What is a hill? I mean, how do you define a hill as opposed to a mountain, or, flat, level ground. The reality is that all surfaces on the planet which aren’t artificial seem to have discontinuities and wrinkles on the macroscale.^{1} Here is a dictionary definition:

A well-defined natural elevation smaller than a mountain.

That is rather vague. Is a small mound of dirt that my cousin made a “hill.” And what is a mountain? Here is a definition:

A natural elevation of the earth’s surface having considerable mass, generally steep sides, and a height greater than that of a hill.

Since a hill does not have explicit bounds (e.g., defined gradients, minimum elevations), then a mountain does not either because its low bound (the hill) is not well defined. **Nevertheless, we use terms like mountains and hills in everyday discourse**. It works operationally because we know what we want to get across. There are no “mountainists” who argue that all hills are really mountains, that hills are a fallacious category. There are no “hillists” who argue that all mountains are really large hills, that mountains are artificial constructs. Hillists don’t talk about “ultra-mountainists” and mountainnists don’t refer to “ultra-hillists.” Each of whom take the concepts of mountains and hills, which all sides accept as valid, to extreme ends that are not justified. People just don’t care about mountains and hills enough to worry about details so long as mountains and hills work in everyday communication.

Sometimes, reading about debates between “gradualists” and “non-gradualists” in evolution strikes me as the same. Most of you likely know the main players, Steven Jay Gould and Richard Dawkins. The origins of this debate can be found as early as the 19th century, when Charles Darwin argued for gradual change due to selection upon small variations, while many others, including T.H. Huxely, argued for larger jumps in phenotype due to the emergence of “sports.” I won’t recapitulate the details, but the theme seems to have reemerged every generation (e.g. Goldschmidt vs. Fisher, etc.). But in the end, I think we are seeing an argument between hillists vs. mountainists. No “gradualist” argues for a constant rate of change over time, and no proponent of punctuated equilibrium today proposes a macromutationist saltationist model. Myself, I am pretty conventional in that I don’t accept some of Gould and Niles Eldridge’s processes which explain the empirical observation of punctuated equilibrium (which I accept). Armand Leroi’s The Scale Independence of Evolution is a good exposition of my general bias toward viewing macroevolution as simply an extension of microevolution, a difference of degree, not kind. Nevertheless, in The Blind Watchmaker I recall Richard Dawkins engaging in some unartfull dodges of the reality that neutral theory and punctuated equilibrium emerged as a response to perplexing false predictions and perception implied by the models promoted by R.A. Fisher (Fisher being the Doyen of Dawkins’ adaptationist school)

Nevertheless, **I do not grant that plain vanilla evolutionary processes can not generate evolutionary discountuities**. Discrete probability distributions converge to continuous ones if you increase the number of trials of an event toward infinite. Similarly, the variations in the rate of change of evolution over time really makes a mockery of conventional semantical currencies that trade in “gradual” or “rapid.” Since when is 10,000 years “rapid.” How can we exactly define “gradual” on the scale of millions of years which most of us have no *gestalt* conception of such a span? (yes, abstractly you know what a million years is, but you do not feel and conceive of it as you can feel and conceive one year)

In the most simple cases evolution is a process of change that occurs contingent upon a variety of parameters. In the real world these parameters are multitudinous, which is why that the process of evolution and the various theories are still being mooted, and an explanation for the periodic emergence of disciplines like evo-devo to shed light on an unexplored domain. Nevertheless, consider a trait like height. It is normally distributed, roughly a bell curve. Why? One explanation^{2} is that height is controlled by hundreds of small loci which cumulatively shape the phenotype.

- Imagine for example that you have 180 loci, n
_{1}to n_{180}. - Imagine that all loci have independent effects (they are additive), and that they have the small effects.
- To simplify it further, imagine that their effects are equal, and consider that each locus can add 1 or 0 centimeters to the total.
- Neglect environment, so the phenotype is totally dependent on the genotype.

This is a discrete distribution, and since I’ve offered only 1 or 0 as the outcomes for each trial (a locus), it is binomial. Nevertheless, it is roughly normal graphically, the famous “bell curve.” Since we are imagining height, we assume it is a person, and I’ll just say there are 90 spots on the genome, two copies each (diploid). Since I stipulated that the loci are independent, if one 0, that doesn’t effect the probability of the next locus. Assume .5 probability of 0 or 1 (again, to make things simple).

“Common sense” should tell you that the probability of **all** loci being 0 is very low (0.5^{180}), as is the probability of all loci being 1. Also, one should assume that the mode will be 90 (about 3 feet tall, perhaps they are hobbits?), because there are **many** ways that the loci could add up to 90, and the number of combinations decreases the closer you get to the total sum of 0 or 180 (where you have one combination that will satisfy the condition). I wanted to present a continuous trait as it might emerge from a discrete (meristic) trait so that you see that gradual microevolution doesn’t just emerge from a statistical soup.^{3} This flux across and between loci is the base that evolution uses to create new extended forms from standing genetic variation, this is the “additive genetic variation” which R.A. Fisher placed at the center of microevolutionary change over time.

Now, let’s imagine that you select from the distribution of individuals all those who are taller than 90 centimeters. What’s going to happen now? Intuitionally you would suspect that the biological reality is that the population derived from this sampling will be taller. Why? I have changed the die and the probability of 0 or 1 is no longer 0.5 for both. Though there are still loci which are zero throughout this population, more of the loci are now 1, so the probability increases for slots in the next generation to contribute height. The higher the seletction cut off, the greater the probability of there being a 1 in the parental slot, and therefore, the probability of a 1 being in a slot in the offspring would also increase.

Therefore, the rate of change between original population and daughter population is proportional to the selection between the populations. If the selection ~ 0, there won’t be any change.

Now, recall that I stipulated that height was totally controlled by the genotype above. That is, all the variation in phenotype is due to the variation in the genotype. Let’s change that, and assume that only 50% of the **variation** in phenotype is due to variation in genotype. The rest is due to other factors, environment, developmental stochasticity, etc., it doesn’t matter. These factors are not correlated with genotype in any way, the the additive genetic variance and other variance components are independent parameters. What happens now?

Go back to our example of a selection cut off of 90 centimemters. Common sense tells us that again the offspring would be taller than the original population because some of the variation in phenotype is due to genotype. But, the key is **some**. Individuals within the selected population will vary, and some will have disfavorable genotypes, but by various other factors gained the phenotype in question (height). So, in comparison to the first scenario the selected population has a less skewed genotype in terms of the frequency of slots filled by 1. So, it stands to reason that the offspring would have fewer slots filled by 1 than in the first scenario, even if the it would be higher than in the original population (i.e., greater than 0.5). The other factors I have stipulated as random, they are **not heritable**, so that parameter should be the same as in the original population. What will occur is that though the offspring of the selected population will be taller than the original population, they will regress to the mean because of the second parameter.

Instead of prolonging this, I will present the rather prosaic “prediction equation” used by animal breeders:

R = *h ^{2}* * S,

where R = response,

*h*= heritability and S = selection coefficient.

^{2}Heritability in this equation usually refers to the additive genetic variance, the proportion of phenotypic variance attributable to genotypic variance. The S parameter simple defines the difference between the original and parental population selected for the new line. R is simply the difference expected in the offspring of the selected population in reference to the original population.

**In plain English, I’m trying to say that the rate of evolution is proportional to the heritability of a trait and the amount of selection that is at work upon a population.** In practice heritability and selection are difficult to pin down in nature (though easier in controlled breeding scenarios). Unlike R.A. Fisher, I suspect there might be other parameters at work over the long term above and beyond additive genetic variance, though nothing exceptionally exotic.

Back to the argument between hillists and mountainists, I think it is kind of ridiculous in that evolution is neither **fundamentally** a constant or erratic process, it is a dynamic that is shaped by parameters at any given time. One could argue that the two camps are trying to describe the empirical character of how evolution plays out, but words are never going to accurately and precisely capture the nuance in the distribution of possible ways that evolution is going to express itself in the space of the tree of life.

A basic understanding of such simple algebra as the breeder’s equation will also be useful if you want to make a calculation of how fast a trait can change given particular selection pressures and heritabilities. I’ve encountered people who discount the possibility of recent human evolution, but the reality is that lactose tolerance has spread to near fixation in northern Europe within the last 10,000 years. The robusticity of the immunological profile of many Eurasians is probably the outcome of selection against disease over the last 10,000 years as agriculture has allowed the increase in population density. Similarly, the heritability of a trait is crucial in determining if a few generations of powerful selection would result in any great change. Evolution happens. Or it doesn’t. There’s no point in arguing over whether a hill is a mountain or vice versa, just paint each individual picture yourself, and pass out the supplies so others can do the same.

1 – I’m not counting the surface of a salt crystal for obvious reasons, which is why I added macroscale.

2 – The genes for height are being elucidated, at least height in the normal range, and so this simple model is not reality, just an illustration.

3 – At more than 4 loci it becomes hard experimentally to differentiate a discrete from a normal distribution in many situations. See Evolutionary Quantitative Genetics for details.