In this chapter we explore various ways of assessing if population are structured (e.g., differentiated). You can think of population structure as identifying clusters or groups of more closely related individuals resulting from reduced gene flow among these groups. Populations can be studied to determine if they are structured by using, for example, population differentiation summary statistics (e.g. \(G_{ST}\)), clustering or minimum spanning networks. Note, that this chapter will utilize many data sets due to the unique features offered by each. Let’s first look at an example of population differentiation based on \(G_{ST}\).

\(G_{ST}\) an example with Felis catus data.

Assessing genetic diversity almost always starts with an analysis of a parameter such as \(G_{ST}\). There are lengthy debates as to what measure of differentiation is better (Meirmans & Hedrick, 2011). Instead of going into that lengthy debate, it would be more worthwhile to point you into the direction of a package dedicated to Modern Methods of Differentiation called mmod. We will use the data set nancycats containing 17 colonies of cats collected from Nancy, France. As cats tend to stay within small groups, we expect to see some population differentiation. In terms of these diversity measures, an index of \(G_{ST} = 0\) indicates no differentiation, whereas \(G_{ST} = 1\) indicates that populations are segregating for differing alleles.

Let’s load the package and the example data set:

library("mmod")
data("nancycats")
nancycats
## /// GENIND OBJECT /////////
## 
##  // 237 individuals; 9 loci; 108 alleles; size: 145.3 Kb
## 
##  // Basic content
##    @tab:  237 x 108 matrix of allele counts
##    @loc.n.all: number of alleles per locus (range: 8-18)
##    @loc.fac: locus factor for the 108 columns of @tab
##    @all.names: list of allele names for each locus
##    @ploidy: ploidy of each individual  (range: 2-2)
##    @type:  codom
##    @call: genind(tab = truenames(nancycats)$tab, pop = truenames(nancycats)$pop)
## 
##  // Optional content
##    @pop: population of each individual (group size range: 9-23)
##    @other: a list containing: xy

Now we will use Hendrick’s standardized \(G_{ST}\) to assess population structure among these populations (Hedrick, 2005).

Gst_Hedrick(nancycats)
## $per.locus
##      fca8     fca23     fca43     fca45     fca77     fca78     fca90 
## 0.4750445 0.2956688 0.2675766 0.2653163 0.4855829 0.1933327 0.3807578 
##     fca96     fca37 
## 0.3913924 0.1609576 
## 
## $global
## [1] 0.3084895

What does this output tell us?

Next we will look at genetic distance to find related groups of individuals.

Genetic Distance

If we wanted to analyze the relationship between individuals or populations, we would use genetic distance measures which calculate the “distance” between samples based on their genetic profile. These distances can be visualized with heatmaps, dendrograms, or minimum spanning networks. In the package poppr, there are several distances available:

Distance Function Marker type Can handle missing data
Bruvo’s distance bruvo.dist microsatellite yes
Edwards’ distance edwards.dist any no
Nei’s distance nei.dist any no
Provesti’s distance provesti.dist any yes
Reynolds’ distance reynolds.dist any no
Rogers’ distance rogers.dist any no
Provesti’s distance bitwise.dist SNP yes

One common way to visualize a genetic distance is with a dendrogram. For this example, we will use the microbov data set (Laloe et al., 2007). This contains information on 704 cattle from both Africa and France over several different breeds. We can create a dendrogram over all 704 samples, but that would be difficult to visualize. For our purposes, let’s take ten random samples and calculate Provesti’s distance, which will return the fraction of the number of differences between samples:

library("poppr")
library("ape") # To visualize the tree using the "nj" function
library("magrittr")
data(microbov)
set.seed(10)
ten_samples <- sample(nInd(microbov), 10)
mic10       <- microbov[ten_samples]
(micdist    <- provesti.dist(mic10))
##              FRBTBDA35243 AFBTSOM9386 FRBTBAZ26396 FRBTGAS9052 AFBIZEB9462
## AFBTSOM9386     0.7500000                                                 
## FRBTBAZ26396    0.6000000   0.6833333                                     
## FRBTGAS9052     0.6333333   0.8500000    0.5333333                        
## AFBIZEB9462     0.7166667   0.6666667    0.7833333   0.8500000            
## AFBTND211       0.6333333   0.5666667    0.6833333   0.7333333   0.8000000
## AFBTSOM9362     0.6000000   0.5333333    0.6666667   0.8166667   0.7000000
## AFBTSOM9360     0.7000000   0.5166667    0.6000000   0.7166667   0.7333333
## FRBTCHA25069    0.6166667   0.7000000    0.5833333   0.6833333   0.8000000
## FRBTBAZ26388    0.5333333   0.7000000    0.5500000   0.6333333   0.7500000
##              AFBTND211 AFBTSOM9362 AFBTSOM9360 FRBTCHA25069
## AFBTSOM9386                                                
## FRBTBAZ26396                                               
## FRBTGAS9052                                                
## AFBIZEB9462                                                
## AFBTND211                                                  
## AFBTSOM9362  0.5333333                                     
## AFBTSOM9360  0.5500000   0.5666667                         
## FRBTCHA25069 0.6500000   0.7333333   0.7166667             
## FRBTBAZ26388 0.6000000   0.6666667   0.6833333    0.6166667

The above represents the pairwise distances between these 10 samples. We will use this distance matrix to create a neighbor-joining tree.

theTree <- micdist %>%
  nj() %>%    # calculate neighbor-joining tree
  ladderize() # organize branches by clade
plot(theTree)
add.scale.bar(length = 0.05) # add a scale bar showing 5% difference.

Notice that the sample names start with either “AF” or “FR”. This indicates their country of origin and we are seeing that the populations cluster correspondingly. Of course, a tree is a hypothesis and one way of generating support is to bootstrap loci. This can be achieved with the poppr function aboot.

set.seed(999)
aboot(mic10, dist = provesti.dist, sample = 200, tree = "nj", cutoff = 50, quiet = TRUE)

## 
## Phylogenetic tree with 10 tips and 8 internal nodes.
## 
## Tip labels:
##  FRBTBDA35243, AFBTSOM9386, FRBTBAZ26396, FRBTGAS9052, AFBIZEB9462, AFBTND211, ...
## Node labels:
##  100, NA, NA, 71.5, NA, NA, ...
## 
## Unrooted; includes branch lengths.

The bootstrap value of 100 on the node separating the French and African samples gives support that the country of origin is a factor in how these breeds are structured. If we wanted to analyze all of the breeds against one another, it would be better to create a bootstrapped dendrogram based on a genetic distance. To do this, we will add 3 stratifications to the microbov data set: Country, Breed, and Species. We will then set the population to Country by Breed, convert the data to a genpop object and then create a tree using aboot with Nei’s genetic distance.

# Setting up the data
strata(microbov) <- data.frame(other(microbov))
microbov
## /// GENIND OBJECT /////////
## 
##  // 704 individuals; 30 loci; 373 alleles; size: 1.1 Mb
## 
##  // Basic content
##    @tab:  704 x 373 matrix of allele counts
##    @loc.n.all: number of alleles per locus (range: 5-22)
##    @loc.fac: locus factor for the 373 columns of @tab
##    @all.names: list of allele names for each locus
##    @ploidy: ploidy of each individual  (range: 2-2)
##    @type:  codom
##    @call: genind(tab = truenames(microbov)$tab, pop = truenames(microbov)$pop)
## 
##  // Optional content
##    @pop: population of each individual (group size range: 30-61)
##    @strata: a data frame with 3 columns ( coun, breed, spe )
##    @other: a list containing: coun  breed  spe
nameStrata(microbov) <- ~Country/Breed/Species

# Analysis
set.seed(999)
microbov %>%
  genind2genpop(pop = ~Country/Breed) %>%
  aboot(cutoff = 50, quiet = TRUE, sample = 1000, distance = nei.dist)
## 
##  Converting data from a genind to a genpop object... 
## 
## ...done.

## 
## Phylogenetic tree with 15 tips and 14 internal nodes.
## 
## Tip labels:
##  AF_Borgou, AF_Zebu, AF_Lagunaire, AF_NDama, AF_Somba, FR_Aubrac, ...
## Node labels:
##  100, 100, 99.8, 93.1, 92.9, 63.8, ...
## 
## Rooted; includes branch lengths.

Now we can see that, in all 1,000 bootstrapped trees, the African and French samples were each in separate clades. Of course, dendrograms are only one type of analysis you can use genetic distances for. Below is a table describing some of the different analyses for which you can utilize genetic distance:

Analysis Function Package Note
Bootstrapped dendrograms aboot poppr
Mantel Test mantel.randtest ade4 To be used with geographic distance matrix
Principle Coordinates Analysis cmdscale stats
DAPC dapc adegenet Convert to matrix with as.matrix
Minimum Spanning Networks poppr.msn poppr requires a distance matrix; cannot handle genpop

K-means hierarchical clustering

A recent study reported that the origin of the potato late blight pathogen Phytophthora infestans lies in Mexico as opposed to South America (Goss et al., 2014). We saw in the previous chapter that South American populations showed signatures of clonal reproduction while Mexican populations showed no evidence rejecting the null hypothesis of random mating. In this section, we will use K-means clustering in combination with bootstrapped dendrograms to see how well this pattern holds up. Clonal populations should have short terminal branch lengths and should cluster according to those branches. Panmictic populations will show no clear pattern. Let’s look at the data:

library("poppr")
data("Pinf")
Pinf
## 
## This is a genclone object
## -------------------------
## Genotype information:
## 
##    72 multilocus genotypes 
##    86 tetraploid individuals
##    11 codominant loci
## 
## Population information:
## 
##     2 strata - Continent, Country
##     2 populations defined - South America, North America

First, we will perform a cluster analysis:

MX <- popsub(Pinf, "North America")
MXclust <- find.clusters(MX)
MX_PCA

MX_PCA

## Choose the number PCs to retain (>=1):
> 50

PC stands for principal components, which are unit-less transformations of your data that explain the variance observed. For the purposes of find.clusters, we can keep as many as we want.

MX_CLUSTER

MX_CLUSTER

## Choose the number PCs to retain (>=2:
> 3

BIC stands for “Bayesian Information Criterion”. The lower the BIC value, the better. On the x axis are the number of clusters. We see that there is a bend at 3 clusters, indicating that the data clusters optimally into three groups.

And now we can see the cluster assignments:

MXclust
## $Kstat
## NULL
## 
## $stat
## NULL
## 
## $grp
##  PiMX01  PiMX02  PiMX03  PiMX04  PiMX05  PiMX06  PiMX07  PiMX10  PiMX11 
##       1       1       3       2       2       2       2       3       3 
##  PiMX12  PiMX13  PiMX14  PiMX15  PiMX16  PiMX17  PiMX18  PiMX19  PiMX20 
##       2       3       3       1       3       1       2       2       2 
##  PiMX21  PiMX22  PiMX23  PiMX24  PiMX25  PiMX26  PiMX27  PiMX28  PiMX29 
##       1       3       1       3       2       1       3       3       2 
##  PiMX30  PiMX40  PiMX41  PiMX42  PiMX43  PiMX44  PiMX45  PiMX46  PiMX47 
##       3       3       1       1       1       1       1       1       1 
##  PiMX48  PiMX49  PiMX50 PiMXT01 PiMXT02 PiMXT03 PiMXT04 PiMXT05 PiMXT06 
##       3       3       3       1       1       2       2       1       2 
## PiMXT07 PiMXt48 PiMXt68 
##       2       2       2 
## Levels: 1 2 3
## 
## $size
## [1] 17 16 15

We will go through the same procedure for the South American population.

SA <- popsub(Pinf, "South America")
SAclust <- find.clusters(SA)
SA_PCA

SA_PCA

## Choose the number PCs to retain (>=1):
> 30
SA_CLUSTER

SA_CLUSTER

## Choose the number PCs to retain (>=2):
> 4

Notice here that there is no local minimum in the curve. This indicates that there might not be enough information in the data set to properly cluster. We will go ahead by choosing the highest number of clusters:

Trees

Now we will build trees. We are using Bruvo’s distance since polyploids bias calculation of other distances:

pinfreps <- c(2, 2, 6, 2, 2, 2, 2, 2, 3, 3, 2)
MXtree <- bruvo.boot(MX, replen = pinfreps, cutoff = 50, quiet = TRUE)

SAtree <- bruvo.boot(SA, replen = pinfreps, cutoff = 50, quiet = TRUE)

We see very long terminal branches in the MX tree. Let’s see how the groups we found with the clustering algorithm match up:

library("ape")
cols <- rainbow(4)
plot.phylo(MXtree, cex = 0.8, font = 2, adj = 0, tip.color = cols[MXclust$grp],
           label.offset = 0.0125)
nodelabels(MXtree$node.label, adj = c(1.3, -0.5), frame = "n", cex = 0.8,
           font = 3, xpd = TRUE)
axisPhylo(3)

You can see that the assigned clusters don’t necessarily group with the dendrogram clusters. Let’s see what happens when we view this with the South American population:

plot.phylo(SAtree, cex = 0.8, font = 2, adj = 0, tip.color = cols[SAclust$grp],
           label.offset = 0.0125)
nodelabels(SAtree$node.label, adj = c(1.3, -0.5), frame = "n", cex = 0.8,
           font = 3, xpd = TRUE)
axisPhylo(3)

Everything clusters together nicely, further supporting a non-panmictic population.

References

Goss EM., Tabima JF., Cooke DEL., Restrepo S., Fry WE., Forbes GA., Fieland VJ., Cardenas M., Grünwald NJ. 2014. The Irish potato famine pathogen Phytophthora infestans originated in central mexico rather than the andes. Proceedings of the National Academy of Sciences 111:8791–8796. Available at: http://www.pnas.org/content/early/2014/05/29/1401884111.abstract

Hedrick PW. 2005. A standardized genetic differentiation measure. Evolution 59:1633–1638. Available at: http://dx.doi.org/10.1111/j.0014-3820.2005.tb01814.x

Laloe D., Jombart T., Dufour A-B., Moazami-Goudarzi K. 2007. Consensus genetic structuring and typological value of markers using multiple co-inertia analysis. Genetics Selection Evolution 39:545–567. Available at: http://dx.doi.org/10.1051/gse:2007021

Meirmans PG., Hedrick PW. 2011. Assessing population structure: \(F_{ST}\) and related measures. Molecular Ecology Resources 11:5–18. Available at: http://onlinelibrary.wiley.com/doi/10.1111/j.1755-0998.2010.02927.x/full