As far as I understand, various abilities like flying, sight, hearing etc. were caused by slow evolution, where those with a greater ability to to these things had better chance of survival. (If this assumption is wrong, then I am happy to delete this question).
Are there, however, any documented examples of by evolutionary leaps being made, over the course of just a few generations? I understand, that some abilities have a tipping point where one gets the ability suddenly, but there is not a lot of physiological change made. An example of this would be the ability to climb a tree, which could suddenly be possible if the body weight is reduced with just a few percent. What my question is about, are sudden changes to the characteristics of a creature.
Answer
@kmm and @shigeta provided you with a nice observational account of sudden leaps in large organisms. However, if you want to look at where this is the norm and try to build a mathematical theory then you need to look at something much smaller; the prime candidate is affinity maturation.
In the human immune system, when exposed to an antigen B cells produce antibodies. If it is your first exposure to the antigen then the antibodies produced will probably have very low binding affinity. However, after some exposure time, your B cells will start to produce antibodies with much higher affinities for the antigen and thus you will be able to better fight off the disease. The cool part, is that the antigen produced is tune via an evolutionary process!
There is differential survival, with only antibodies with the highest affinity being able to survive. Variability is introduced by a very high mutation rate in the complementarity determing region (CDR). (Tonegawa, 1983). The length of this evolutionary process is very short, typically a local equilibrium is found after only 6-8 nucleotide changes in CDR (Crews et al., 1981; Tonegawa, 1983; Clark et al., 1985), so you need only a few point mutations to quickly develop a drastically better tuned antibody.
The standard mathematical model for this is Kauffman's NK model. With a protein sequence on $N$ sites, we say that evolution is fast (and we have a sudden leap) if after our fitness landscape changes, we can get to a new local equilibrium in a number of generations that scales with $\log N$. Kauffman & Weinberger (1989) showed how this model can be used to study affinity maturation, and showed that to achieve a sudden leap we need high epistasis and low correlations between pointwise mutants. In particular, their model suggests that typical epistasis in the CDR is on the order of 40 proteins (out of the total 112 proteins in the CDR).
References
Clark, S.H., Huppi, K., Ruezinsky, D., Staudt, L., Gerhard, W., & Weigert, M. (1985). Inter- and intraclonal diversity in the antibody response to influenza hemagglutin. J. Exp. Med. 161, 687.
Crews, S., Griffin, J., Huang, H., Calame, K., & Hood, L. (1981). A single V gene segment encodes the immune response to phosphorylcholine: somatic mutation is correlated with the class of the antibody. Cell 25, 59.
Kauffman, S. and Weinberger, E. (1989) The NK Model of rugged fitness landscapes and its application to the maturation of the immune response. Journal of Theoretical Biology, 141(2): 211-245
Tonegawa, S. (1983). Somatic generation of antibody diversity. Nature 302, 575.
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