## Rate of evolytion of quantitative traits, units

For quantitative traits, either morphological or other traits, the rate of their evolution can be measured, for example, in units (e.g. in millimetres) by which the particular trait changes for an average member of the particular species over 1 million years. Because, in small structures, growth by 1 millimetre corresponds to a far more substantial evolutionary change than for large structures, it is more common to give the rate of evolution in percentage or in multiples of the standard deviation over a certain time interval. The best known unit used for measuring the rate of evolution of quantitative traits is the **Darwin**, where a rate of 1 Darwin corresponds to the rate of change of a structure that changed approximately 2.7182 times over 1 million years, i.e. by a multiple of the base of the natural logarithm. The rate of an evolutionary change in Darwins can thus be calculated using the equation

v = (ln X1 – lnX2) / t

where *X*_{1} is the original size of the structure, *X*_{2} is the new size of the structure and *t* is the time interval (in million years) during which evolution of this structure occurred. As the change is calculated as the difference in natural logarithms, it actually corresponds to the ratio of the original and new sizes of the structure and consequently the calculated rate does not depend on the units employed or on the absolute size of the studied structures. Understandably, a different rate would be measured if we were to monitor the rate of a change in the length of a certain structure, its area or its volume. Thus, if we want to compare the rates of evolution of fleas and elephants, we can employ the Darwin rate unit, but we must be careful and ensure that we do not compare a change in length with a change in volume.

The **Haldane** is another well-known unit used to express the rate of evolutionary changes in quantitative traits (Hendry & Kinnison 1999). A rate of 1 Haldane corresponds to a change in the size of the trait by one standard deviation per generation. Rates given in this unit also take into account the variability in the studied trait and the generation time of the studied species, so that they are more useful for comparing the abilities of species to respond to selection with a certain intensity. In addition, Darwins cannot be used to measure the rate of evolution of quantitative traits expressed on an interval scale. If, for example, we measure the rate of evolution of body temperature expressed in degrees Celsius or Kelvins, a different number is obtained in each case. However, if this rate is measured in Haldanes, the same number is obtained for both scales. If we compare the rates of evolution of two species calculated in Darwins and Haldanes, quite the opposite results can frequently be obtained. Consequently, where possible, it is mostly preferable to give the evolutionary rate in both units (Hendry & Kinnison 1999).