Research on Rates of Evolution

Introduction

Evolution is a process that takes place from one generation to the next in living lineages of plants and animals. Evolutionary results are sometimes visible on the time scale of an individual generation, but we usually see and study these cumulatively on longer experimental, ecological/historical, and geological scales of time. Rates of evolution are important because they are a key indication of how the evolutionary process works— rates quantify evolutionary change in relation to time.

A process is a sequence of operations that produces a result. Reading is a process. Readers read words, and the process can be described by a simple rate, usually represented as words per minute. If a student reads 100 words per minute, we expect that it will take him or her a little over an hour and a half (100 minutes) to read a 10,000-word book, and 16-17 hours (1000 minutes) to read a 100,000-word book. The rate has to be determined experimentally, of course, by counting the number of words read in a fixed amount of time, or by measuring how long it takes the student to read a text of some known length. Then a simple rate is calculated by reducing the resulting quotient: e.g., 10,000 words per 100 minutes reduces to 100 words per minute, and the result is assumed to be independent of both the numerator and denominator. This assumption of independence, rarely tested, is implicit in our everyday use of simple rates.

The tricky part about a rate is that it is a ratio with a numerator and a denominator (rate and ratio have the same etymological root). In the example here, the numerator is words and the denominator minutes. When we reduce the ratio we assume that readers read at the same rate no matter how long the text. This could be true, but even reading might be a little more complicated.

How long does it take a mouse lineage to evolve to double its size? The answer depends, of course, on the rate— how fast is evolution?

Empirical observations

Rates of evolution are generally calculated in terms of proportional change, ln (x₂ / x₁) = ln x₂ − ln x₁, divided by elapsed time. The reasons for logging measures of morphology are reviewed in Gingerich (2000).

J. B. S. Haldane (1949) proposed a rate calculation in a unit called the darwin that requires two quantities:

(1) the difference between means of two samples of natural-logged measurements, d = y₂ − y₁;

(2) the time interval between the samples, I = t₂ − t₁, measured or estimated in millions of years.

The resulting rate in darwins is:

D  =  d  /  I

A rate in darwins is expressed in terms of factors of e (base of the natural logarithms) per million years, neither of which are intuitive units: e and millions of years are perfectly arbitrary in this context; the units do not appear in genetic models; there is an erroneous suggestion, or even implication, that evolution takes place on million-year time scales (see below); and rates in darwins cannot be compared for measurements that have different or unknown dimensionality (Gingerich, 1993).

It makes much more sense to follow a lesser known suggestion of Haldane and calculate rates of evolution in terms of proportional change divided by elapsed time, in a unit called the haldane (Gingerich, 1993). This calculation requires three quantities:

(1) the difference between means of two samples of natural-logged measurements,d = y₂ − y₁;

(2) the pooled standard deviation of the samples,s_p = √(s_p)², where (s_p)² = [(n₁−1) (s₁)² + (n₂−1) (s₂)² ] / (n₁ + n₂ − 2), where s₁ and s₂ are the standard deviations of the samples of natural-logged measurements; and

(3) the time interval between the samples, I = t₂ − t₁, counted or estimated in generations.

The resulting rate in haldanes is:

H_{log I}  =  Z  /  I , where Z = d / s_p

Here the result is expressed in terms of phenotypic standard deviations per generation, and the subscripted log I is a reminder that the result is dependent on time scale (rates of most interest are H₀ where I = 1 generation and log I = 0).

Quantification in haldanes requires knowledge about phenotypic variability but this is really necessary in any case in an evolutionary study because, as Haldane himself wrote, variation is the raw material of evolution. Standard deviations are components of selection intensity and response. A generational time scale, rather than millions of years, is the time scale on which evolution takes place. And finally, rates in haldanes are independent of the dimensions of the underlying measurements. These are all advantages of haldane rate units over darwins.

Empirically, when rates of evolution are calculated on different time scales, the rates decline systematically the longer the time interval being studied (Gingerich, 1983, 1993, 2001; Figure 1). This is not an assertion– it is an empirical observation that anyone willing to compile rates over a range of time scales can easily demonstrate for himself or herself. A simple evolutionary rate of measured change and time does not produce a number independent of the denominator.

Reactions

Reactions to this observation have been surprising. Paleontologists starting with the late Stephen Jay Gould (1984) dismissed the observation of an inverse relationship of rates to their denominators as a 'psychological and mathematical artifact' of plotting a rate against its denominator. Sheets and Mitchell (2001) call the dependence of rate on interval 'spurious self-correlation,' as if it somehow isn't real. Roopnarine (2003) calls the inverse relationship between rate and timescale a 'mathematical artifact predictable on the basis of the behavior of random walks.'

In response I reiterate, we calculate rates because we expect rates to be independent of their denominators, but independence is rarely what we see for evolutionary rates. There is nothing psychological or artifactual or spurious about plotting a ratio against its denominator to test their independence— and arguing that an inverse relationship of rates and intervals exists as an artifact is tantamount to arguing for independence when the opposite has just been demonstrated!

Rates are not independent of their denominators, and consequently they have to be evaluated in light of the dependence, as was done, for example, by Mandelbrot (1967) in another context— evaluating the length of the coastline of Britain.

Bookstein (1987) interpreted the dependence of evolutionary rates on their denominators to mean that evolutionary rates do not exist, just as, he argued, the length of a coastline does not exist independent of its scale of measurement. But existence dependent on scale is still existence. I agree that rates of evolution are meaningless independent of the time span over which they were calculated, but they continue to have meaning as long as the time scale is known and specified. Further, the natural focal time scale for evolutionary studies— the generation-to-generation or one-generation time scale— is the only time scale of interest for evolution as a process. There is no shorter time scale, and results on longer time scales (e.g., million-year time scales) are cumulative results reflecting evolutionary history but not the evolutionary process directly.

How Fast is Evolution?

Most of the rates of evolution that have been calculated and published over the years, including those of Haldane (1949), were calculated based on fossil lineages. Observed changes were very slow, and the rates were very low (indeed Haldane had to invent the term "millidarwin" to express the rates he reported for fossil lineages). Such emphasis on rates calculated from the fossil record misled all of us into thinking that evolution itself is very slow.

What do rates of evolution in the fossil record tell us about evolution on the generation-to-generation time scale of the process? Panel D in Figure 1 answers this question: rates calculated from the fossil record are very low: on the order of 10^-7 to 10^-3 haldanes or standard deviations per generation. But when considered as they scale relative to their denominators, rates calculated from the fossil record project to rates of 10^-1 to 10⁰ standard deviations per generation on a time scale of one generation. Such rates are so high that they might not be believable if they were not consistent with what we observe in the laboratory and what we infer by scaling rates calculated on shorter time scales (panels A-C in Figure 1). For the question— how fast is evolution in the face of selection?— the simple answer is 'fast' and the number to go with this is about one-tenth of a standard deviation per generation. This is more or less an upper bound of course, but it also expresses what we commonly see (Figure 1).

Accepting rates from the fossil record at face value— failing to rescale to a timescale of one generation— explains several anomalous results in the earlier literature. Lande (1976) inferred that change in fossil lineages can be explained by as few as one selective death per million individuals per generation, and Lynch (1990) inferred that rates of morphological change in fossil lineages are substantially below the minimum neutral expectation. Both results are surprising, but result from erroneously assuming that rates calculated on geological scales of time represent evolution on the time scale of the evolutionary process (Gingerich, 2001). Natural selection operates statistically, moment-to-moment, day-to-day, generation-to-generation, on the developing or mature phenotype (and hence too on the underlying genotype) in front of it. The process has no memory nor anticipation. Selection doesn't read a genetic code to act on some preceding or somefuture phenotype— it always acts here and now. Hence the only rates of evolution that should be substituted in genetic models like those of Lande and Lynch are generation-to-generation rates calculated on one-generation time scales.

A Rate Perspective on Punctuated Equilibria
and the Red Queen vs. Stasis Debate?

If the process of evolution is so dynamic on a generational scale of time, why does it appear virtually stationary on longer scales of time? If you have read this far, then you are ready for a thought experiment that helps to clarify both 'punctuated equilibria' (Eldredge and Gould, 1972) and the debate over Red Queen evolution vs. stasis? (Stenseth and Maynard Smith, 1984). The thought experiment is illustrated in Figure 2, which is explained in the text on pp. 141-143 of Gingerich (2001). Have fun!

Hint: Earth history and geological time have been orders-of-magnitude longer than the evolutionarily-equivalent functional range of variation seen in even diverse groups of organisms. Morphology is highly constrained relative to the time that has been available for evolutionary experimentation.

Gingerich Publications on Rates of Evolution (chronological order)

Gingerich home page

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