spatial autocorrelation analysis of migration and selection · (1) spatial variation patterns...

Copyright 0 1989 by the Genetics Society of America

Spatial Autocorrelation Analysis of Migration and Selection

Robert R. Sokal,* Geoffrey M. Jacquez* and Michael C. Wooten?

*Department of Ecology and Evolution, State University of New York, Stony Brook, New York 1 1 794-5245, and ?Department of Zoology and Wildlife Science, Auburn University, Auburn, Alabama 36849-5414

Manuscript received May 3 1 , 1988 Accepted for publication December 8, 1988

ABSTRACT We test various assumptions necessary for the interpretation of spatial autocorrelation analysis of

gene frequency surfaces, using simulations of Wright’s isolation-by-distance model with migration or selection superimposed. Increasing neighborhood size enhances spatial autocorrelation, which is reduced again for the largest neighborhood sizes. Spatial correlograms are independent of the mean gene frequency of the surface. Migration affects surfaces and correlograms when immigrant gene frequency differentials are substantial. Multiple directions of migration are reflected in the correlograms. Selection gradients yield clinal correlograms; other selection patterns are less clearly reflected in their correlograms. Sequential migration from different directions and at different gene frequencies can be disaggregated into component migration vectors by means of principal components analysis. This encourages analysis by such methods of gene frequency surfaces in nature. The empirical results of these findings lend support to the inference structure developed earlier for spatial autocorrelation analysis.

S PATIAL autocorrelation analysis has been used increasingly in recent years for making inferences

concerning the microevolutionary factors that under- lie observed patterns of spatial variation in allele frequencies. These processes are mutation, selection, migration, and isolation by distance, as well as factors related to the organization of the genetic material. A general framework for such analyses has been developed by SOKAL and ODEN (1978a), SOKAL (1979a), SOKAL and WARTENBERG (198 1) and SOKAL (1983, 1986). These techniques have been applied to numer- ous data sets from animals, plants and humans [see SOKAL, SMOUSE and NEEL (1 986), SOKAL, ODEN and BARKER (1 987), SOKAL e t al. (1 987) and the references cited therein]. However, it is necessary to test whether any of the above-mentioned factors can be distinguished with reasonable certainty in examples in which their relative roles are known. Otherwise, inferences drawn from spatial autocorrelation analysis of field observations must remain unproven. To this end, we have undertaken a series of studies to examine the effect on spatial autocorrelation of changes in the population genetic parameters of simulated populations. In an earlier study (SOKAL and WARTENBERG 1983) the isolation-by-distance model was employed. These authors found that stochastic generating processes resulted in surfaces with characteristics that are functions of the process parameters, such as parent vagility and neighborhood size. Differences in these parameters were detectable as differences in spatial correlograms after only a few generations of simulation. Separate realizations with identical parameters

Genetic\ 121: 845-83.5 (April. 1989)

yielded similar spatial correlograms. These studies confirmed the inference structure of spatial autocorrelation analysis in the characterization of isolation- by-distance processes.

SOKAL and WARTENBERG (1983) based spatial autocorrelation analysis on the following four assertions: (1) spatial variation patterns (spatial response surfaces) can be summarized and characterized by a “signature” obtained through spatial autocorrelation analysis, spectral decomposition, and related techniques; (2) similar deterministic forces result in similar spatial response surfaces; (3) stochastic processes with the same parameters yield independent and different spatial surfaces, but these surfaces will have similar signatures, suggesting similar generating processes; (4) changes in these parameters will be reflected by changes in the signatures. In their earlier study, SOKAL and WARTENBERC (1983) examined the first, third, and fourth of these assertions. In the present study, all four are tested explicitly. These authors also assumed that when many variables, especially different gene loci, are studied simultaneously, they are not all likely to track the same environmental factor across geographic space, and the ones that do are unlikely to respond in the same way to the same environmental factor. We examine the consequences of this assump- tion, which is not susceptible to direct tests by simulation, elsewhere (R.R. SOKAL and G.M. JACQUEZ unpublished data).

In this publication, we continue our studies of the consequences of changes in microevolutionary parameters on gene frequency surfaces. We extend our

846 R. R. Sokal, G. M. Jacquez and M. C . Wooten

investigation of isolation by distance, and we examine the effects of migration from one or more directions, of spatially patterned selection under two regimes, and of the effects of successive migrations from different sources and directions. The specific questions addressed are: (1) What are the consequences for spatial autocorrelation under isolation by distance when neighborhood sizes are increased beyond values previously studied? (2) Are the shapes of correlograms resulting from steady migration pressure affected more by intensity of migration t.han by gene frequency differential between immigrant.s and residents? ( 3 ) How strong must migration be to overcome the spatial autocorrelation generated by isoIation-by-distance? (4) Can the effects of migration from one, two, and four directions be distinguished? ( 5 ) Will spatially patterned selection result in spatial autocorrelations? (6) Can we distinguish the effects of migration from those of selection? (7) Can we recognize the results of re- peated migrations from different directions and sources and separate these into their individual components? As a by-product of these analyses, various observations concerning population structure under these influences emerge, which we also report.

MATERIALS AND METHODS

In this study, we continued the use of the Monte Carlo simulation program developed by ROHLF and SCHNELL (1 97 1) and adapted by SOKAL and WARTENBERG (1983) for autocorrelation analysis of the isolation-by-distance model. This program simulates a population of 10,000 individuals arranged on a 100 X 100 lattice, one individual at each grid intersection. The individuals are considered monecious dip- loids capable of self-fertilization.

Initially, this grid is settled at random by individuals representing the genotype for a one-locus, two-allele system sampled from a population usually of gene frequency 0.5 at Hardy-Weinberg equilibrium, although this initial frequency was changed in some of the simulations reported below. Once per generation each individual is replaced by one offspring from parents chosen at random from within a defined neighborhood centered on that individual. Each combination of conditions was replicated as five independent runs, each lasting 200 generations.

In those cases, used largely as controls in this study, in which we subjected the lattice only to isolation by distance, the neighborhood sizes (WRIGHT 1969, p. 295) chosen were N = 9, 25, 49, 81 and 121. These represent square sublat- tices of dimensions 3, 5, 7 , 9 and 11 centered on the individual to be replaced. Both parents were considered vagile. The first two neighborhood sizes correspond to sets 3 and 4 of SOKAL and WARTENBERG (1 983), respectively. Because sampling was from the full sublattice including the center, the program permitted the occasional sessile parent, and because sampling was with replacement, even more rarely, such a parent would self-fertilize.

We mimicked the effect of migration by an extension of the boundaries of the lattice. Individuals located along one or more of the margins of the lattice were permitted to be replaced by offspring from parents within the lattice as well as from outside its border. The source population of immigrants was not spatially structured but was a Hardy-Wein-

berg population of specified gene frequency. Preliminary studies had shown that migration could not be detected when applied to isolation-by-distance lattices of neighborhood sizes 9 and 2 5 . The effective migration rates (proportion m of genotypes entering the lattice from without, per generation) ranging from 0.003344 to 0.023856 for various combinations of neighborhood sizes and directions of migration were insufficient to overcome the inbreeding effects of the small effective neighborhood sizes (4Nm 5 2.39). Only at N = 49 was migration detectable by both visual inspection of the surfaces as well as through correlogram analysis. This is therefore the neighborhood size at which we ran all the migration experiments. Simulations of per- sistent migration were conducted for three sets of directions: I-sided (from the north), 2-sided (from north and west) and 4-sided (from all sides bounding the lattice). This yielded migration rates of 0.008635, 0.017133 and 0.033992, respectively.

Migration from changing directions was carried out by simulating the first 100 generations under one set of parameters, and then changing to a second set for the next 100 generations. The results reported below are always of gene frequency surfaces at generation 200, except where special mention is made of surfaces from earlier generations.

Selection was modeled by distorting the Mendelian ratios producing the offspring in the direction indicated by the fitness coefficients. This process, similar to zygotic selection, operated at a life history stage corresponding to gametic selection, since individuals once produced and placed on the lattice point, survived and were potentially reproductive. Two selection regimes were simulated: selection against recessives with an intermediate heterozygote [ W ( D ) = 1, W(H) = 1 - s, W ( R ) = 1 - 2s] and selection for heterozygotes [W(D) = 1 - s, W ( H ) = 1, W ( R ) = 1 - 11. In these formulas W ( D ) W ( H ) and W ( R ) are the fitnesses of AA, Aa and aa, respectively, and s and t are selection coefficients. Each regime used four spatial patterns in the magnitudes of the selection coefficients. One was a gradient along an inclined plane running north-south, produced by making J

a linear function of the Y-axis of the 100 X 100 lattice. The function was s = 0.05/99 - 0.05/99Y for selection against recessives. This yielded a range of s-values from 0 to 0.05. With selection for heterozygotes the function wass = 0.0495 + 0.05Y/99, t = 0.15 - s, yielding a range of s-values from 0.05 to 0.10, and of t-values from 0.10 to 0.05. The second pattern was a parabola with low values of s in the north and south and high values in the middle. Its function was s = 0.05 - (50.5 - Y)' (0.05/49.5') for selection against recessives, yielding a range of s from 0.0 to 0.05. When selection was for heterozygotes, the same function for s was used, with t = 0.15 - s. This produced a range of s values from 0.05 in the middle to 0.10 in the north and south, with t ranging from 0.05 to 0.10. The third pattern consisted of five irregularly shaped patches i n the lattice (Figure 1). The fourth pattern consisted of 3 smaller square patches meas- uring 30 X 30 lattice units arranged diagonally across the lattice. For both patch patterns the selection regime was applied within the patches using s = 0.05 for selection against recessives and s = 0.10, t = 0.05 for selection in favor of heterozygotes. Selection was absent outside the patches.

We summarized the results of each run every fifth generation by dividing the 100 X 100 lattice into 400 quadrats of size 5 X 5, and computing the gene frequency for each of these quadrats containing 25 individuals. Additionally, we computed, for every fifth generation, variances of gene frequencies among quadrats and Wright's F-statistic of individuals with respect to the entire lattice (WRIGHT 1969,

Spatial Autocorrelation 847

I;IGL~RE 1 ."lllustl;ltion of arr;lngement of 5 irregularly shaped Ixuc1lc.s 0 1 1 111r 1 0 0 X 100 lattice. Selection \vas practiced against rl1csc. ~"llcllrs.

p. 294). This quantity, formulated as F = (4DR-H2)/(4DR- H? + 2H), where D, H and R are the observed proportions of AA, An and aa, respectively, corresponds to FIT. For each run we graphed the gene frequency surfaces every fifth generation, starting at generation 0 and terminating at generation 200. To summarize the very large number of surfaces (for generation 200 alone there are 800) we computed spatial correlograrns for all of them based on the 400 qwdrats i n the 20 x 20 grid (for details see CLIFF and ORD 198 1 ; SOKAL ;lnd ODEN 1978a. b: SOKAL and WARTENRERC 1983). Correlogranls based on Moran's I-coefficients are reported below. The distance classes for the correlograms were chosen as in SOKAL and M'ARTENRERC (1983); the upper class limits for the first 5 classes are 1.5, 2.5, 3.5, 4.5 and 5.5 grid units, with the succeeding two classes having upper cl;lss limits of 1025 and 20.5 grid units. The eighth class (20.5-30.5 grid units), while conlputed and illustrated, is not featured i n the analyses that follow; previous work had shown it to lead to results that varied too much to be inter1"etable-presllnlably due to edge effects and the reduced number of observations in the highest distance classes. The computations for spatial autocorrelations were carried out by the SAAP program developed by Daniel Wartenberg. The analyses reported in this study are based solely on the resulrs obtained a t generation 200.

Significmce testing of spatial correlograrns is carried out i n several ways. Individual autocorrelation coefficients can be tested for significance by well-established procedures described in CLIFF and ORD (1981). We followed their approach using the less restrictive randomization hypothesis fi)r e\.;lluating the st;lnd;lrd error of autocorrelation coefficients. For entire correlogr;lms we employed the Bonferroni technique suggested by ODEN (1984). The above two procedures test individual coefficients or entire correlograms against the n u l l hypothesis of no spatial autocorrelation. All corrclogr-ams resulting from simulations in this study, except i n a very few cases that will be expressly mentioned, differ significantly from the n u l l hypothesis of no spatial structure at anv distance class. To test dlether two correlogranls, both indic;ltingsignificant spatial structure, differ from each other, we employed Mantel tests (MANTEL 1967; SOKAI. 1979b; HUBERT 1987) following the procedure developed

FIGURE 2.-Aver;lge spatial correlogranls for isolation bv distance a~ generation 'LOO for 5 neighhorhood sixs, N = 9, 25, 49. X 1 and 12 1. FLICII IIICWI corre1ogr;m is 1xlsc.d on 5 rep1ic;ltes. Ordinate: spatial autocorrehtinn coefficirnt (Moran's I ) . Abscissa: tlist;lnre i n quadrat units.

by SOKAL and WARTENRERC (1983). This tests whether Manhattan distances (SNEATH and SOKAL 1973) among all pairs of correlograms for the five replicate runs of each treatment combination are less than Manhattan distances computed for all parts of correlogranls representing replicates from different treatments. This test is not very powerful and is unable to show significance for some differences between mean cor-relograms that exhibit clear trends. How- ever, because this test is conservative, significant differences established by it can be relied upon.

RESULTS

Isolation by distance

To extend the earlier results of SOKAL and WAR- TENRERG (1 983) and to serve as controls for the m i - gration and selection experiments reported below, we carried out isolation-by-distance runs at neighborhood sizes 25, 49, 81 and 12 1 . We also include in our analysis the five replicate runs at neighborhood size 9 reported by these authors. The results are shown in Figure 2 as mean correlograms (over the 5 replicate runs). The findings at N = 25 compare well with those obtained i n the earlier study. The three larger neighborhood sizes first simulated in the present study behave as predicted by SOKAL and WARTENRERC (1983): the increase in spatial autocorrelation with neighborhood size levels off as N increases further, since the distinctness of the now larger, homogeneous areas decreases. Neighborhood size 49 shows even stronger autocorrelation than N = 25 for the first 6 distance classes. The results for N = 81 are again lower than for N = 49 and in fact are not unlike those for N = 25. For neighborhood size 121 the average autocorrelation coefficient for the first distance class (1.5 grid units) is 0.406, lower than the corresponding value of 0.596 for neighborhood size 9, but neighborhood size I2 1 has slightly higher autocorrelation coef-

848 R. R. Sokal, G. M. Jacquez and M. C. Wooten

ficients for distance classes 3.5 to 10.5 grid units than N = 9. The overall correlogram is clearly higher than that for neighborhood size 9. For the limiting case of panmixia reported in the earlier study (SOKAL and WARTENBERC 1983), autocorrelation was uniformly not significantly different from zero for all distance classes.

The intersections of the mean correlograms with the abscissa (the X-intercepts) reflect the size of homogeneous areas on the frequency surfaces (SOKAL 1979a). As neighborhood size increases, the size of homogeneous areas produced by isolation-by-distance increases and the X-intercepts shift to the right. But, as noted earlier, the largest neighborhoods result in less distinct patches, hence the intercepts for N = 81 and 12 1 are again lower than those for N = 49. Mean distances represented by these intercepts are 7.45, 9.65, 1 1.39, 1 1.07 and 10.50 grid units for neighborhood sizes 9, 25, 49, 81 and 12 1, respectively. The mean values of FIT decrease as predicted by theory: they range from 0.257 for N = 9 to 0.0247 for N = 121.

When the five replicate correlograms of all possible pairs of the 5 isolation-by-distance neighborhood sizes were tested for differences by means of Mantel tests, only the mean correlograms for N = 9 and N = 49 differ significantly ( t = 2.510, P = 0.0060). Adjusted for multiple testing this probability becomes 0,0584, slightly above the critical 5% value based on Sidik's multiplicative inequality (SOKAL and ROHLF 198 1). Given the clear trends of the mean correlograms, there is little doubt of the significance of the results. Had each combination of parameters been replicated more than five times, the significance of the differences would have been established with certainty.

Other isolation-by-distance runs at neighborhood size 49 with initial gene frequencies at p = 0.25 and 0.10 were intended as controls for migration simulations with these resident gene frequencies. But these runs also permit comparisons of correlograms based on the same neighborhood size but different initial gene frequencies. SOKAL and WARTENBERC (1981, 1983) had assumed that such gene frequency surfaces would produce similar correlograms, but had not demonstrated this point. The results of these comparisons are shown in Figure 3. The mean correlograms for initial p = 0.50, 0.25 and 0.10 are quite similar and not significantly different by the Mantel test ( P = 0.7256). Mean X-intercepts and FIT values are also in the same range. For any reasonable level of replication these correlograms are unlikely to differ. Thus, these findings support the belief that identical correlograms are produced by the same stochastic process, regard- less of the magnitude of the initial gene frequencies.

Migration from the same direction($ We discuss the results of the migration simulations

from two points of view: gene-frequencies differentials

I O 1

-0.5 i 0 lo 20 30

DISTANCE

FIGURE 3,"Average spatial correlogram for isolation by distance at generation 200 for neighborhood sise 49 with starting allele frequences of p = 0.50 , 0 .25 and 0.10. Each mean correio- gram is based on 5 replicates. Ordinate: spatial autocorrelation coefficient (Moran's I ) . Abscissa: distance in quadrat units.

and directions of migration. Gene-frequency differentials are the differences between native and immigrant gene frequencies. Direction of migration has already been described in MATERIALS AND METHODS.

Gene-frequency differentials: The mean correlograms for a resident gene frequency P R = 0.50 with one-sided migration are shown in Figure 4a. The results for the four immigrant gene frequencies p l = 0.55, 0.60, 0.75 and 0.90 are quite similar to each other and to the isolation-by-distance control. Of 12 contrasts tested, only that between immigrant gene frequencies 0.55 and 0.90 is significant ( t = 2.565, P = 0.0052), but when the probability is adjusted for multiple testing it becomes 0.0606, slightly above the conventional significance level. Thus we cannot safely conclude that the gene-frequency differentials employed affected the spatial structure of the lattice with residents at p R = 0.50.

By introducing the same immigrant gene frequencies into resident populations of gene frequencies and p R = 0.25 and 0.10, we were able to increase immigrant gene frequency differentials. For resident gene frequency p , = 0.25 (Figure 4b) all correlograms representing immigration are higher than the isolation-by-distance control up to distance class 10.5 grid units, lower beyond. By the most conservative criteria-an experimentwise error rate of less than 0.005 for all comparisons of interest-there is a significant overall heterogeneity among treatments (immigration gene frequency differentials), the mean correlogram for immigrant p I = 0.90 differs from all other treatments but p , = 0.75, and the isolation-by-distance controls as well as $1 = 0.55 differ from P I = 0.75 and 0.90. The effects of migration are clearly apparent. Visual inspection of maps of the surfaces reveals clines at least for p I = 0.90 and these are corroborated by

Spatial Aut(

N = 49 = o.55 = Li,6; """"

p = 0.75 p = 0.90

-0.5-

-1.2 0 10 20 30

a DISTANCE

t o ' N = 49

= o,55 """"""

= 0.60 ------" p = 0.75 p = 0.90

-0.5' \\\

% \

-12 0 10 20 30

DISTANCE b

N = 49 = o,55 """"""

= 0.60 """"

-12 0 10 20 30

C DISTANCE

FIGURE 4.-Average spatial correlograms for migration from a single compass direction. Effects of varying immigrant gene frequency differentials. Each mean correlogram is based on 5 replicates. Ordinate: spatial autocorrelation coefficient (Moran's I ) . Ab- scissa: distance in quadrat units. a, Resident gene frequency p R = 0.50. h, Resident gene frequency p~ = 0.25. c. Resident gene frequclnc-v pH = 0.10

>correlation 849

an increase of mean monotonicities of correlograms from 7.0 distance classes for the controls to 8.0 (the maximum possible) for $1 = 0.90, as well as by the corresponding lengthening of the mean X-intercept from 11.08 to 13.70 grid units.

Further increasing the immigration gene frequency differential by introducing the same immigrant gene frequencies to a resident population of gene frequency p , = 0.10 (Figure 4c) resulted in nearly diagrammatic results. AI1 immigrant gene frequencies, including PI = 0.55 and 0.60, produced dramatic clines. With an experimentwise error rate < 0.005, there is significant overall heterogeneity among treatments, and the isolation-by-distance controls differ from each treatment. Immigrant gene frequencies 0.90 and 0.60 differ from each other at P < 0.025, although this is difficult to see at the resolution of Figure 4c. Mean monoto- nicity is uniformly 8.0 distance classes for all treatments (7.0 for the controls) and the mean X-intercept has lengthened from 10.40 grid units for the controls to 14.09 for p1 = 0.90. The five replicate correlograms bunch up into tight sheaves not reflected in the treatment mean correlograms of Figure 4, b and c, much tighter than in the isolation-by-distance controls or in the case of the populations with resident gene frequency of 0.50. This low variability of replicates is responsible for the significant difference between the correlograms for p1 = 0.90 and 0.60 noted above, despite the small difference between their means.

Note that for immigrant gene-frequency differentials ( p , - P I ) 5 0.40 clines are not detected, whereas significant differences from the isolation-by-distance controls and marked clines occur with the differential 2 0.45. The clines are characterized by monotonic declining correlograms, lengthened X-intercepts, and significant negative autocorrelations at the highest distance classes.

Directions of migration: We illustrate the effects of altering the direction of immigration from 1-sided (from north) to 2-sided (from north and west) to 4- sided (from all four compass directions) using immigrant gene frequency P I = 0.90 (Figure 5). Migration from one side produces a clear but marginally significant cline, as already discussed. Two-sided migration produces a gene frequency surface representing an inclined plane tilted from a high in the northwest corner of the lattice to a low in the southeast corner. Such a pattern increases the number of similar quadrats at low distances along the east-west and north- south axes and at high distances comparing the north and west sides and the south and east sides. In conse- quence, the mean correlogram of the 2-sided migration surface is higher than that for 1-sided migration at distance classes 1.5 through 10.5 grid units and at 30.5. The correlogram based on 4-sided migration is indistinguishable from that for 1-sided migration or


10'

\

N = 49 , side """"""

sides """"

-0.5 4 0 10 20 30

DISTANCE

FIGURE 5.-Average spatial correlograms for directional irnmi- gration. Immigration from one, two and all four directions is compared with isolation-by-distance controls. Resident gene frequency p R = 0.50, immigrant gene frequency p , = 0.90. Each mean correlogram is based on 5 replicates. Ordinate: spatial autocorrelation coefficient (Moran's I ) . Abscissa: distance in quadrat units.

the isolation-by-distance controls for the first six distance classes. But at the last distance class there is again strong positive autocorrelation which clearly reflects the "square bowl" shape of the gene frequency surface. Since the rims of these bowls represent the maximum effect of the high immigrant gene frequencies, we would expect the highest distances to include rims on opposite sides, which would be quite similar to each other. Thus, 4-sided immigration is clearly differentiable from 1- and 2-sided immigration. These findings are supported by Mantel tests where with an experimentwise error 5 0.005 we can demonstrate overall heterogeneity among I-, 2-, and 4-sided migration, significant differences between the correlograms for 1- and 2-sided migration, as well as between 2-sided and 4-sided migration. Recall that these tests exclude the eighth distance class (30.5 grid units), hence do not even involve the most characteristic aspect of the correlogram based on 4-sided migration. The mean monotonicities of the correlograms decrease from 7.8 to 6.6 distance classes as the number of sides involving migration increases, whereas the mean X-intercept changes from 12.72 grid units (1- sided) to 13.41 (2-sided) and to 9.08 (4-sided). Note that in the last case, the square bowl, the descent into the low gene frequency center is quite steep, making for smaller homogeneous areas.

The mean gene frequencies of the surfaces for 2- sided and 4-sided migration reflect the increased gene flow over one-sided migration. At generation 200, p = 0.614, 0.679 and 0.788 for the three treatments, respectively. Migration from changing directions

A given gene frequency surface may be the conse- quence of multiple migration events. These may occur

simultaneously or over different time periods. In this section we investigate the consequences of two sequential migration events. We ask the following questions:

What is the effect of sequential migration with changing parameters (immigrant gene frequency, preference weights for immigrant parents and direction of immigration) on the spatial distribution of gene frequencies? Is it possible to disaggregate the sequential migration events and infer the separate directions of migration?

T o simulate the effect of sequential migration from two different directions, we proceeded as follows. Since the migration rate was at times too low to overcome the gene-frequency fixing counteraction of the underlying isolation-by-distance process, we made provisions for reinforcing the migration trend by as- signing preference weights to parents located in the direction of the immigrant source population. Thus a preference weight of 3 indicates that for any point on the lattice a parent located in the direction of the immigrants would be three times more likely to be chosen than a parent in any other direction. We produced 6 initial surfaces which resulted from sub- jecting a lattice with a starting resident gene frequency p , = 0.50 at generation 0 to the 6 possible combinations of preference weights 1 and 3, and immigrant gene frequencies PI = 0.55, 0.75 and 0.90. In all cases the immigration was from the north and lasted for 100 generations. We made 15 copies (not separate realizations) of each of the 6 surfaces at generation 100 and each of these copies was subjected to one of 15 treatments. These are the possible combinations of 5 new immigrant gene frequencies ( P I = 0.10,0.25, 0.50, 0.75 and 0.90) with the 3 following pairings of preference weights and directions of immigration, lW, 3W and 3NW. These new conditions were applied for a subsequent 100 generations. The 6 initial surfaces multiplied by the 15 subsequent conditions yield a total of 90 simulated surfaces. Only single replicates were run for each combination of parameters. These surfaces can be arranged into 6 groups by criteria of initial and subsequent preference weights for immigrants and directions of migration. These are, respectively, lN:lW, 3N:3W, 1N:3W, 3N:lW, 1N:JNW and 3N:3NW. The abbreviations indicate the preference weight and direction of the first 100 generations, followed by these parameters for the second 100 generations. In the first four combinations the sequential migration events occur at right angles to each other (orthogonal sequential migration), in the last two combinations they are at 45" (oblique sequential migration). Permitting the 15 gene frequency surfaces in each group to vary in their immigrant gene frequencies furnished a degree of realism

Spatial Autocorrelation 85 1

N

E W

S

o m m a m m m m m m FIGURE 6,"Gene-frequency surface of 20 X 20 lattice after 100

generations of migration from the north at immigrant frequency p, = 0.9, resident gene frequency p~ = 0.5, preference weight = 3. Shading respresents 10 equal intervals along a gene-frequency scale ranging from 0.22 to 0.96.

to the simulation, since we could not expect that in nature immigrants would differ from residents by a constant amount in the allele frequencies for different loci.

The gene frequency surfaces at generation 100 and at generation 200 were mapped and subjected to spatial autocorrelation analysis. Pairwise correlations over the 400 quadrats were computed among the 15 gene frequency surfaces at generation 200 for each of the 6 groups and decomposed by principal components analysis (SNEATH and SOKAL 1973) to deter- mine if the surface patterns could be resolved into the separate migration events. The first and second factor scores were mapped onto the geographic space and the resulting maps examined for trends. Both raw factor scores and those obtained from rotation to simple structure were examined.

In both the orthogonal and the oblique migration simulations, migration is from the north for the first 100 generations. As described earlier, unidirectional migration will produce clinal surfaces, although the slope and definition of the cline depend on the gene frequency differential of the migrants. We had found in previous simulations that a differential of at least 0.40 is required to produce clinal structure by generation 200. As a sample of the six surfaces resulting at generation 100 we show the most pronounced one in Figure 6. The correlograms of all six correspond to those described earlier.

It is not possible to illustrate the 90 different surfaces reflecting 100 additional generations of migration or their correlograms. We limit our illustrations to the 4 surfaces shown in Figure 7, which feature the highest immigrant gene frequency ( p , = 0.90) for both the first and second 100 generations. The surfaces are the results of the following combinations of preference weights and directions of migration: 1N:3W, 3N:3W, IN:3NW and 3N:3NW. The orthogonal se-

quential migration surfaces (1 N:3W and 3N:SW) are similar to one another, as are the oblique ones (1N:SNW and 3N:3NW). All four surfaces show evi- dence of their past migration history in that higher allele frequencies p are observed in the north and west for the orthogonal sequential migration surfaces and in the north and northwest for the oblique sequential migration surfaces. The most recent migrations are most strongly manifested, with the western edges of the orthogonal sequential migration surfaces and the northwest corners of the oblique sequential migration surfaces showing the highest frequencies of p . Differ- ences in preference weights for the two sequential migration events produced little change in the spatial pattern of distribution of the gene frequencies. When the surfaces created by immigrants with gene frequency PI = 0.55 were examined (not shown here) no clines were evident, in line with other simulations involving this slight immigrant gene frequency differential.

From an inspection of the surfaces and correlograms of all the sequential migration simulations we arrive at the following conclusions. First, the recent migration is usually more strongly manifested on the resulting surface, given that the immigrant gene frequency differential is sufficient to yield any result at all. Second, the effect of earlier migration events is evident even after 100 subsequent generations of isolation by distance combined with a different migration event.

Maps of the raw factor scores for the four groups representing orthogonal migration (1 N: 1 W, 3N:3W, 1 N:3W, 3N: 1 W) reflect the sequential migration pattern, with scores for the first factor corresponding to the most recent migration during generations 10 1 to 200 of the simulation and with scores for the second factor indicating the earlier migration during generations 1 to 100. We illustrate the results for 3N:3W in Figure 8a. The results are more complex for the oblique migration simulations. Although migration from the northwest is reflected in the map of the first factor scores, the map of the second factor scores does not reflect migration from the north. The results for 3N:3NW are shown in Figure 8b. Inspection of similar maps based on rotation to oblique simple structure (not shown), convinced us that factor rotation does not increase our ability to discriminate between the two sequential migration events. Different preference weights for migration appear to have little effect on the factor score maps. The proportion of the variation explained by the first factor varies from 0.23 for group 3N:lW to 0.36 for group 3N:3NW. That explained by the second factor varies from 0.1 1 to 0.19. The proportion of the variation explained by the 3rd factor was always below IO%, except for group 3N:lW, for which it was 11%.

852 R. R. Sokal, G . M. Jacquez and M. C. Wooten

1N:3W 3N3W

N N

periods of 100 generations of immigration in cl

different directions and at different strengths. Immigrant gene frequency f i t = 0.90, resident El3 ' .

.I .I

gene frequency p R = 0.5. The four surfaces are identified by their combinations of preference weights and directions of immigration for the first and second hundred generations. Shading

. I .

1N3NW

S

3N3NW

N

S S

For the sequential migration analysis we conclude with the following observations. First, the patterns observed on the factor score maps (synthetic gene frequency maps of MENOZZI, PIAZZA and CAVALLI- SFORZA 1978) generally reproduce the sequential migration patterns modeled in the simulations, with the spatial pattern of distribution of the first factor score reflecting the axis of migration for the most recent migration event. This holds true for both orthogonal and oblique migration simulations. Second, the synthetic frequency map for the second factor appears to include components from both the first and the second migration events. We see, for example, in data set 3N:3W (Figure Sa) that the synthetic frequency map shows an eastern border of low factor score values in addition to an overall north-south cline. The use of any but the first factor score for inferring individual migration events may therefore be suspect, since the second factor score loads on both the first and second migration events. Third, rotation to oblique simple structure does not improve the correspondence between the spatial distributions of the factor scores and the modeled migration events, and in some cases tends to obscure it. As an example, the synthetic frequency

map of the second factor in data set 1N:3NW shows, correctly, an overall north-south cline in unrotated factor space. After rotation, the map of the second factor shows a northwest-southeast cline that does not correspond to the migrations modeled in the simulation.

Selection

The correlograms for the four spatial patterns of the selective agent-linear gradients, parabolas, irregular patches and regular patches-each with selection imposed against recessives or for heterozygotes, are shown in Figure 9. An overall Mantel test among the correlograms was highly significant. All significance statements in this paragraph are supported at an experimentwise error rate I 0.005, except for one whose rate is only 5 0.025. The linear gradients resulted in clearly defined clines. The mean correlogram representing selection for heterozygotes is shal- lower than that representing selecting against recessives, as might be expected. However, this difference cannot be shown to be significant. Because the control mean correlogram (isolation-by-distance, neighborhood size 49) is intermediate between the two linear

N

Spatial Autocorrelation

N

853

a

b

W

S

N

E

W E FIGURE 8."Surfaces of raw factor scores

based on principal components analyses of correlations between 15 simulated gene frequency surfaces varying in preference weights and immigrant gene frequency differential. For detailed

1 are at left, those for factor 2 at right. a, Surface subject to 100 generations of immigration from ' €m El ' the north with preference weight 3, followed by 100 generations from the west with preference weight 3. b, Surface subject to 100 generations

S description of design see text. Scores for factor

N of immigration from the north with preference

S S

I O 1 I ',. N = 4 9 - cline """"". cline """- parabo1U"-

patches"""

.. .- "

-10 4 0 lo 20 30

DISTANCE FIGURE 9.-Average spatial correlograms for eight selection re-

gimes described in the text and an isolation-bydistance control ( N = 49). For each different pattern the first listed correlogram is based on selection against recessives with an intermediate heterozygote and the second on selection for heterozygotes. The first two patches are the irregular pattern, the second two the diagonal square patches. Each mean correlogram is based on five replicates. Ordinate: spatial autocorrelation coefficient (Moran's I). Abscissa: distance in quadrat units.

gradient correlograms, it cannot be shown to differ from them. All other patterns of selective agents yielded correlograms below the control correlogram up to distance class 10.5 grid units. They do not differ

E

weight 5; followed by 100 generations of immigration from the northwest with preference weight 3. Shading represents 5 equal intervals along a scale of factor scores ranging as follows: -1.55 to 2.790, -2.26 to 2.83, -1.59 to 2.74 and -2.55 to 2.44, respectively, for a, left and right, and b, left and right.

m .

significantly among themselves, although they do differ significantly from the control and the linear gradients. The parabolas are characterized by apprecia- ble positive autocorrelation in the last distance class (30.5 grid units), reflecting similar gene frequencies between grid points at the two ends of the parabola. The selection patterns for the two kinds of patches result in less pronounced correlograms than the isolation-by-distance controls, since the selection coefficients employed were not sufficiently strong to over- ride the isolation-by-distance effects. The overlay of patches on top of the relatively homozygous neighborhoods produced by the isolation-bydistance processes may have resulted in a finer partition of the surfaces into small homogeneous regions.

The correlogram produced by the linear gradients are of the same general form as those produced by one-sided migration. Although the mean correlogram for linear-gradient selection against recessives is above all correlograms obtained for one-sided migration when resident gene frequency is p R = 0.5, it is no longer distinguishable from those of migration processes when p R = 0.10. Thus the distinction between migration and selection in actual data must be made on the basis of correlograms for more than one gene frequency.


DISCUSSION AND CONCLUSIONS

The simulated gene-frequency surfaces of this study are relatively stable, though not stationary. SOKAL and WARTENBERC (1 983) used the term quasi-stationarity for such surfaces. In a like manner, gene frequency surfaces of natural populations are likely to be stable, but not stationary. Stationarity of gene frequency surfaces in nature must be very rare.

With respect to the questions posed in the INTRO- DUCTION, the present study was able to confirm and extend the earlier findings of SOKAL and WARTEN- BERG (1983): surfaces subject only to isolation by distance are uncorrelated with each other, but each is similarly autocorrelated as long as the parameters underlying the stochastic process are the same. These authors also predicted that autocorrelations for large neighborhood sizes would be lower than those for intermediate sizes because, while the diameters of the homogeneous areas would increase, their amplitude would decrease. This prediction is confirmed in our study by the reduction of autocorrelations for neighborhood sizes 81 and 121 (question 1).

Migration from the same direction results in clinal surfaces and correlograms. Although such results had been expected widely (ENDLER 1977, 1983; RENDINE, PIAZZA and CAVALLI-SFORZA 1986), this is the first empirical demonstration in connection with spatial autocorrelation analysis. In response to question 2, it is clear that within the range of parameters employed in this study, immigrant gene frequency differentials are far more important in shaping resulting surfaces and correlograms than are migration intensities. In- creasing preference weights for immigrant parents did not produce marked changes in surfaces or correlograms. This is not to say that if migration rates had been increased to a level swamping the resident population, these effects would not have predomi- nated. However, in such a case the resulting surface would not have assumed the characteristics of a dif- fusion cline. The lesson to be learned is that migration phenomena can be detected from gene frequency surfaces only when the immigrant and resident population differ substantially in gene frequency (by more than 0.40 for this study). The frequency differential must be large to offset stochastic variation introduced by isolation by distance (question 3). In the absence of isolation by distance the frequency differential can be much smaller and still produce clinal correlograms. Because conventional autocorrelation coefficients standardize the variable analyzed, a shallow cline yields the same correlogram as a steep cline provided the slopes are a scalar function of one another. How- ever, autocorrelated noise introduced by isolation by distance obscures the correlograms of shallow clines, making them indistinguishable from those produced solely by isolation by distance.

The effects of differences in directions of immigration (question 4) reflect the nature of the surfaces produced. The diagonally tilted trough of 2-sided immigration and the square bowl of 4-sided immigration create configurations which yield characteristic responses by the correlograms. These observations drive home the fact that correlograms are efficient summaries of the surfaces on which they are based.

The effects of selection were marked only for the linear gradients (question 5). For the patches it would appear that selection did not furnish a sufficient differential between the gene frequencies of the patches and that of the surrounding matrix to be picked up by the spatial autocorrelation analysis. Two hundred generations of selection in the patches with the selection coefficients described earlier should result in gene frequencies of 1.0000 for selection against recessives and 0.3335 for asymmetric selection against heterozygotes. Because of migration into the patches resulting from the vagility of individuals in the isolation-bydistance model, the actually observed values are, respectively, 0.5500 and 0.4783. As long as the patches are sufficiently differentiated from the surrounding area (by a combination of strong selection and weak migration), a clear and significant spatial pattern will emerge. This is demonstrated in work by G. BARBU- JAN', N. L. ODEN and R. R. SOKAL (unpublished data) in which an artificial surface of the same dimensions (20 x 20 grid) was created by adding a constant value of 0.5 to a system of regular patches on a surface consisting of white noise only. The patches became clearly visible on surface maps of the gene frequencies and yielded characteristic correlograms.

In the case of the linear gradient it is not possible from the correlogram of a single surface alone to distinguish between gradients caused by migration and those due to selection (question 6). We anticipate, however, that in a natural situation such a distinction can be made when the study is conducted using a number of different loci. It is expected that migration will be reflected in most of these loci (absent only in those whose gene frequency differential with the residents is too small). By use of synthetic gene frequencies (see below), this trend can be reinforced, as was observed in our experiments where we featured different immigrant gene frequencies for the same patterns of compass directions. In contrast, selection patterns are unlikely to be similar for different loci since presumably different allozymes or other biochemical genetic markers will track different environmental factors and in different ways. Thus we would expect correlation among gene frequency surfaces produced by migration and similarity among their correlograms. N o such correlation is expected for selection-induced surfaces nor should there necessarily be similarities among their correlograms.

Spatial Autocorrelation 855

We can recognize the results of different sequential directions of migrations at the end of 200 generations since the surfaces usually show both components (question 7). As expected, the last migration usually has the strongest effect. In the case of orthogonal migration we were able to disaggregate the effects by using principal components analysis, since the correlation structure of the gene frequencies will reflect the differential contribution to it by various immigrant gene frequencies in several directions. We cannot, however, be certain that all components will emerge. As we rotated the angle of immigration of the recent migrants to only 45 degrees from that of the previous migrants, the two effects seem to blend one into the other. One may be able to obtain more precise resolutions by plotting directional correlograms (ODEN and SOKAL 1986) of the factor scores.

The differences among correlograms produced by different parameters for isolation by distance, migration, or selection are not as significant as they might be in view of the conservative testing method applied, which lacks statistical power. As more powerful methods for testing differences between correlograms are developed, we are certain that the current conservative findings can be extended to show considerable heterogeneity of the correlogram structure in response to changing parameter values. This in turn will give rise to more powerful inferences to be made in actual field situations.

Contribution No. 700 in Ecology and Evolution from the State University of New York at Stony Brook. We are indebted to Dr. Kent Fiala for initiating the simulations and obtaining some of the results on isolation by distance. Junhyong Kim wrote the program for mapping the gene frequencies, Donna DiGiovanni prepared the illustrations and Cheryl Daly word processed the manuscript. Dr. Neal Oden read a draft of the paper and offered constructive criticism. We thank them all. This research was supported by grant GM28262 from the National Institutes of Health and grant BSR8614384 from the National Science Foundation to Robert R. Sokal.

LITERATURE CITED

CLIFF, A. D, and J. K. ORD, 1981 Spatial Processes. Pion, London. ENDLER, J. A., 1977 Geographic Variation, Speciation, and Clines.

Princeton University Press, Princeton, N. J. ENDLER, J. A., 1983 Testing causal hypotheses in the study of

geographical variation, pp. 424-443 in Numerical Taxonomy, edited by J. FELSENSTEIN. Springer-Verlag, New York.

HUBERT. L. , 1987 Assignment Methods in Combinatorial Data Analy- sis. Marcel Dekker, New York.

MANTEL, N., 1967 The detection of disease clustering and a generalized regression approach. Cancer Res. 27: 209-220

MENOZZI, P., A. PIAZZA and L. CAVALLI-SFORZA, 1978 Synthetic maps of human gene frequencies in Europeans. Science 201: 786-792.

ODEN, N. L., 1984 Assessing the significance of a spatial correlogram. Geogr. Anal. 16: 1-16

ODEN, N. L., and R. R. SOKAL, 1986 Directional autocorrelation: An extension of spatial correlograrns to two dimensions. Syst. Zool. 35: 608-617.

RENDINE, S., A. PIAZZA and L. L. CAVALLI-SFORZA, 1986 Simulation and separation by principal components of multiple demic expansions in Europe. Am. Nat. 128: 681-706.

ROHLF, F. J., and G. D. SCHNELL, 1971 An investigation of the isolation by distance model. An. Nat. 105: 295-324.

SNEATH, P. H. A , , and R. R. SOKAL, 1973 Numerical Taxonomy. W. H. Freeman, San Francisco.

SOKAL, R. R., 1979a Ecological parameters inferred from spatial correlograms, pp. 167-196 in Contemporary ~uantitativeEcology and Related Ecometrics, edited by G . P. PATIL and M. L. ROSEN- ZWEIG. International Cooperative Publishing House, Fairland, Md.

SOKAL, R. R., 1979b Testing statistical significance of geographic variation patterns Syst. Zool. 28: 227-23 1 .

SOKAL, R. R., 1983 Analyzing character variation in geographic space, pp. 384-403 in Numerical Taxonomy, edited by J. FEL- SENSTEIN. Springer-Verlag, New York.

SOKAL, R. R., 1986 Spatial data analysis and historical processes, pp. 29-43 in Data Analysis and Informatics, IV, edited by E. DIDAY, Y. ESCOUFIER, L. LEBART, J. PAGES, Y. SCHEKTMAN and R. TOMASSONE. North-Holland, Amsterdam.

SOKAL, R. R., and N. L. ODEN, 1978a Spatial autocorrelation in biology. 1 . Methodology. Biol. J. Linn. SOC. 10: 199-228.

SOKAL, R. R., and N. L. ODEN, 1978b Spatial autocorrelation in biology. 2. Some biological implications and four applications of evolutionary and ecological interest. Biol. J. Linn. SOC. 1 0

SOKAL, R. R., and F. J. ROHLF, 198 1 Biometry, Ed. 2. W. H.

SOKAL, R. R., and D. E. WARTENBERG, 198 1 Space and population structure, pp. 186-2 13 in Dynamic Spatial Models, edited by D. GRIFFITH and R. MCKINNON. Sijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands.

SOKAL, R. R., and D. E. WARTENBERG, 1983 A test of spatial autocorrelation using an isolation-by-distance model. Genetics 105: 219-237.

SOKAL, R. R., N. L. ODEN and J. S. F. BARKER, 1987 Spatial structure in Drosophila buzzatii populations: simple and two- dimensional spatial autocorrelation. Am. Nat. 1 2 9 122-142.

SOKAL, R. R., P. E. SMOUSE and J. V. NEEL, 1986 T h e genetic structure of a tribal population, the Yanomama Indians. XV. Patterns inferred by autocorrelation analysis. Genetics 114: 259-28 1 .

SOKAL, R. R., I. A. LENGYEL, P. A. DERISH, M. C. WOOTEN and N. L. ODEN, 1987 Spatial autocorrelation of A B 0 serotypes in mediaeval cemeteries as an indicator of ethnic and familial structure. J. Archaeol. Sci. 14: 615-633

WRIGHT, S., 1969 Evolution and the Genetics of Populations, Vol. 2: The Theory of Gene Frequencies. University of Chicago Press, Chicago.

229-249.

Freeman, San Francisco.

Communicating editor: M. T . CLEGG

spatial autocorrelation analysis of migration and selection · (1) spatial variation patterns...

Documents