# 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,sp = √(sp)2, where (sp)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:

Hlog I  =  Z  /  I , where Z = d / sp

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.

 Figure 1. Rates of evolution from different sources, calculated in haldanes on different scales of time, and here scaled temporally against their denominators to make them comparable: (A) selection experiments in laboratory and field; (B) 'microevolution'; (C) historical field study; (D) rates from the fossil record. All predict rates on the order of 10-1 to 100 (0.1 to 1) standard deviations per generation on a time scale of one generation (100; the time scale of the evolutionary process). Positive and negative rates are distinguished to emphasize that these define upper and lower bounds, respectively, of a distribution that is artificially hollow because very low rates cannot be distinguished operationally from zero. Figure from Gingerich (2001: fig. 8; sources of data are listed in original publication).

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.

 Figure 2. Heuristic time-form lattice to explain how evolution can be so dynamic in the short term, with H0 rates on the order of 0.1 standard deviations per generation on the scale of one generation, and so static over longer intervals. (A) If H0 rates are on the order of 0.1 standard deviations per generation, then evolution takes place on a time-form lattice where a 0.1 standard deviation step or difference in morphology corresponds roughly to a one-generation step or difference in evolutionary time. The smallest living mammal is the least shrew weighing about 3 or e1.1 g, and the largest living mammal is the blue whale weighing about 100 metric tonnes or e18.4 g. The standard deviation of body weight in mammals is about 0.15 units on a natural logarithmic scale. Hence the largest and smallest mammals living today differ by approximately 100 standard deviations or 103 0.1-standard-deviation units. These are physiological limits and mammals have never been much smaller or much larger: thus the time-form lattice for mammalian evolution is about 103 units wide. The generation time for an average living mammal is on the order of one year, and the Cenozoic history of the modern orders of mammals as we know them goes back 55–65 million years, which is conservatively about 10 million or 107 generations. Thus the time-form lattice for Cenozoic mammal evolution is about 107 units long temporally. (B) The lattice is not square, but some four orders of magnitude longer temporally than it is wide in form. Mammals starting at some average size at the beginning of the Cenozoic can be expected to have diffused and filled the lattice in less than 105 generations [(500/1.96)2 ˜ 65000 generations] – less than one percent of their subsequent Cenozoic history. Then they were constrained to evolve within the lattice for the remaining 99 percent of their history. Rates of evolution on the time scale of the process are so high that lineages rapidly find and fill most niches within their physiological limits. Then they change little until the system is perturbed. This, in other terms, is the evolutionary stasis model of Stenseth and Maynard Smith (1984). Figure from Gingerich (2001: fig. 9).

## Gingerich Publications on Rates of Evolution (chronological order)

Gingerich, P. D. 1983. Rates of evolution: effects of time and temporal scaling. Science, 222: 159-161. Online or Request PDF/reprint 138

Gingerich, P. D. 1984. Punctuated equilibria-- where is the evidence? Systematic Zoology, 33: 335-338. Online or Request PDF/reprint 151

Gingerich, P. D. 1984. Smooth curve of evolutionary rate: a psychological and mathematical artifact-- reply to Gould. Science, 226: 994-996. Online or Request PDF/reprint 158

Gingerich, P. D. 1986. Temporal scaling of molecular evolution in primates and other mammals. Molecular Biology and Evolution, 3: 205-221. Online or Request PDF/reprint 173

Gingerich, P. D. 1987. Evolution and the fossil record: patterns, rates, and processes. Canadian Journal of Zoology, 65: 1053-1060. PDF or Request PDF/reprint 185

Gingerich, P. D. 1991. Fossils and evolution. In S. Osawa and T. Honjo (eds.), Evolution of Life: Fossils, Molecules, and Culture, Springer-Verlag, Tokyo, pp. 3-20. PDF or Request PDF/reprint 224

Gingerich, P. D. 1993. Quantification and comparison of evolutionary rates. In P. Dodson and P. D. Gingerich (eds.), Functional morphology and evolution, American Journal of Science, 293A (Ostrom volume): 453-478. PDF or Request PDF/reprint 258

Gingerich, P. D. 1993. Rates of evolution in Plio-Pleistocene mammals: six case studies. In R. A. Martin and A. D. Barnosky (eds.), Morphological Change in Quaternary Mammals of North America, Cambridge University Press, Cambridge, pp. 84-106. PDF or Request PDF/reprint 265

Clyde, W. C. and P. D. Gingerich. 1994. Rates of evolution in the dentition of early Eocene Cantius: comparison of size and shape. Paleobiology, 20: 506-522. Online or Request PDF/reprint 285

Gingerich, P. D. and G. F. Gunnell. 1995. Rates of evolution in Paleocene-Eocene mammals of the Clarks Fork Basin, Wyoming, and a comparison with Neogene Siwalik lineages of Pakistan. Palaeogeography, Palaeoclimatology, Palaeoecology, 115: 226-247. Online or Request PDF/reprint 291

Gingerich, P. D. 1996. Rates of evolution in divergent species lineages as a test of character displacement in the fossil record: tooth size in Paleocene Plesiadapis (Mammalia, Proprimates). In M. Godinot and P. D. Gingerich (eds.), Paléobiologie et Evolution des Mammifères Paléogènes: Volume Jubilaire en Hommage à Donald E. Russell, Palaeovertebrata, Montpellier, 25: pp. 193-204. PDF or Request PDF/reprint 316

Gingerich, P. D. 2000. Arithmetic or geometric normality of biological variation: an empirical test of theory. Journal of Theoretical Biology, 204: 201-221. Online or Request PDF/reprint 357

Gingerich, P. D. 2001. Rates of evolution on the time scale of the evolutionary process. In A. P. Hendry and M. T. Kinnison (eds.), Contemporary Microevolution: Rate, Pattern, and Process, Kluwer Academic Publishers, Dordrecht, Genetica, 112/113: 127-144. PDF (Deep Blue) or Request PDF/reprint 383

Gingerich, P. D. 2003. Mammalian responses to climate change at the Paleocene-Eocene boundary: Polecat Bench record in the northern Bighorn Basin, Wyoming. In S. L. Wing, P. D. Gingerich, B. Schmitz, and E. Thomas (eds.), Causes and consequences of globally warm climates in the early Paleogene, Geological Society of America, Special Papers, 369: 463-478. PDF or Request PDF/reprint 402

Gingerich, P. D. 2003. Land-to-sea transition of early whales: evolution of Eocene Archaeoceti (Cetacea) in relation to skeletal proportions and locomotion of living semiaquatic mammals. Paleobiology, 29: 429-454. Online or Request PDF/reprint 405

Gingerich, P. D. 2006. Environment and evolution through the Paleocene-Eocene thermal maximum. Trends in Ecology and Evolution, 21: 246-253. Online or Request PDF/reprint 441

Wood, A. R., M. L. Zelditch, A. N. Rountrey, T. P. Eiting, H. D. Sheets, and P. D. Gingerich. 2007. Multivariate stasis in the dental morphology of the Paleocene-Eocene condylarth Ectocion. Paleobiology, 33: 248-260. Online or Request PDF/reprint 458

Gingerich, P. D. 2007. Punctuated equilibrium (book review). Paleontologia Electronica, 10: 1-2. Online or Request PDF/reprint 469

Gingerich, P. D. 2009. Rates of evolution. Annual Review of Ecology, Evolution, and Systematics, 40: 657-675. Online/PDF or Request PDF/reprint 498

Other Publications

Bookstein, F. L. 1987 . Random walk and the existence of evolutionary rates. Paleobiology, 13: 446-464.

Eldredge, N. and S. J. Gould. 1972. Punctuated equilibria: an alternative to phyletic gradualism. In T. J. M. Schopf (ed.), Models in Paleobiology, Freeman, Cooper and Company, San Francisco, pp. 82-115.

Gould, S. J. 1984. Smooth curve of evolutionary rate: a psychological and mathematical artifact. Science, 226: 994-995.

Haldane, J. B. S. 1949. Suggestions as to quantitative measurement of rates of evolution. Evolution, 3: 51-56.

Koch, P. L. 1986. Clinal geographic variation in mammals: implications for the study of chronoclines. Paleobiology, 12: 269-281. JSTOR

Lande, R. 1976. Natural selection and random genetic drift in phenotypic evolution. Evolution, 30: 314-334.

Lynch, M. 1990. The rate of morphological evolution in mammals from the standpoint of the neutral expectation. American Naturalist, 136: 727-741.

Mandelbrot, B. B. 1967. How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science, 156: 636-638.

Roopnarine, P. D. 2003. Analysis of rates of evolution. Annual Review of Ecology and Systematics, 34: 605-632.

Sheets, H. D. and C. E. Mitchell. 2001. Uncorrelated change produces the apparent dependence of evoutionary rate on interval. Paleobiology, 27: 429-445.

Stenseth, N. C. and J. M. Smith. 1984. Coevolution in ecosystems: Red Queen evolution or stasis? Evolution, 38: 870-880.