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The Population Genetic Structure of Clonal Organisms Generated by Exponentially Bounded and Fat-Tailed Dispersal

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The Population Genetic Structure of Clonal Organisms Generated by Exponentially Bounded and Fat-Tailed Dispersal Copyright Ó 2007 by the Genetics Society of America DOI: 10.1534/genetics.107.077206 The Population Genetic Structure of Clonal Organisms Generated by Exponentially Bounded and Fat-Tailed Dispersal Luzie U. Wingen,*,1 James K. M. Brown* and Michael W. Shaw† *Department of Disease and Stress Biology, John Innes Centre, Norwich NR4 7UH, United Kingdom and †School of Biological Sciences, University of Reading, Reading RG7 6AS, United Kingdom Manuscript received June 12, 2007 Accepted for publication July 10, 2007 ABSTRACT Long-distance dispersal (LDD) plays an important role in many population processes like colonization, range expansion, and epidemics. LDD of small particles like fungal spores is often a result of turbulent wind dispersal and is best described by functions with power-law behavior in the tails (‘‘fat tailed’’). The influence of fat-tailed LDD on population genetic structure is reported in this article. In computer simulations, the population structure generated by power-law dispersal with exponents in the range of À2 to À1, in distinct contrast to that generated by exponential dispersal, has a fractal structure. As the power- law exponent becomes smaller, the distribution of individual genotypes becomes more self-similar at different scales. Common statistics like GST are not well suited to summarizing differences between the population genetic structures. Instead, fractal and self-similarity statistics demonstrated differences in structure arising from fat-tailed and exponential dispersal. When dispersal is fat tailed, a log–log plot of the Simpson index against distance between subpopulations has an approximately constant gradient over a large range of spatial scales. The fractal dimension D2 is linearly inversely related to the power-law exponent, with a slope of À2. In a large simulation arena, fat-tailed LDD allows colonization of the entire space by all genotypes whereas exponentially bounded dispersal eventually confines all descendants of a single clonal lineage to a relatively small area. HE importance of long-distance dispersal (LDD) 2003; Davies et al. 2004; Bialozyt et al. 2006). Com- T for the distribution and evolution of organisms puter models of the process explained the patchy has long been recognized and was acknowledged as genetic patterns, observed in modern oak populations, early as 1859 by Darwin (Darwin 1859). Until recently, by LDD founder events (Davies et al. 2004; Bialozyt however, population genetic and evolutionary studies et al. 2006). have concentrated mainly on short-distance dispersal Dispersal by wind is a major mechanism of LDD. that is easier to measure but nonetheless has important Fungal spores causing severe agricultural diseases are consequences for local population dynamics. Owing to dispersed in rare events over hundreds or even thou- better methodology for assessing LDD and increased sands of kilometers (Brown and Hovmøller 2002). awareness of its importance, interest in it has risen Transport through the air is profoundly affected by again in the last 15 years (Nathan et al. 2003). turbulence over a wide range of spatial scales (Aylor LDD plays an important role in colonization of 2003; Nathan et al. 2005). islands (Cain et al. 2000 and references therein; There is an ongoing debate about what kind of Gittenberger et al. 2006), in range expansion (Cain dispersal kernel, the function that describes the prob- et al. 1998; Clark 1998), and in the rates of population ability that a propagule will be deposited at a given expansion and spread of epidemics (Shaw 1995; Kot distance, is best suited to describe LDD. Several studies et al. 1996). However, LDD is rare and difficult to analyze have modeled LDD of insects or seeds by a mixture of in detail in the field. Modeling of LDD has thus become two exponential dispersal distributions, one with a short an instrument to investigate its importance in evolu- median dispersal distance and one with a very long one tionary and ecological processes. One example is the (e.g.,Nichols and Hewitt 1994; Bialozyt et al. 2006). range expansion of oak trees to the north during the Inferring true dispersal curves from small, wind- postglacial recolonization of Europe. LDD plays an dispersed biological objects like spores or pollen is dif- important role in explaining the speed of the expansion ficult. Measured dispersal distributions are frequently (Le Corre et al. 1997; Austerlitz and Garnier-Ge´re´ leptokurtic or fat tailed, meaning that they have greater density in their shoulders and tails than a Gaussian distribution with the same variance (references in Kot 1Corresponding author: Department of Disease and Stress Biology, John Innes Centre, Colney, Norwich NR4 7UH, United Kingdom. et al. 1996). Many pollen dispersal data are best fitted E-mail: [email protected] with an inverse power-law function (Bullock and Clarke Genetics 177: 435–448 (September 2007) 436 L. U. Wingen, J. K. M. Brown and M. W. Shaw 2000; Austerlitz et al. 2004; Devaux et al. 2005; Klein genetic structure of populations with a roughly stable pop- et al. 2006; Shaw et al. 2006). For fungal spores, the ulation size. This scenario corresponds to a natural population question of the best-fitting dispersal function is hindered that is restricted in growth, e.g., by limited space or limited nutrients. The simulation arena used was very large and so the by the necessity of large experimental plots free from too occupying population should be very large as well. As com- much background infection. A recent study addressing puter memory was inevitably limited, only sample lineages the above problems showed that dispersal of the wheat were simulated. A large population was assumed to be present stripe or yellow rust fungus (Puccinia striiformis) fitted a in the background of these individuals, competing for re- power-law model well if enough sufficiently distant spore sources and thus limiting the expansion of the simulated ackett undt individuals. A birth process with Poisson-distributed progeny traps were used (S and M 2005). Moreover, number of one individual per parent and a fixed death age of although several physical processes underlie wind dis- one generation were used to simulate these sample lineages in persal, theoretical arguments strongly propose that LDD a fluctuating, nonexpanding population. of small objects can be modeled by a single function that General model settings: Simulations were initiated with 30,000 will have inverse power-law behavior in the tails (Shaw individuals, initially all of different genotypes, each repre- ot tockmarr ylor sented by a 32-bit number, in effect 32 biallelic loci. The initial 1995; K et al. 1996; S 2002; A 2003; individuals were placed randomly in a simulation arena of Shaw et al. 2006). Uplift is the most important factor for 108 3 108 square units in size, with 1 unit corresponding to the heavier propagules but many factors are equally impor- closest distance allowed between two individuals. Individuals tant for smaller objects such as spores (Nathan et al. gave birth to offspring at the beginning of each time step, 2005). A simplification of the stochastic dispersal process which were immediately dispersed according to the chosen dispersal function. The genotype of a new individual was either by a single negative power-law dispersal function is a the same as that of the parent or mutated by conversion of one useful basis for theoretical modeling. random bit of the 32-bit genotype. Mutation took place at random Recent simulation studies have addressed either the with a frequency set by the mutation rate m ¼ 10À4. Individuals influence of LDD, modeled as a dual exponential func- died after 1 time step and thus had the chance to produce tion, on population genetic structure (Bialozyt et al. progeny only once in their lifetime. Offspring were not placed outside of the simulation arena or closer to other offspring than 2006) or the influence of power-law dispersal function the minimal interaction distance. If the simulation generated on spatial distribution of species (Cannas et al. 2006). such an event, a new location was calculated until a legitimate one This article investigates the influence of LDD, mod- was found. Individuals possibly adjacent to a given point were eled as a negative power law, on population genetic found quickly using the indexing algorithm in Shaw (1996). The structure of populations in quasi-equilibrium. Inverse simulations were assumed to be a part of a huge population of uniform density. All ‘‘background’’ lineages were assumed to power-law functions with exponents in the range of 1 , disperse in the same way as the simulated lineages and thus b # 2 were used as dispersal functions to simulate fat- result in a similar population genetic structure. The main tailed dispersal. The resulting population structures simulations were aimed to run for 50,000 generations. were compared to those generated by a negative expo- Simulations that were used mainly to calculate the fractal nential dispersal function or a global dispersal (uniform dimensions were run for 10,000 generations only. Dispersal functions: The dispersal of the spores was modeled random) function. Widely used statistics from ecology by an inverse power-law probability density function with the and population genetics were applied to the resulting spore concentration t(r j u) at distance r from the source along populations. Some of them were more suitable than a given bearing u given by others to describe the population structures and to dis- 1 tinguish the outcomes of different modes of dispersal. tðr j uÞ } ð1Þ 1 1 r b The simulations reported here used an arena several orders of magnitude larger than the median dispersal with b . 1(Shaw et al. 2006). Of special interest were values distance of the dispersal function. We used a novel sim- of b # 2. Theoretical and experimental results suggest this is the relevant range in wind dispersal of small particles like ulation strategy that allowed us to investigate a range of fungal spores and pollen (Mccartney 1987; Mccartney and spatial scales covering nine orders of magnitude.
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