Aesthetics, mathematics, physics and biology

Tags

art, biology, evolution, Jackson Pollock, mathematics, Natural selection, physics, Quantitative genetics, science, Stephen Jay Gould

One of Namuth’s many photos of Jackson Pollock painting with his “drip” method. (Photo credit: Wikipedia)

I’ve just started reading Paul J. Nahin’s book Doctor Euler’s Fabulous Formula about one of Leonhard Euler‘s famous equations. The preface establishes the author’s thesis that the great thing about Euler’s formula is that it is beautiful because it is the result of skill (or as the author says “disciplined reason”). He goes on to compare the beauty of a mathematical formula, and specifically mathematical formulae as expressions of concepts in physics, to the work of great artists. Beauty in art is something we can all relate to. What surprised me is that he based this comparison on the dichotomy between “disciplined artists,” such as Michelangelo and “undisciplined artists,” such as Jackson Pollock. To call what “two-year olds routinely do” on a daily basis art, he says, “is delusional or at least deeply confused…” (xix).

Nahin was wrong about Jackson Pollock. He knows he is wrong, i.e. I don’t think he really believes this, but this false dichotomy is highly illustrative. He happens to be wrong in a way that we can learn a lot from.  I would like to show that he’s illustrating the particular value of mathematics in evolutionary biology.  The value of mathematics in biology can only be seen if the level of application of models matches the proper place of the forces it models. If our application of broad theories is too narrow, then we will not find beautiful theories useful. On the other hand, if we view selection as a very broad force, as Stephen Jay Gould did in Wonderful Life, then we can see the “usefulness” of evolutionary theory.

NYC – MoMA: Jackson Pollock’s The She-Wolf (Photo credit: wallyg)

Jackson Pollock was disciplined and skilled. Consider several facts about his career, training and technique. Firstly, he was a trained artist. He worked hard to develop his techniques. If you look at his early work (e.g. “She-wolf”), it’s certainly not what he’s most remembered for, but it is distinctive. If he could have just splattered paint then why would he bother to develop all that technique? People took him seriously as an artist before he developed his splattering technique. Next consider the drip paintings that he’s remembered for. These paintings were the result of careful consideration and a disciplined technique. Just a few examples: he positioned his canvas on the floor; he carefully mixed the paint to a particular consistency to achieve the kind of “splatter” that he needed; he used carefully chosen brushes and other devices. Finally he chose colors that made sense in order to make a particular artistic statement.

Life, the subject of biology, is a Jackson Pollock painting. I enjoy a mathematical argument because it is logical, internally consistent, one thing builds on another. I can see what it is made out of. I know the goals the entire time the argument is building. Wouldn’t it be great to apply that kind of thinking to understanding living things and their origins? Yeah, that would be great, but it’s incredibly hard for two reasons. Since life is a Jackson Pollock painting, the constraints are incredibly broad; so broad that life has huge leeway to accomplish the same thing in different ways. “Higher fitness” is not very specific. The paint spatters in different ways every time, regardless of its consistency and color palette being the same. The other reason is a confusion about where the predictions of theory are supposed to lie. Because selection is very very broad, we need to make predictions that are similarly broad. We can use the theory we have now to predict that selection would favor earlier age of maturity. That doesn’t mean we can predict whether that age is six months or two years. There are also different developmental ways to accomplish that. The theory we have doesn’t predict whether faster cell growth or slackened mate preferences will produce earlier age of maturity.

Biological mathematical models, if they are to be beautiful in the same way that physical models are, need to be broad, in the same way that Jackson Pollock’s constraints were broad. Take the most beautiful equation in biology, the Price Equation, for example. The Price Equation is beautiful because it is simple, so easy to derive that it seems it must be true, and you can do all kinds of things with it. You can derive almost any model of evolution from the Price Equation (try it!). You can also deal with almost every kind of selection at any level, and genetic drift. Why? Because it is so broad that actually applying it to a population would be ridiculous. The Price Equation is exact, which means that the equation includes more information than we can ever hope to measure in a natural population. The only way to make the application of the equation exact is to dial down the constraints so much that the prediction the equation makes is almost worthless. In other words, to make the equation exact, you have to impose such strong selection that you can predict the result without the equation. Mike Wade might disagree with me about this, but I can bet he won’t disagree about the equation’s beauty. The Price Equation is so broad that Martin Nowak has gone so far as to call it a “mathematical tautology” (also available at arXiv).

Another example, slightly less contentious, but not quite as obvious, are the predictions of life-history theory. Russ Lande and Brian Charlesworth‘s equations (derived from Guess What) make definite, broad predictions about the evolution of age at first reproduction, reproductive output and other life-history traits. The breadth of these predictions matches the strength of selection to enact changes at the genotypic level. What I mean is that as long as evolution is proceeding in the particular direction predicted by Lande and Charlesworth, then the model is justified. The actual genotypic or even morphological changes, or the diversity of adaptations at a particular level, doesn’t matter to the predictions of the theory. It also doesn’t matter whether you’re talking about yeast, fruit flies, elephant birds or dinosaurs. That’s the level of prediction we have with the theory of selection. It’s not precise like Rembrandt or even early Picasso. It’s a lot more like Jackson Pollock. It’s a mess, but it’s a carefully constrained, disciplined mess that is beautiful itself.

Jackson Pollock; Mural Detail II (Photo credit: notanyron)

This morning I’ve realized yet again that the reason I love mathematics is largely aesthetic. I love math for the same reason that I love listening to a great piece of music. Beautiful models and theories are useful, Nahin and I contend, not because they are inherently “correct,” but because they inspire people to work hard at verifying and refining their logical consistency. Nahin points out that Newton and Einstein’s theories of graviation are “wrong” in the sense of making predictions that still hold true. But they are beautiful. We all say “yuck!” when we see someone try to fit a logistic curve to real data, but we don’t always know why we cringe. We cringe because we all know that the logistic wasn’t so well-studied because it was correct. It was so favored because it’s a beautiful equation. It’s as beautiful as the Mandelbrot set or Euler’s Formula, and it’s easy to teach to undergraduates. It makes sense. However, demographers and ecologists all know that it doesn’t really describe human populations. A.J. Lotka didn’t live to see that, but that doesn’t mean his time was wasted (see Lotka, Alfred J. 1998. Analytical theory of biological populations. New York: Plenum Press).

Advertisements

High School Students Visit the Lab

Tags

biology, candy, evolution, Genetic drift, high school, Natural selection, teaching

Recently I got the chance to host some high-school students in the lab and show them what I do as a graduate student.  My motivation was mainly that I enjoy teaching, and I know that when I was a high-school student I would have enjoyed seeing what a theorist actually does.  This project was part of UNC’s Academic Day: according to the organizer, biology was the most popular major to explore during this event that hosts high school students from around the state to show them what universities are all about.

I told them what I do day-to-day, what the track is for someone getting a Ph.D. and the nature of my research.  I then explained some basics of population genetics, like the Hardy-Weinberg trinomial and a few other things. Then we conducted an activity where the goal was to show the action of natural selection and genetic drift. This was the fun part.

The Activity

I had eight students, all female, and all but one with brown eyes. This in itself could have provided data for an activity, but I had brought “populations” of Skittles. The students started each with a population of twenty Skittles and measured the frequencies of each color (the “phenotype”). Having a population of twenty makes calculating frequencies easy: just multiply by five and you’ve got a frequency out of 100. Most of the students had a predominance of one color. I provided them with paper and pencil, since they were visiting.

First I asked the students to come up with a way of coding (and hence measuring) the phenotype. I had already come up with a method that the students found acceptable, which was to score the colors from 1 to 5 starting with green, so that green = 1, yellow = 2, orange = 3, red = 4 and purple = 5. Then the students measured the average phenotype by multiplying the frequencies and color scores. Then came the eating: students on the left side of the table ate 10 Skittles randomly; the students on the right of the table at according to some rule that they were free to concoct. The left side represented genetic drift, and the right side selection. The goal of the experiment was to see how much the average trait value would change between the parent population (before eating) and the offspring population (after eating).

The results were interesting: I really didn’t know if this would work, or if the students would care enough to actually try it. The mean phenotypes on the genetic drift side of the table changed very little, but several of the students lost a color in the process, which represents the loss of alleles that can happen with drift. The means on the selection side of the table did change, but what was more interesting was that the students all chose different rules. Some started by eating green and progressed to yellow when they ran out of green. This represents directional selection, or selection against one extreme phenotype. Some started at the other end with purple, which is also directional selection. Others ate from both ends of the spectrum (green and purple, progressing to yellow and red). This corresponds to stabilizing selection, or favoring intermediate phenotypes. Others started with orange; this results in disruptive selection, that penalizes intermediate phenotypes. I did not plan on telling them about these modes of selection at the outset. This was a very pleasant accident.

Thoughts and conclusions

The biggest obstacle was time: I only had about forty minutes with these young ladies, and so telling them about graduate school and doing the experiment was kinda tight. Another interesting way to do things might have been to use their own physical characteristics (e.g. hair and eye color) as the data, but I didn’t think of that until I saw the Founder Effect that had landed in our lab: they all had brown eyes, except for one student. I explained that if they were to form a parthenogenetic colony on an island somewhere, that after a few years either all their offspring would have blue eyes or brown eyes. This joke might have been a little more uncomfortable to hear had there been any dudes in the room other than me.

Sewall Wright, who couldn’t make it on a Thursday (Photo credit: Wikipedia)

I was really glad to have the chance to show these students that theory is a part of science. Without showing them any high-tech gear I was able to show them what I do as a scientist. People, especially young people, often confuse measurement for science, and the techniques of science with the actual intellectual exercise of science. Honestly I was surprised to be surrounded by eight eager teenagers as soon as I held up my hand and said “evolutionary theory.” I expected everyone would want to hear about more sexy things like developmental biology.

The cost of reproduction in birds

Tags

biology, birds, evolution, Genetic drift, Natural selection, research, science

The concept of trade-off is paradigmatic in life-history theory. an organism can only acquire a finite amount of energy in its lifetime, so it must “choose” how to allocate that energy to growth and survival or reproduction. Reproduction is assumed to be costly so that individuals who spend more on reproduction, for example by laying more eggs, will not survive as well. We suppose that over evolutionary time, natural selection will act on genetic variation for these allocation decisions, so that the sequence of decisions over an individual’s lifetime will represent an optimal allocation of resources.

Unfortunately this intuitively appealing idea has been very hard to find in nature. In fact, many studies have come up with positive correlations: animals that reproduce more tend to survive better. A recent study by Eduardo Santos and S. Nakagawa found that this trade-off was almost impossible to detect in most studies, or non-existent altogether. In a meta-analysis of brood supplementation studies (researchers added eggs to the nests of breeding birds), they found little impact on survival. Their result held across all the major taxonomic groups of birds, the biggest division being between passerines (songbirds, crows, flycatchers, etc) and non-passerines (ducks, loons, parrots, woodpeckers). Regardless of overall “lifestyle” the birds tested in most studies were able to withstand the hypothesized survival cost of additional eggs dumped on them by researchers.

Why would this be the case? As always there is the possibility that the studies were poorly designed, or that brood supplementation is not a good way to test for a trade-off. Particularly, brood supplementation only taxes the parents of their ability to defend and feed offspring; it does nothing to the energy that females put into egg production. The other possibility is that adult birds just don’t put that much effort into reproduction in the first place. Perhaps survival is far more important. The trade-off is still there, but it’s just not important for most birds.

The hypothesis that life is just not as Malthusian as we have often supposed in evolutionary biology intrigues me greatly. If evolution acted in the “well-oiled machine” manner that many laypeople and professional scientists find appealing, then we’d expect selection to push annual reproduction right up to the level allowed by the trade-off. What studies have found is birds putting minimal effort into reproduction, parenting or anything that affects their survival. This means that selection is a lot weaker than we expect: this gives genetic drift a lot more room to account for polymorphism. It also makes sexual selection more plausible: if most species have fairly conservative lifestyles and selection for survival is not that strong, then males (or females) can afford costly ornaments.

An unrelated study also appeared this week that is getting a lot of press: researchers in Iceland found a strong relationship between the age of fathers and mutations passed to their offspring. This is the first study to quantify the per-year effect of paternal age on offspring mutations in humans, so it’s a pretty big deal. I will talk more about this in a future posting since it’s related to my dissertation research, but in the meantime, go read the article and enjoy the flurry of debate surrounding it.

E. S. A. Santos, S. Nakagawa (2012). The costs of parental care: a meta-analysis of the trade-off between parental effort and survival in birds Journal of Evolutionary Biology, 25, 1911-1917 DOI: 10.1111/j.1420-9101.2012.02569.x

Related articles

• Kiwis in ‘severe reproductive bottleneck’ (stuff.co.nz)
• Parent-offspring Conflict: Time to Listen to the Argument (psychologytoday.com)
• Better looking birds have more help at home with their chicks (esciencenews.com)
• Older fathers pass on more mutations (newscientist.com)

Drift and selection: the epic battle?

Tags

biology, evolution, Genetic drift, genetics, Natural selection, science

I learned a lot at last week’s Nescent Academy on Quantitative Genetics. I saw a lot of material that I wouldn’t have seen in other forums, like the Ornstein-Uhlenbeck models of genetic change under microevolution and macroevolution. During the last half of the week, which focused on macroevolution, I confirmed my impression that when talking about evolutionary history, genetic drift is really the name of the game. When population geneticists talk about the history of particular genes (for example, a gene implicated in a human disease), they rarely speak about selection. There was also a lot of good information on research findings from natural populations.

There were three points in particular that struck me:

1. When looking at natural populations, there is abundant genetic variation in almost any trait
2. Selection is generally weak and generally stabilizing selection
3. Stochastic processes, such as genetic drift, can account for a lot of diversification seen in nature

These findings were interesting to me because I study selection, usually using deterministic models and because I’ve seen the perspectives of other researchers about the relative roles of selection and drift. I have tended to assume, along with many other researchers, that most diversification is due to selection, and that for any “real” differences to matter over evolutionary time that selection must be involved. Why is this?

Selection and adaptation are appealing concepts and they are simple to understand. Darwin’s (three or four) postulates give us all we need to understand how adaptation comes about. Adaptation is a really nice idea: things become more efficient, better, over time. Not only is that aspect appealing, but it’s easy to understand how it could happen: selection eliminates the less efficient, and promotes the more efficient. All you need after that is inheritance. This is so easy that most people get it the first time. Leaving aside the appeal of this from the social perspective (read the first chapter of The Dialectical Biologist), it’s just easy. I teach evolution and ecology to undergraduates and most of them come in getting the basic idea of selection. It’s not hard.

Take genetic drift on the other hand. If you’re like most biologists who’ve tried to teach about genetic drift, you know that genetic drift is the opposite of selection from a teaching perspective. Genetic drift, like selection, removes variation from populations. Only mutation can bring it in. Under genetic drift alleles just disappear: by random chance they fail to make it into the next generation. This only happens in finite populations, that is every single real population. How? Think of it this way: you know that if you flip a coin a thousand times you will get close to 500 heads and the rest will be tails. Do this instead: flip a coin ten times ten times and record the number of heads you get for each ten coin flips. You could then make a graph depicting the number of times you get five heads, six heads and so on. You should get a nice looking histogram: close your eyes and put your finger on a spot on the graph. Your real population is that spot. It could be the one where you got zero heads.

There are two important things about genetic drift: one is how it leads to diversification, and the other is how it accounts for polymorphism. Drift leads to variation between populations because when populations are separated they randomly undergo drift, possibly with different end results: if there are two alleles A and B at a locus under drift, one population could lose allele B and the other could lose allele A. Repeat that over many loci and your get very different looking animals that don’t recognize each other when they get a chance to make babies. The second property is that when you observe polymorphism (genetic variation), it is probably due to drift. Drift over time removes genetic variation from a population, but before that happens the frequency of the allele in question bounces all over the place by random chance. The time window over which that happens is incredibly large, much longer than that for selection. Therefore the large amounts of genetic variation in natural populations are probably due to weak selection, strong drift and lots of mutation across the genome.

Here’s my explanation of the above findings: selection is always happening, but is generally weak. Selection is weak both because drift is happening at the same time, and because life is just not as hard as Darwin and Malthus had in mind. Selection in nature is usually stabilizing selection, meaning that there is some intermediate value that is favored most, and extreme values are selected against: the typical example is birth weight in industrialized societies. Small babies are prone to infection and pulmonary dysfunction and large babies are at greater risk for perinatal complications. However, in most cases, it appears that deviations from the optima that we can detect in nature are not heavily penalized. Especially in large-bodied, iteroparous organisms like birds, ungulates and primates, life just doesn’t seem that hard. This means drift has plenty to work with. Most of the organic diversity we see is probably due to drift randomly sending populations closer to new optima where stabilizing selection takes over again. This is basically Sewall Wright‘s shifting balance theory.

Wright in 1954 (Photo credit: Wikipedia)

This positively demonic process could account for most of the organic diversity we see out there, but it is not an appealing idea. I think most intellectuals go through a phase where they attribute everything to randomness — and I’m not suggesting we all get on that bus — but there’s also a Conspiracy to remove slack from the world. People generally don’t like the idea of random forces to explain things. Especially since a lot of biologists, including myself, don’t understand genetic drift (how could you?) it’s really hard to get behind the idea. However, especially when analyzing real populations, such as the evolutionary history of humans, and testing ideas about sexual selection, we have to consider the role of drift. Much of the persistent, between population variation we see that looks adaptive could be due to genetic drift.

Related articles

• Sexual selection in humans: some interesting recent work (lxmx.wordpress.com)
• Genetic variation within a population (ahschoolapbio2013.wordpress.com)
• Adaptation vs Drift at Evolution Ottawa 2012 (sandwalk.blogspot.com)
• A free online course in genetics and evolution by Mohamed Noor (whyevolutionistrue.wordpress.com)
• How do you do evolutionary theory? (lxmx.wordpress.com)