Identification of spatial genetic boundaries using a multifractal model in human population genetics.

There are two purposes in displaying spatial genetic structure. One is that a visual representation of the variation of the genetic variable should be provided in the contour map. The other is that spatial genetic structure should be reflected by the patterns or the gradients with genetic boundaries in the map. Nevertheless, most conventional interpolation methods, such as Cavalli-Sforza's method in genography, inverse distance-weighted methods, and the Kriging technique, focus only on the first primary purpose because of their arbitrary thresholds marked on the maps. In this paper we present an application of the contour area multifractal model (CAMM) to human population genetics. The method enables the analysis of the geographic distribution of a genetic marker and provides an insight into the spatial and geometric properties of obtained patterns. Furthermore, the CAMM may overcome some of the limitations of other interpolation techniques because no arbitrary thresholds are necessary in the computation of genetic boundaries. The CAMM is built by establishing power law relationships between the area A (> or =rho) in the contour map and the value p itself after plotting these values on a log-log graph. A series of straight-line segments can be fitted to the points on the log-log graph, each representing a power law relationship between the area A (> or =rho) and the cutoff genetic variable value for rho in a particular range. These straight-line segments can yield a group of cutoff values, which can be identified as the genetic boundaries that can classify the map of genetic variable into discrete genetic zones. These genetic zones usually correspond to spatial genetic structure on the landscape. To provide a better understanding of the interest in the CAMM approach, we analyze the spatial genetic structures of three loci (ABO, HLA-A, and TPOX) in China using the CAMM. Each synthetic principal component (SPC) contour map of the three loci is created by using both Han and minority groups data together. These contour maps all present an obvious geographic diversity, which gradually increases from north to south, and show that the genetic differences among populations in different districts of the same nationality are greater than those among different nationalities of the same district. It is surprising to find that both the value of p and the fractal dimension alpha have a clear north to south gradient for each locus, and the same clear boundary between southern and northern Asians in each contour map is still seen in the zone of the Yangtze River, although substantial population migrations have occurred because of war or famine in the last 2,000 or 3,000 years. A clear genetic boundary between Europeans and Asians in each contour map is still seen in northwestern China with a small value of alpha, although the genetic gradient caused by gene flow between Europeans and Asians has tended to show expansion from northwestern China. From the three contour maps another interesting result can be found: The values of alpha north of the Yangtze River are generally less than those south of the Yangtze River. This indicates that the genetic differences among the populations north of the Yangtze River are generally smaller than those in populations south of the Yangtze River.