rise and fall of mountains and oceans, all of which can destroy or bury or hide any fossils unfortunate enough to be in the way. As a result, “We continually overrate the perfection of the geological record and falsely infer, because certain genera or families have not been found beneath a certain stage, that they did not exist before that stage.” (Jones, p. 269) “The Cambrian Explosion [beginning about 545 million years ago], so called, is a failure of the geological record rather than of the Darwinian machine. Its radical new groups reflect not a set of exceptional events but something more banal: the first appearance of animals with parts capable of preservation. Before then, there were soft creatures that decayed as soon as they died. Why shells appeared all of a sudden is not certain.” (Jones, p. 274)

The rarity of fossilization can be seen by the fate of the passenger pigeon. Estimates put the number of them at over nine billion alive during the time of the Mayflower, more than the total of all the birds in the US today. They were still flourishing in the US as late as the times of the Civil War but the species went extinct in 1914 and not a single fossilized specimen has ever been found. If not for the existence of written records about this bird, we would not have known it even existed. (Jones, p. 266)

What gets preserved as fossils tend to be big and hard things like dinosaurs and mammoths, just like it is easier to find earlier versions of cars and ships than of slide rules. But thanks to the new techniques of DNA mapping, we do not have to depend exclusively on fossils to learn about how our ancestors evolved and diverged. DNA can interpolate the gaps in the record much better than even fossils can.

POST SCRIPT: Family Guy‘s Stewie and Brian go to Iraq

Evolution-17: How species diverge

(Please see here for previous posts in this series.)

When my daughter was quite young, about five or so, the question of where people came from came up in a mealtime conversation. Naturally we told her that human beings had evolved from ancestors who were monkey-like and then became human-like. She sat there for a while silently digesting this interesting bit of new information and mulling it over in her mind. It seemed clear that she was not at all disgusted or even bothered by the thought that we were related to the monkey family. That kind of revulsion seems to be something that has to be acquired, often nurtured by religions.

But something was bothering her and she finally articulated it, asking “But when that happened, wouldn’t the mother monkey notice that her child looked different?”

She had hit upon an issue that many skeptics of evolution raise. They argue that there is a contradiction if we assume that we had evolved from an ancestor species that was so different from us that we could not interbreed with that species. Surely, the argument goes, doesn’t evolution imply that if species A slowly evolves into species B, then there must be a time when the parent is of species A while the child is of species B? Isn’t it a ridiculous notion for parent and child to belong to different species?

The answer is that it is perfectly possible that as we go from generation to generation, for each child to be the same species as the parent, while of a different species from a distant ancestor. In fact, we have living examples of such a phenomena,
In The Ancestor’s Take (p. 302), Richard Dawkins uses the example of the ring speciation of the herring gull and lesser black-backed gull to illustrate how this happens. In Britain, these two kinds of birds don’t hybridize even though they meet and even breed alongside one another in mixed colonies. Thus they are considered different species. But he goes on to say:

If you follow the population of herring gulls westward to North America, then on around the world across Siberia and back to Europe again, you notice a curious fact. The ‘herring gulls’, as you move around the pole, gradually become less and less like herring gulls and more and more like lesser black-backed gulls, until it turns out that our Western European lesser black-backed gulls actually are the other end of a ring-shaped continuum which started with herring gulls. At every stage around the ring, the birds are sufficiently similar to their immediate neighbors in the ring to interbreed with them. Until, that is, the ends of the continuum are reached, and the ring bites itself in the tail. The herring gull and the lesser black-backed gull in Europe never interbreed, although they are linked by a continuous series of interbreeding colleagues all the way around the other side of the world.

Dawkins gives a similar example of this kind of ring speciation with salamanders in the Central Valley of California, which is ringed by mountains. If you start with salamanders at one end of the valley and proceed clockwise around the mountain range to the opposite side of the valley, the salamanders change slowly, at each stage being able to interbreed with the neighbors. The same thing is true when you go counterclockwise from the same starting point. But when you arrive at the opposite point of the valley where the two chains of evolution arrived at by going in different senses meet, you find they are now two different species.

The herring gulls and salamanders are the examples separated in space (which we can directly see now) of the same argument separated in time (which we can only infer) that says that as descendants are produced, they form a continuum. Each generation, while differing slightly, can interbreed with its previous generation, but over a long enough period of time, the two end points of the continuum need not be able to interbreed.

Thus it is possible for one species to evolve into another and for an organism to be intermediate between two species.

POST SCRIPT: Family Guy on why Congress voted for the Iraq war

Evolution-16: The evolution of the eye

(Please see here for previous posts in this series.)

The eye is one organ almost invariably brought out by creationists to argue against evolution. How could something so complex have possibly evolved incrementally, they ask?

Darwin himself suggested the way that the eye could come into being. Due to the fact that eyes don’t fossilize and thus leave a permanent record, it is hard to trace back in time and see the various stages in the evolution of the eye as linear developments. So he looked instead at the eyes of currently existing different organisms at intermediate stages of development, and concluded (On the Origin of Species, 1859, p. 188):

With these facts, here, far too briefly and imperfectly given, which show that there is much graduated diversity in the eyes of living crustaceans, and bearing in mind how small the number of living animals is in proportion to those which have become instinct, I can see no very great difficulty (not more than that in the case of many other creatures) in believing that natural selection has converted the simple apparatus of an optic nerve merely coated with pigment and invested with transparent membrane, into an optical instrument as perfect as is possessed by any member of the great Articulate class.

Steven Pinker (How The Mind Works, 1997, p. 159) describes how Darwin established how the eye could have evolved, according to the step-by-step process that I have described earlier, each step having a low probability for an individual but becoming likely when large numbers of organisms are involved over long times.

By looking at organisms with simpler eyes, Darwin reconstructed how that could have happened. A few mutations made a patch of skin cells light sensitive, a few more made the underlying tissue opaque, others deepened it into a cup and then spherical hollow. Subsequent mutations added a thin translucent cover, which subsequently was thickened into a lens, and so on. Each step offered a small improvement in vision. Each mutation was improbable, but not astronomically so. The entire sequence was not astronomically impossible because the mutations were not dealt all at once like a big gin rummy hand; each beneficial mutation was added to a set of prior ones that had been selected over the eons.

Still think it is implausible? Once again, using mathematics and computer simulations based on strict natural selection principles and starting, as Darwin himself suggested, with a light-sensitive nerve, it is possible to estimate how long the process of eye evolution took (Pinker, p.164):

The computer scientists Dan Nilsson and Susanne Pelger simulated a three-layer slab of virtual skin resembling a light-sensitive spot on a primitive organism. It was a simple sandwich made up of a layer of pigmented cells on the bottom, a layer of light sensitive cells above it, and a layer of translucent cells forming a protective cover. The translucent cells could undergo random mutations of their refractive index: their ability to bend light, which is real life often corresponds to density. All the cells could undergo small mutations affecting their size and thickness. In the simulation, the cells in the slab were allowed to mutate randomly, and after each round of mutation the program calculated the spatial resolution of an image projected onto the slab by a nearby object. If a bout of mutations improved the resolution, the mutations were retained as the starting point for the next bout, as if the slab belonged to a lineage of organisms whose survival depended on reacting to looming predators. As in real evolution, there was no master plan or project scheduling. The organism could not put up with a less effective detector in the short run even if its patience would have been rewarded by the best conceivable detector in the long run. Every change had to be an improvement.

Satisfyingly, the model evolved into a complex eye right on the computer screen. The slab indented and then deepened into a cup; the transparent layer thickened to fill the cup and bulged out to form a cornea. Inside the clear filling, a spherical lens with a higher refractive index emerged in just the right place, resembling in many subtle details the excellent optical design of a fish’s eye. To estimate how long it would take in real time, rather than compute time, for an eye to unfold, Nilsson and Pelger built in pessimistic assumptions about heritability, variation in the population, and the size of the selective advantage, and even forced the mutations to take place in only one part of the “eye” each generation. Nonetheless, the entire sequence in which flat skin became a complex eye took only four hundred thousand generations, a geological instant.

In his book The Ancestor’s Tale, Richard Dawkins points out (p. 388) that after the evolution of light-sensitive cells in worms about 600 million years ago, the kinds of image-forming optics that we now call the eye is estimated to have independently evolved more than 40 different times in various parts of the animal kingdom. Vastly different eye forms like the human eye and the compound eye of the crustaceans evolved differently and independently from a primitive common light sensitive cell that formed a proto-eye.

So far from being an event of unimaginably breathtaking improbability, the evolution of the eye is relatively mundane, although the organ itself is quite remarakable.

That is exactly the point that those opposed to natural selection refuse to acknowledge when they act as if all the parts of the eye must have come together almost at once. What is highly improbable to happen in one fell swoop becomes possible when it happens gradually.

Richard Dawkins in his book Climbing Mount Improbable looks at case after case of things that seem to be very complex and how they could have come about by natural selection. But Darwin did not need Dawkins to be convinced. In his own day, he had enough evidence to satisfy him. “If it could be demonstrated that any complex organ existed, which could not possibly have formed by numerous, successive, slight modifications, my theory would absolutely break down. But I can find no such case.” (Darwin, p. 189)

He further added (Darwin, p. 109): “Slow though the process of selection may be, if feeble man can do so much by his powers of artificial selection, I can see no limit to the amount of change, to the beauty and infinite complexity of the coadaptations between all organic beings, one with another and with their physical conditions of life, which may be effected in the long course of time by nature’s power of selection.”

POST SCRIPT: Great moments in the evolution of technology

Have you seen the the sideways bike?

Evolution-15: How species evolve

(Please see here for previous posts in this series.)

The final feature that needs to be addressed is the probability of mutations cumulating to produce new organs and species.

This question lies at the heart of many people’s objections to evolutionary ideas. They cannot envisage how infinitesimal changes, each invisible to the eye, can add up to major changes. That is because they tend to think that the two foundations for this to occur (the occurrence of successful mutations and the mutations then spreading throughout the population) are both highly unlikely, and so that the chance of a whole sequence of such processes occurring must be infinitesimally small.

What this series of posts has shown is that given large population sizes and the long geological times available, not only are those two things not unlikely, they are almost inevitable. The mathematical results discussed in the previous posts have shown two very important results. The first is that when the large sizes of populations and the long times involved are taken into account, the probability of favorable mutations occurring is quite high. The second is that the probability of a single favorable mutation spreading to every organism in the population is also high. These two results undercut the arguments of those who simply throw their hands up in the air and declare that evolution is extremely unlikely to have occurred.

Once that is understood, it is not hard to see how successive mutations that each have a selection advantage can pile one on top of another to produce new species leading to the diverse and complex biological system that we see today. The history of twentieth century biology is the discovery of case after case of the evolutionary history of organisms. Richard Dawkins’s book Climbing Mount Improbable describes many such cases.

In order to understand how speciation comes about, we need to understand better what constitutes different species. As pointed out by biologist Steve Jones, “Species are divided from each other in many different ways – by space, by time, by mating preference, by the inability to fertilize an egg or produce healthy young, or by the sterility of the offspring.” (Almost Like a Whale, p. 231)

The most famous historical example is that of the finches of the Galapagos Islands. During his five year trip on the Beagle Darwin had periodically shipped various specimens back to England to specialists to study them, and a respected ornithologist John Gould had agreed to study the birds. Soon after his return to England in 1836, Darwin met with Gould who gave him the surprising news that birds that Darwin had thought were different species like wrens and finches and orioles were actually all different kinds of finches. Furthermore, Gould had also found among Darwin’s specimens three mockingbird species, each set of which came from a different island in the Galapagos. (The Reluctant Mr. Darwin, David Quammen, (2006), p. 24)

This fueled Darwin’s suspicions that these birds had evolved from a single species that had somehow made the difficult journey to the Galapagos from the South American mainland, and after spreading out to each island had then become isolated from the others and evolved into separate species under the different selection pressures they experienced on the different islands.

Darwin decided to devote himself to the study of speciation and in order to do so, started with the breeding of animals in captivity. He extensively studied another bird (the pigeon) that was highly popular among breeders in England at that time and for which there existed a huge number of breeds. By studying pigeons (the first chapter of On the Origin of Species deals exhaustively with them), Darwin satisfied himself that all the species of pigeons were descended from a single type, the rock pigeon. “Great as the differences are between the breeds of pigeons, I am fully convinced that the common opinion of naturalists is correct, namely, that all have descended from the rock-pigeon (Columba livia).” (On the Origin of Species, Charles Darwin, (1859), p. 23)

In an interesting follow up to the Darwin finch story, Rosemary and Peter Grant spent twenty years in the Galapagos studying the finches and they actually observed the evolution that Darwin could only speculate about. For example, Darwin had noticed that the finches’ beaks “resembled different kinds of pliers: heavy duty lineman’s pliers, high-leverage diagonal pliers, straight needle-nose pliers, curved needle-nose pliers, and so on. Darwin eventually reasoned that one kind of bird was blown to the islands and then differentiated into thirteen species because of the demands of different ways of life on different parts of the island, such as stripping bark from trees to get at insects, probing cactus flowers, or cracking tough seeds.” (How the Mind Works, Steven Pinker (1997), p. 163)

Darwin felt that such changes would occur too slowly to be observed. But what the Grants did 150 years later was to show that such changes could in fact be observed in real time. They “painstakingly measured the size and toughness of the seeds in different parts of the Galapagos at different times of the year, the length of the finches’ beaks, the time they took to crack the seeds, the numbers and ages of the finches in different parts of the islands, and so on – every variable relevant to natural selection. Their measurements showed the beaks evolving to track changes in the availability of different kinds of seeds, a frame-by-frame analysis of a movie that Darwin could only imagine.” (Pinker, p. 163)

Evolution is a fact. Natural selection is far and away the only theory that comes even close to explaining in detail how it happened, although will always be many things as yet unexplained. As biologist Steve Jones writes, “Although biologists still argue about how the process works, fossils make it impossible for anyone, biologist or not, to deny that it happened.” (Almost like a Whale, 1999, p. 306)

Next in the series: The evolution of the eye

POST SCRIPT: The Darwin Year 2008-2009

This series of posts was inspired by the fact that 2009 will be the 200th birth anniversary of Charles Darwin and the 150th anniversary of the publication of On the Origin of Species. Case is planning a year long celebration of this event for the 2008-2009 academic year.

As part of this, the Common Book Reading committee has selected David Quammen’s The Reluctant Mr. Darwin as the book that all incoming first-year students in fall 2008 read, and will also try to expand the program to make it a campus-wide reading experience.

Quammen has agreed to be the fall convocation speaker in September 2008. There are also plans to have E. O. Wilson (one of the most eminent evolutionary biologists and author of Sociobiology), Sean B. Carroll (The Making of the Fittest), and Rosemary and Peter Grant also give talks during that year.

Evolution-14: How a single mutation spreads everywhere

In the previous post, we saw that if we start with a trait that is present in just 0.1% of the population (i.e., f=0.001), and if this has a small selection advantage of size s=0.01, this will grow to 99.9% (F=0.999) in just under 1,400 generations, which is a very short time on the geological scale.

But in a population of one million, an initial fraction of f=0.001 means that we are starting with about 1000 organisms having the favorable mutation. But it could be argued that new mutations usually start with just a single new kind of organism being produced in one single organism. How does that affect the calculation?

Suppose that you have a population of organisms of size N and they all start out having the same gene at a particular position (called the ‘locus’) on one of the chromosomes that make up the DNA. Now suppose a random mutation occurs in just one organism, the way that it was described in an earlier post in this series describing the shift from violet to UV sensitive sight in some birds. Most of the time, even a favorable mutation will disappear because of random chance because (say) that mutated organism died before it produced any offspring or it did produce a few and that particular gene was not inherited. But on occasion that mutation will spread. How likely is it that such a single mutation will spread to every single organism (i.e., become ‘fixed’ in the population)?

When one is not dealing with deterministic systems involving smoothly varying numbers (as was done in the previous case), a different kind of calculation (based on probabilities and known as ‘stochastic’) has to be done, and in this case the expectation value for the number of generations T taken for the single new mutation to spread all over and become fixed in the population (i.e. to spread to 100% of the organisms) is given by T=(2/s)ln(2N) generations, where ‘ln’ stands for natural logarithms. (Molecular Evolution, Wen-Hsiung Li, 1997, p. 49)

Even if s is taken as a very small advantage of size 0.01, for a population of N=one million, the average time taken for just a single mutation to become fixed is just 2,900 generations. So we see that mutations occurring in a single organism can become universal in a very short period on the geological time scale.

There are two important points that need to be emphasized.

There first is that even a very small selection advantage is sufficient to have that mutation dominate the species. This means that the advantage may not be even visible in the organism itself, which may look like every other organism in the species. For example, an eye mutation that works better by just a tiny bit may look like every other eye. Thus we should not think in terms of big changes for natural selection to work.

The second point is that even starting from a single mutation, as long as it takes hold (which has a probability of 2s of happening) and does not disappear and has an selection advantage however small, the mutation can spread surprisingly rapidly in the population and become universal and form the basis for future mutations.

It is interesting that even if there is no survival advantage to the new gene (i.e., s=0 and the mutation is said to be ‘neutral’), the mutation can on occasion still spread and become fixed, except that now the average time taken is much longer and given by T=4N generations. So that for a population of one million it would take on average about 4 million generations for a neutral mutation to spread everywhere, as compared to just 2,900 generations for a selection advantage of 0.01.

Darwin did not have access to this kind of mathematical analysis, which came long after his death. It is a tribute to his genius that he intuitively sensed the power of cumulative change over long time scales.

So far, I have shown how the first two items in the three components of natural selection, although seeming to have small probabilities of occurring, actually are quite likely. The third aspect of natural selection that has to be looked at is how the cumulative effects of small changes lead to big changes.

POST SCRIPT: Some real fact-checking

In yesterday’s post, I spoke about how the media does almost no fact-checking on Bush. Well, except for Jon Stewart who catches Bushies making up stuff about Iraq.

Evolution-13: Differential rates of survival

(Please see here for previous posts in this series.)

Of the three stages of natural selection outlined before, the only one that occurs purely by chance is the first one, that of the occurrence of mutations. I discussed how although the chances of producing a favorable mutation by changes in any individual site in the DNA (called ‘point mutations’) on an individual member of the species is very small, when the number of individuals in a species and the long times available for the changes to occur are factored into the calculation, the result is that such mutations are not only likely, they are almost inevitable to occur and furthermore are likely to occur many times.

The Hardy-Weinberg law showed that if natural selection was not at work (along with some other conditions), populations settled into stable equilibrium values after just one generation of random mating. The next question to be addressed is to see how the populations change when natural selection is allowed to act. How likely is it that favorable mutations produced in the set of genes (the genotype) that characterize the organism (the phenotype) will end up with that actual organism predominating in the species? (Recall that natural selection acts on the phenotype and not the genes directly, while mutations act only on the genes and inheritance is passed on purely via the genes. A genetic change that has no effect on the phenotype will not influence the fitness of the organism.)

It is not the case that this happens every time. Most mutations are deleterious and do not spread and even favorable changes usually disappear by chance before they can spread to become a significant number in the population. But when the favorable mutations do take hold, that particular variety becomes widespread and dominates the population.

There are many such cases that have occurred in nature. The most famous and widely quoted example of this kind of growth of a favored phenotype is the case of the peppered moths in industrial areas of England and America. As a result of the pollution that created dark backgrounds on the lichen covering the trees where the moths rested, the darker varieties of the moths were camouflaged better from predators than the lighter ones and thus had a significant survival advantage. From 1848 to 1896, the darker forms grew to as much as 98% of the population. Subsequently, with the advent of pollution control measures that cleaned up the environment and reduced the soot pollution, the dark moth population decreased to as low as 10%. In The Making of the Fittest (2006, p. 52-53), Sean B. Carroll points out that peppered moths are not the only such examples, that many similar changes in coloration due to selection pressures have occurred in land snails, ladybird beetles, desert mice, and other species.

The way that even a very small natural selection advantage can result in that variety dominating a species can be appreciated using the more familiar example of compound interest. Suppose a parent gives each of two children $1,000 at the same time. One of the children invests in a bank that offers an interest rate of 5.0% while the other, being slightly more thrifty, shops around and invests in a different bank at 5.1%. Although they start out with the total money being split 50-50, in 7,000 years the second child (or rather that child’s descendents) will have 99.9% of the total money, thanks to that very small advantage in the annual rate of return.

It is exactly this kind of differential survival rate that plays such an important role in natural selection. Even minute differences in fitness can result, over the long term, in the runaway domination of a preferred variety. To see how fast this can happen, population geneticists have carried out calculations.

Suppose one variety has a mutated gene that has a very slight fitness advantage over the existing gene. ‘Fitness advantage’ can be quantified by defining the fitness w as the measure of the individual’s ability to survive and reproduce. (The concept of fitness is a combination of the organism’s ability to survive for any length of time (at least until its reproducing age is over) and its fecundity in terms of the number of offspring it produces.) Suppose the original gene has fitness w=1 and the new mutation has fitness w=1+s, where s is the selection advantage.

The selection advantage is a measure of how much more likely it is that that particular variety will propagate itself in future generations when compared with the standard type. So if, on average, the new mutated variety produces 101 fertile adult descendents while the same number of the standard organism produces 100, then s=0.01.

When this selection advantage is included in the calculation, the number of generations T it will take for a mutation to increase its frequency in the population from an initial value of f to a final value of F is given by the formula T=(1/s)ln[F(1-f)/f(1-F)], where ‘ln’ stands for the natural logarithm. (Molecular Evolution, Wen-Hsiung Li, 1997, p. 39)

So if we start with a trait that is present in just 0.1% of the population (i.e., f=0.001), and if this has a small selection advantage of size s=0.01, this variety will grow to 99.9% (F=0.999) of the population in just under 1,400 generations, which is a very short time on the geological scale.

But in a population of one million, an initial fraction of f=0.001 means that we are starting with about 1000 organisms having the favorable mutation. In reality, we are likely to begin with just one mutation in one organism. How does that affect the calculation?

That will be discussed in the next post in the series.

POST SCRIPT: Lil George and evolution

This clip from a cartoon show I had never heard of (probably because it is on cable) is pretty funny.

Evolution-12: Population genetics and the Hardy-Weinberg law

(Please see here for previous posts in this series.)

In the previous post, I discussed the puzzle posed by a naïve understanding of Mendelian genetics, which was that one might expect that organisms that displayed recessive gene traits would slowly disappear in a population while those with dominant gene traits would grow in number. But if that were true that would prevent new mutations from gaining a foothold in the population and growing in number, if it happened to be a recessive trait.

The crucial work that formed the breakthrough that revived the theory of natural selection was done in 1908 by G. H. Hardy (a Cambridge University mathematician and author of a fascinating book A Mathematician’s Apology) and Wilhelm Weinberg (a German physician), working independently. What is nice is that the result is quite simple to derive, and surprising.

The main result is that whatever the distribution of gene pairs AA, Aa, and aa you start with in a population, after just one generation the number of people with those distributions will reach an equilibrium value that will never subsequently change. In other words, the numbers of the different types of genes in a population are stable. So traits, once they appear, do not disappear simply because of the accidents of random mating. This counters the ‘blending inheritance’ objections to Darwin’s theory.

The proof of this result assumes that certain conditions apply so that only mating effects are at play: that the total population is large enough (effectively infinite for statistical purposes) to avoid the phenomenon of genetic drift, whereby the ratio of a particular gene varies purely due to statistical fluctuations (i.e., say the population with a particular gene happens to breed disproportionately, thus causing that gene’s frequency to change), is diploid, that the population reproduces sexually and that mating within the population is totally random, that natural selection is not working to change the distributions of the genotypes, and that other factors like genetic mutations and migrations in or out of the population are not occurring (i.e., no gene flow).

Here’s the result. Suppose that you start with a population in which AA types occur with probability p, Aa types occur with probability 2q (where the 2 is inserted just to make the arithmetic a little simpler), and aa types account occur with probability r. Since the total population must add to 100%, this means that the total probabilities p+2q+r=1.

Under the conditions given above, the Hardy-Weinberg result says that: (1) after just one generation, the AA types occur with probability P where P=(p+q)2, Aa with probability 2Q where Q=(p+q)(q+r), and aa with probability R where R=(q+r)2; and (2) these new probabilities will remain unchanged with each succeeding generation.

As an example, if we started with the population of AA being 50% (p=0.5), Aa being 40% (2q=0.4), and aa being 10% (r=0.1), then after just one generation, the Hardy-Weinberg result predicts that the proportions will be P=0.49 or 49% for AA, 2Q=0.42 or 42% for Aa, and R=0.09 or 9% for aa, and remain fixed at these values forever afterwards.

The proof of this result is quite simple and elegant and here it is:

If there is random mating, then the probability of any particular mating combination is just the product of their individual probabilities.

The probability of an AA mating with another AA is p2. The offspring will get just one gene from each parent, and in this case the result will always be AA.

The probability of an AA mating with an Aa is 4pq (the extra factor of 2 comes from the fact that this mating combination can occur two ways, that either the father could be AA and the mother Aa, or the father could be Aa and the mother AA) and there is a 50% chance that the offspring will be an AA and 50% chance of being an Aa.

Similarly, the probability of an AA mating with an aa is 2pr. The offspring will get just one gene from each parent, and in this case the result will always be Aa.

The probability of an Aa mating with an Aa is 4q2 and there is a 25% chance that the offspring will be AA, 50% chance of being an Aa, and 25% of being an aa.

The probability of an Aa mating with an aa is 4qr and there is a 50% chance that the offspring will be Aa and 50% chance of being an aa.

The probability of an aa mating with another aa is r2. The offspring will get just one gene from each parent, and in this case the result will always be aa.

When you add all the probabilities for each type of offspring together, the probabilities of getting AA and Aa and aa are just the expressions for P, 2Q, and R given above.

What Hardy and Weinberg noticed was that if, by some chance, the starting values p, q, and r were such that they satisfied the equation q2=pr, then after one generation, P=p, Q=q, and R=r. In other words, if the starting values satisfied that particular relationship, the probabilities are unchanged from one generation to the next.

Of course, the values of p, q, and r we actually start with for a random population can have any value, as long as p+2q+r=1. But after the first generation of mixing, the values P, Q, and R actually do satisfy the relationship Q2=PR, irrespective of the starting values of p, q, and r.

Since the values of P, Q, R that are obtained after one generation become the starting values to calculate the distributions for the subsequent generation, and since P, Q, R satisfy the required relationship Q2=PR, these values will remain the same for every succeeding generation after the first.

What I found particularly surprising is that usually equilibrium conditions tend to be approached gradually and even asymptotically. Here, whatever the starting point, you get equilibrium after just one iteration.

The stability of population distributions under conditions of random mating is an important result. It implies that gene distributions do not change due to mating but only under some kind of pressure to do so..

From the year 1908 onwards, mathematical biologists proceeded to make rapid advances in the embryonic field now known as population genetics. The names of R. A. Fisher, Sewall Wright, and J. B. S. Haldane are the ones associated with the birth of this field and by the 1930s or so, their work had put Darwinian natural selection and Mendelian genetics on a firm scientific and mathematical footing (William B. Provine, The Origins of Theoretical Population Genetics, 2001).

In the next post in the series, I will look at how natural selection causes the population distributions to shift.

POST SCRIPT: Iraq war lies

Watch this video to see the brazenness with which the country was lied into war.

Evolution-11: The rise of population genetics and the neo-Darwinian synthesis

(Please see here for previous posts in this series.)

The joining of Darwin’s theory of natural selection with the Mendelian theory of genetics is one of the great triumphs of biology, now called somewhat grandly the ‘neo-Darwinian synthesis’. It forms the basis of all modern biology, and was strengthened by the discovery of DNA as the structure of genetic information and which explained how Mendelian genetics worked on a microscopic scale. The modern ability to map out the entire genome of humans and other species has produced overwhelming evidence in support of Darwin’s theory of how organisms evolve and branch out into different forms. The rough tree of life that Darwin sketched out in his book based on the anatomy of biological species has now been made more precise and detailed by the mapping of the DNA of species, showing ever more clearly how species are related to one another and when they separated from a common ancestor.

Mendel showed that genes were discrete objects that retained their identity as they were handed down from generation to generation and that thus any changes in genes, however small, did not get blended away in a regression towards the mean. So you would have thought that the rediscovery and rapid popularization of Mendel’s ideas in 1900 would have signaled a resurgence of Darwin’s idea that natural selection worked on very small, almost continuous changes, and the defeat of those who argued that one needed discontinuous changes for evolution to occur.

Ironically, the exact opposite happened. Because Mendelian genetics was a discrete mechanism with the genetic information seeming to occur in small lumps that remained intact, it superficially seemed to support the discontinuous model of natural selection, and the proponents of discontinuous changes were able to co-opt Mendel’s theory to their cause. By around 1908 or so, it seemed like Darwin’s own favored model of small continuous changes leading to large changes was in almost total retreat, actually doomed by the arrival of Mendel. While there was a sprinkling of mathematicians like Udny Yule (what a wonderful name!) who argued that Mendel’s theory was compatible with Darwin’s model of continuous evolution, their voices were lost in the volume of controversy generated by the competing biological schools. (The Origins of Theoretical Population Genetics, William B. Provine, 2001, p. 85)

Part of the problem was that scientists were still struggling to understand the workings of both Darwin’s theory and Mendelian genetics and many misunderstanding of each were then prevalent. For example, one thing that was puzzling about genetics (and puzzled me for a long time too) was this whole business of dominant and recessive genes and how it affected population distributions.

It was Mendel’s work that argued for the existence of these two types of characteristics, the actual mechanism of which became better understood with the discovery on DNA and increased understanding of the way that chromosomal information was handed down from parent to child.

Simply put, each person has pairs of genes, one from each parent on the respective inherited chromosome. To be concrete, we can look at the gene for eye color. Each gene may be of a dominant type (denoted by A, for say brown eyes) or a recessive type (denoted by a, for say blue eyes). So a person would have one of the pairs AA, Aa, or aa on the pair of chromosomes that contain the genes for eye color. Since A is the dominant one, it always wins, and so those people with either AA or Aa will have the characteristic A (brown eyes) manifest itself in their features, and only the person who has aa will display the characteristic a (blue eyes). Each parent will also have AA, Aa, or aa, and will randomly pass on just one of the pair of genes it possesses to the child.

It seems intuitive that if a population starts out with some distribution of AA, Aa, and aa types, and there is random mating in the population, then the number of people displaying the dominant characteristic A will steadily increase in the population, while the manifestation of the recessive characteristic a will decrease and perhaps eventually even disappear altogether, since only someone possessing the relatively unlikely combination aa will manifest it. Since regressive characteristics did not seem to be disappearing in real life populations, and in fact seem quite stable in their numbers, early geneticists had some doubts about whether they were interpreting Mendel’s model correctly.

But starting around 1908, things started to change as better experiments were done and more mathematical versions of the two theories started being used. Mendelian and Darwinian theories started to get quantified and people began to realize that Darwin’s version of natural selection with continuous changes was in fact compatible with Mendel’s theory. By 1918, the reversal was complete and Darwin was ascendant and has remained so ever since. This was largely due to the rise of the field now known as population genetics, whose practitioners developed mathematical models that looked at the consequences of Mendelian genetics in natural selection.

What started the shift was the result now known as the Hardy-Weinberg law, which will be discussed in the next posting in this series.

POST SCRIPT: American beliefs about evolution

Gallup has done one of its periodic surveys about Americans views on evolution

These results show that:

  • 24% of Americans believe that both the theory of evolution and the theory of creationism are probably or definitely true.
  • 41% believe that creationism is true, and that evolution is false.
  • 28% believe that evolution is true, but that creationism is false.
  • 3% either believe that both are false or have no opinion about at least one of the theories.

That first group of 24% is definitely confused, since there is no way that evolution could have occurred in the 10,000 years or less allowed by creationism. The survey creators speculate that these were people who believed that god influenced evolution and somehow wanted to incorporate that view and responded in contradictory ways depending on which question was asked.

Also:

  • The reasons for rejecting evolution were mostly religious.
  • The more regularly you attended church, the less likely you were to believe in evolution.
  • Republicans were less likely to believe evolution (30%) than Democrats (57%).

Summary:

It is apparent that many Americans simply do not like the idea that humans evolved from lower forms of life. This appears to be substantially based on a belief in the story of creation as outlined in the Bible — that God created humans in a process that, taking the Bible literally, occurred about 10,000 years ago.