Many thanks to all those who responded to my question concerning the the relationship between 's' and rate of migration of an allele. The original question, plus a list of responses is given below: My cursory conclusion were (remembering that this was just to write a paragraph in a review) were Rate of migration (over geographic distance) can get confused with migration (proportional exchange between subpopulations), so its probably best to use 'wave of advance' or 'rate of spread' for the former. Its a bit clumsy, and probably not the way non-specialist would use it, but it is at least unambiguous. There is nothing technically wrong with Fishers' equation or approach. These papers use simulation to look at rate of spread in sub-populations: Slatkin, M. (1976). The rate of spread of an advantageous allele in a subdivided population. In "Population genetics and ecology" (S. Karlin and E. Nevo, Eds.), pp. 767-780, Academic Press, London. Slatkin, M. and Charlesworth, D. (1978). The spatial distribution of transient alleles in a subdivided population: a simulation study. Genetics 89: 793-810. An excellent, accessible review is Morjan, C. L. and Rieseberg, L. H. (2004). How species evolve collectively: implications of gene flow and selection for the spread of advantageous alleles. Molecular Ecology 13: 1341-1356. Anyone more interested should look at the models of ecological invasion as analogous processes ********original question********* I have two questions concerning the relationship between the selective advantage of an allele and the rate at which it migrates over geographic distance [for those of you interested in the background: we can model or measure the selective advantage associated with alleles conferring drug resistance in malaria, so have estimates of 's', but the alleles appear to occur very rarely and have subsequently migrated around the world]. Intuitively there must be a positive relationship but has it been quantified? On the basis that Fisher/Haldane/Wright sewed up most of population genetics in the 1920s to 1940s I tried there and was not disappointed: Fisher, R. A. (1937). The wave of advance of advantageous genes. Ann.Eugen 7: 355-369. who found that rate of advance was proportional to the square root of 's'. This paper appears to be cited only rarely (at least in this context; it is used a lot in chemistry as a basis of diffusion of reactions) so my first question is whether it was flawed, overlooked, or was simply superseded by later work? My second question pertains to island and stepping stone models: has anyone investigated the relationship between 's' and migration rate from these models? Apologies if they have, but I haven't located it because most of the work looks at more complex questions like maintenance of clines, genetic variability etc. I realise I could probably do it myself using stepping models but there is the (non-negligible) risk I would get it wrong, it is unnecessary if it has been done already, but most importantly I just need to cite the result in a review and don't want to have to prove it in an appendix. As a technical point, malaria is haploid so we need not worry about dominance. Thanks to those who can comment. Regards, Ian. *******Responses from Evoldir (in order of reciept) ******* Dear Ian, I am not sure if I am exactly replying to your question, but you could probably get some idea from two of our publications if you look them carefully. 1. Vogl et al. 2003 Genetics 165: 1385-1395 2. Das et al. 2004 Genetics 168: 1975-1985 3. Baines et al. 2004 Genetics 168: 1987-1998. In Baines et al. we have dealt with the migatrion of the selected allele with an experimental setup and also found evidence of cline of haplotypes across latitudinal transects. Best wishes, Aparup ******* Dr. Aparup Das, Faculty Fellow Population Genomics and Evolution Laboratory Department of Biology Poornaprajna Institute of Scientific Research Post Box No. 18, Devanahalli BANGALORE - 562 110, INDIA Tel: +91-80-27647333 (O), 27647555 (lab) Fax: +91-80-27647444 E-mail: aparup@poornaprajna.org; adas@uni-muenchen.de Homepage: http://www.poornaprajna.org/aparup.htm Ian Hastings -- Fisher's Wave of Advance work has not been ignored. It has frequently been revisited by mathematicians, who love to ring changes on it. It is mentioned (and a simplified derivation given) in my course notes which are available on the web: http://evolution.gs.washington.edu/pgbook/pgbook.html (see pages 158-159). It is also covered, I think, in Warren Ewens's recent book (2003). I think I cited it in my 1976 Annual Review of Genetics review of migration work. I wouldn't call over 800 citations in the literature being ignored or overlooked. As for stepping stones, your question is ill-posed. There is no relation between m and s, of course. You meant to ask whether the rate of advance of a wave could be related to these quantities. There the most interesting paper is by Monty Slatkin and Deborah Charlesworth (Genetics, 1978) which gets very different speeds from the Fisher result. Not much has been done analytically with the stepping-stone case. J.F. ---- Joe Felsenstein joe@gs.washington.edu Department of Genome Sciences and Department of Biology, University of Washington, Box 357730, Seattle, WA 98195-7730 USA Look up work of Mark Kot Russell Lande Miller Research Professor 2004-05 Dept. of Integrative Biology University of California Berkeley, CA 94720-3140 Hi Ian, This is a very interesting question, and to my knowledge (which, admittedly, doesn't go very far), it hasn't been addressed in theoretical population genetics other than the Fisher's paper. However, ecologists have taken up a very similar problem--the growth of a population over a range, starting at with some small, localized population. This is well described on p. 21-26 of _An Illustrated Guide to Theoretical Ecology_ by Ted Case. I think it would be relatively straightforward to adapt the ecologists' results to the spread of an advantageous mutation by considering the relationship between s and the intrinsic growth rate. Best Wishes, Scott Williamson Ian, I'm not sure if this is exactly what you are looking for, but Slatkin talks about the rate of spread of advantageous alleles in subdivided populations in this chapter: SLATKIN, M., 1976 The rate of spread of an advantageous allele in a subdivided population, pp. 767-780 in Populations Genetics and Ecology, edited by S. KARLIN and E. NEVO, New York. Loren Rieseberg and have a more recent discussion of the topic here: Rieseberg, LH and JM Burke. 2001. The biological reality of species: gene flow, selection, and collective evolution. Taxon 50: 47-67. I've attached a copy of this latter article to this message for your convenience John John M. Burke, Ph.D. Ian -- I think that the following two papers may be of interest to you in this regard: Rieseberg, L. H. and Burke, J. M. The biological reality of species gene flow, selection, and collective evolution. Taxon. 2001; 50(1)47-67. Rieseberg, L. H. and Burke, J. M. A genic view of species integration. J. Evol. Biol. 2001; 14(1)883-886. Fred Allendorf Ian, I heard a talk by Peter Kareiva about 10 years ago where he discussed Fisher's paper that you mention. I recall that he believed it to be "under cited" as well. He used it in context of a discussion on the potential for spread of escaped transgenes in feral populations. I don't think I would categorize it as flawed, rather it is a bit simplistic. If someone were clever enough to extend the model to include two dimensional spread in a patchy environment I think it might be noticed a bit more. I should have paid more attention in my undergraduate calculus courses! Cheers, ******* Paul E. Arriola, Ph.D. Associate Professor of Biology voice: (630) 617-3109 Genevieve Staudt Chair fax: (630) 617-3735 Elmhurst College e-mail: paula@elmhurst.edu 190 Prospect Avenue http://www.elmhurst.edu/~bio/arriola Elmhurst, IL 60126-3296 Dear Ian, I looked into a similar question with Loren Rieseberg - we asked how important migration rates were compared to "s" in the spread of advantageous alleles across subdivided populations. We also did a large literature review on the strength of selection of advantageous alleles (as estimated from QTL data) and gene flow estimates. You may find our review article relevant (it is attached as a PDF file). We also ran some simulation models, which were greatly inspired by previous (and much overlooked) work by Slatkin (1976). His was a stepping-stone model, but we were more interested in the spread of alleles across subpopulations arranged on a grid and using a realistic dispersal function (e.g., leptokurtic). If you are interested in those, I can provide you with some more details; but our results were fairly similar to a publication on "s" and stepping-stone models that came out in the meantime by Whitlock (2003). Also, Cherry and Wakely (2003) looked at the spread of s in an island model. Cherry JL, Wakely J (2003) A diffusion approximation for selection and drift in a subdivided population. Genetics, 163, 421-428. Slatkin M (1976) The rate of spread of an advantageous allele in a subdivided population. In: Population Genetics and Ecology (eds Karlin S, Nevo E), pp. 767-780. Whitlock MC (2003) Fixation probability and time in subdivided populations. Genetics 164, 767-779. Good luck! Carrie Morjan Aurora University Aurora, IL ******* Ian Hastings, Liverpool School of Tropical Medicine, Pembroke Place, Liverpool L3 5QA phone: 0151-705-3183 fax: 0151-705-3371 Ian Hastings