# R/simcross User Guide

R/simcross is an R package to simulate genotypes from an experimental cross. The aim is flexibility rather than speed.

Meiosis is simulated following the Stahl model (Copenhaver et al. 2002) with the interference parameter being an integer. In this model, chiasma locations are the superposition of two processes: a proportion p come from a process exhibiting no interference (that is, a Poisson process) and the remainder (proportion 1 – p) come from a process following the chi-square model (Foss et al. 1993; Zhao et al. 1995). Thus, with p=0, the model reduces to the chi-square model. The chi-square model has a single parameter, m, which is a non-negative integer and controls the strength of interference. m=0 corresponds to no interference. Broman et al. (2002) estimated m=10 for the level of interference in the mouse (derived assuming p=0). The chi-square model is a special case of the gamma model (McPeek and Speed 1995) which has a positive parameter ν; the chi-square model corresponds to the case that m = ν – 1 is an integer.

In many organisms, there is always at least one chiasma for each pair of homologous chromosomes at meiosis. Simulations with R/simcross may be performed with this assumption; the same model is used, but with rejection sampling to get results conditional on there being at least one chiasma. Note that this isn’t an assumption of an obligate crossover; if there is exactly one chiasma, then a random meiotic product will have 0 or 1 crossovers, with probability 1/2 each. Also note that, with the obligate chiasma assumption, chromosomes must be greater than 50 cM.

## Basic functions

There are two basic functions for simulating a cross: create_parent and cross.

create_parent is for generating a parent object, either an inbred individual or the F1 offspring of two inbred individuals. It takes two arguments: the length of the chromosome in cM, and the allele or pair of alleles.

Here’s how to generate two inbred individuals, one from each of strain 1 and strain 2, and an F1 individual, with a 100 cM chromosome.

library(simcross)
p1 <- create_parent(L=100, allele=1)
p2 <- create_parent(L=100, allele=2)
f1 <- create_parent(L=100, allele=1:2)

You don’t need to specify the arguments by name (e.g., in the last line above, you could just write create_parent(100, 1:2)); I’m doing so here just to better document the names of the arguments.

The cross function is used to generate one offspring from the cross between two individuals. The input is a pair of individuals (e.g., as produced by create_parent), the interference parameter m (default m=10), the parameter p for the Stahl model (default p=0), xchr to indicate whether the chromosome being simulated is the X chromosome (FALSE, the default, to simulate an autosome), and male to indicate whether the offspring is to be male (which only matters if xchr=TRUE). Further, one may use obligate_chiasma=TRUE (default FALSE) to require at least one chiasma on the 4-strand bundle).

Here’s an example, to generate an F2 individual. (The outer parentheses cause the result to be printed.)

(f2 <- cross(f1, f1))
## $mat ##$mat$alleles ## [1] 2 ## ##$mat$locations ## [1] 100 ## ## ##$pat
## $pat$alleles
## [1] 1 2 1
##
## $pat$locations
## [1]  19.41483  51.33810 100.00000

The output is a list with two components: the maternal and paternal chromosomes. Each chromosome is a list with the allele in a set of intervals, and the locations of the right endpoints of the intervals.

In the example above, the maternal chromosome is a non-recombinant 2 chromosome. The paternal chromosome has two crossovers, with the the 1 allele up to 19.4 cM, the 2 allele in the interval 19.4 – 51.3, and the 1 allele for the remainder of the chromosome.

By default, we use m=10, p=0, and obligate_chiasma=FALSE. The chromosome length is taken from the input parent objects (f1 in this case). If we wanted to do the simulation with no crossover interference, but with an obligate chiasma, we’d use:

f2 <- cross(f1, f1, m=0, obligate_chiasma=TRUE)

Behind the scenes, there are two additional functions, sim_crossovers, for simulating crossover locations on a chromosome, and sim_meiosis, for simulating a meiotic product from an individual, but in general the user need not bother with these. The cross function calls sim_meiosis twice (once for each parent) and then combines the results into a single individual. sim_meiosis calls sim_crossovers to generate the meiotic product.

## Generating pedigrees

While one could simulate any experimental cross from a series of calls to create_parent and cross, it is generally more efficient to first develop a table that describes pedigree for the cross, and then simulate from the pedigree with the function sim_from_pedigree (see the next section).

### Example AIL pedigree matrix

To define a pedigree, we use a numeric matrix (or data frame) with four columns: individual ID, mom, dad, and sex (coded as 0=female, 1=male). The R/simcross package includes a sample pedigree for advanced intercross lines, AILped, taken from the QTLRel package. Here is the top of that dataset:

head(AILped, n=10)
##    id mom dad sex generation
## 1   1   0   0   1          0
## 2   2   0   0   0          0
## 3   3   2   1   1          1
## 4   4   2   1   0          1
## 5  15   4   3   0          2
## 6  16   4   3   0          2
## 7  17   4   3   0          2
## 8  14   4   3   1          2
## 9  11   4   3   1          2
## 10 12   4   3   1          2

### Checking a pedigree matrix

The function check_pedigree can be used to check that a pedigree matrix conforms to R/simcross’s requirements: founders have mom == dad == 0, all other individuals have both parents present in the pedigree, and parents always precede any of their children. The check_pedigree function returns TRUE if the pedigree matrix is okay; otherwise, it throws an error.

check_pedigree(AILped)
## [1] TRUE

R/simcross includes a set of functions for generating pedigree matrices for different cross designs: sim_ril_pedigree, sim_4way_pedigree, sim_ail_pedigree, and sim_do_pedigree.

### RIL pedigree

sim_ril_pedigree generates a pedigree matrix for a single recombinant inbred line derived from 2k founder lines for some k>0. (Examples include the Collaborative Cross (Threadgill and Churchill 2012) and MAGIC lines (Kover et al. 2009).) The arguments are ngen (number of generations of inbreeding), selfing (default FALSE, for sibling mating), parents (a vector of integers for the parents; the length must be a power of 2 (i.e., 2, 4, 8, 16, etc.) and corresponds to the number of founder lines), and firstind, the ID number to attach to the first individual following the parents (so that the pedigree matrices for multiple RIL may be rbind-ed together). Note that there’s no real simulation here; the result is entirely deterministic.)

(ril <- sim_ril_pedigree(ngen=4, parents=1:4))
##    id mom dad sex gen
## 1   1   0   0   0   0
## 2   2   0   0   1   0
## 3   3   0   0   0   0
## 4   4   0   0   1   0
## 5   5   1   2   0   1
## 6   6   3   4   1   1
## 7   7   5   6   0   2
## 8   8   5   6   1   2
## 9   9   7   8   0   3
## 10 10   7   8   1   3
## 11 11   9  10   0   4
## 12 12   9  10   1   4
## 13 13  11  12   0   5
## 14 14  11  12   1   5
## 15 15  13  14   0   6
## 16 16  13  14   1   6

The gen column in the output is the generation number, with 0 corresponding to the founders. The generations are simply sequential and so don’t correspond to the numbering scheme used for the Collaborative Cross (see, for example, Broman 2012).

### 4-way cross pedigree

sim_4way_pedigree generates a pedigree matrix for an intercross among four inbred lines. The arguments are ngen (which must be 1 or 2) and nsibs. We start with four inbred individuals, and cross them in two pairs to generate a pair of heterozygous individuals. If ngen==1, we then just generate a set of sum(nsibs) F1 offspring. If ngen==2, we generate length(nsibs) pairs of F1s and intercross them to generate a set of F2 sibships; in this case, the input vector nsibs determines the sizes of the sibships.

The following generates two F2 sibships with 3 offspring in each.

(fourway <- sim_4way_pedigree(ngen=2, nsibs=c(3, 3)))
##    id mom dad sex gen
## 1   1   0   0   0   0
## 2   2   0   0   1   0
## 3   3   0   0   0   0
## 4   4   0   0   1   0
## 5   5   1   2   0   1
## 6   6   3   4   1   1
## 7   7   5   6   0   2
## 8   8   5   6   1   2
## 9   9   5   6   0   2
## 10 10   5   6   1   2
## 11 11   7   8   0   3
## 12 12   7   8   1   3
## 13 13   7   8   0   3
## 14 14   9  10   1   3
## 15 15   9  10   0   3
## 16 16   9  10   1   3

### AIL pedigree

Advanced intercross lines (AIL) are generated by crossing two inbred lines to form the F1 hybrid, intercrossing to form the F2 generation, and then performing repeated intercrosses with some large set of breeding pairs. At each generation, the mating pairs are chosen at random, often with an effort to avoid matings between sibling pairs. I would prefer the term “advanced intercross populations,” as it’s a set of heterozygous, genetically distinct individuals; they aren’t really lines.

sim_ail_pedigree generates a pedigree matrix for 2-way advanced intercross lines. Unlike sim_ril_pedigree and sim_4way_pedigree, this is actually a simulation, as the mating pairs at each generation are chosen at random. The arguments are ngen (number of generations), npairs (number of mating pairs at each generation), nkids_per (for number of kids per sibship in the last generation), and design ("nosib" to avoid matings between siblings, or "random" to form the matings completely at random). At each generation, each mating pair gives two offspring (one male and one female) for the next generation. At the last generation, there are nkids_per offspring per mating pair, to give a total of npairs*nkids_per at the last generation.

Here’s an example of the use of sim_ail_pedigree:

ailped <- sim_ail_pedigree(ngen=12, npairs=100, nkids_per=5)
nrow(ailped)
## [1] 2504
table(ailped$gen) ## ## 0 1 2 3 4 5 6 7 8 9 10 11 12 ## 2 2 200 200 200 200 200 200 200 200 200 200 500 With 100 breeding pairs and 5 kids per pair in last generation, we have 200 individuals for most generations and 500 at the end. ### Diversity Outbred pedigree The Diversity Outbred population (DO; Svenson et al. 2013) is like AIL, but starting with eight inbred lines rather than two. Actually, the mouse DO started with partially-inbred individuals from Collaborative Cross lines (the so-called preCC; intermediate generations in the development of eight-way RIL). Heterogeneous stock (HS; Mott and Flint 2002) can be viewed as a special case, but starting with the eight founder lines. sim_do_pedigree generates a pedigree matrix for a DO population. The arguments are just like those of sim_ail_pedigree, but with one addition: ccgen, which is a vector with the numbers of generations of inbreeding to form the initial preCC lines that are then used to initiate the DO. (The default for ccgen is taken from Figure 1 of Svenson et al. 2013.) Use ccgen=0 to simulate a pedigree for HS. The length ccgen should be npairs, the number of breeding pairs; we take two individuals (one female and one male) from each preCC line to begin the outcrossing generations. doped <- sim_do_pedigree(ngen=12) nrow(doped) ## [1] 6772 table(do=doped$do, gen=doped$gen) ## gen ## do 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ## FALSE 16 576 288 288 288 288 288 288 246 118 70 50 40 22 12 6 ## TRUE 0 288 288 288 288 288 288 288 288 288 288 288 720 0 0 0 The output contains the usual id, mom, dad, and sex, plus gen (for generation number) and do (1 indicates part of the DO population, 0 indicates part of the earlier generations). As you can see from the table above, the generation (gen) has a different meaning, according to whether do is 0 or 1. When do==0, it is the number of generations following the initial founder lines; when do==1, it is the generation number of the outbreeding DO population. Also note that we start with sixteen lines rather than 8: to properly handle the X chromosome, we need to consider a male and female from each of the eight founder lines. These are numbered 1–8 for the females, and 9‐16 for the males. Also note that the preCC lines are formed from a cross among the eight founders, with the order of the crosses chosen at random (four females and four males, one from each of the eight founder lines). ## Simulating from a pedigree The point of the construction of a pedigree matrix for a cross design, as in the previous section, was in order to simulate genotype data for the cross design. The R/simcross function for this is sim_from_pedigree. Its arguments are pedigree (the pedigree matrix), L (length of chromosome), xchr (TRUE or FALSE, according to whether to simulate the X chromosome or an autosome; default FALSE), and then the parameters governing the crossing over process: m, p, and obligate_chiasma, with defaults m=10, p=0, and obligate_chiasma=FALSE. (L can also be a vector of chromosome lengths, for simulating multiple chromosomes at once. In this case xchr should be either a logical vector, of the same length as L, or a character string with the name of the chromosome in L that correspond to the X chromosome. An example of simulating multiple chromosomes is in the final section in this vignette.) The sim_from_pedigree function calls create_parent for all founding individuals and cross for any offspring; the output is a list with each component corresponding to one individual and having the form output by cross: a list with mat and pat, for the maternal and paternal chromosomes, respectively, each of which has alleles (allele present in each interval) and locations (location of right endpoint of each interval). Here’s an example, for simulating an AIL to generation 8. ailped <- sim_ail_pedigree(ngen=8, npairs=30, nkids_per=5) xodat <- sim_from_pedigree(ailped, L=100) Here’s the result for the last individual: xodat[[length(xodat)]] ##$mat
## $mat$alleles
## [1] 2 1 2
##
## $mat$locations
## [1]  19.20702  43.17502 100.00000
##
##
## $pat ##$pat$alleles ## [1] 1 2 1 2 ## ##$pat$locations ## [1] 17.78088 35.51860 71.15012 100.00000 We can plot the average number of breakpoints across the individuals’ two chromosomes, by generation, as follows. n_breakpoints <- sapply(xodat, function(a) sum(sapply(a, function(b) length(b$alleles)-1)))
ave_breakpoints <- tapply(n_breakpoints, ailped$gen, mean) gen <- as.numeric(names(ave_breakpoints)) plot(gen, ave_breakpoints, xlab="Generation", ylab="Average no. breakpoints", las=1, pch=21, bg="Orchid", main="AIL with 30 breeding pairs") The function where_het will show the regions where an individual is heterozygous. where_het(xodat[[length(xodat)]]) ## left right ## 1 0.00000 17.78088 ## 2 19.20702 35.51860 ## 3 43.17502 71.15012 We can plot the average proportion of the chromosome that is heterozygous, by generation, as follows. prop_het <- sapply(lapply(xodat, where_het), function(a) sum(a[,2]-a[,1])/100) ave_prop_het <- tapply(prop_het, ailped$gen, mean)
plot(gen, ave_prop_het,
xlab="Generation", ylab="Average proportion heterozygous", las=1,
pch=21, bg="Orchid", main="AIL with 30 breeding pairs")
abline(h=0.5, lty=2)

Note: if we’d greatly restricted the number of breeding pairs per generation, we’d see evidence of inbreeding, as a reduced proportion of heterozygosity. There’s considerably more noise, though, since we’ve got just 6 individuals per generation.

ailped2 <- sim_ail_pedigree(ngen=8, npairs=3, nkids_per=50)
xodat2 <- sim_from_pedigree(ailped2, L=100)
prop_het2 <- sapply(lapply(xodat2, where_het), function(a) sum(a[,2]-a[,1])/100)
ave_prop_het2 <- tapply(prop_het2, ailped2\$gen, mean)
gen2 <- as.numeric(names(ave_prop_het2))
plot(gen2, ave_prop_het2,
xlab="Generation", ylab="Average proportion heterozygous", las=1,
pch=21, bg="Orchid", main="AIL with 3 breeding pairs")
abline(h=0.5, lty=2)

## Generating marker data

R/simcross simulates the locations of crossovers as continuous values in the interval (0,L). This is precise and compact, and it allows detailed study of the breakpoint process, but it can be cumbersome to work with and is often not what you want from simulations. In most cases, one is interested in individuals’ genotypes at a set of markers.

There are two functions for getting marker genotypes on the basis of the the detailed crossover location data: get_geno and convert2geno.

The function get_geno will grab the genotype at a specified location on the chromosome, returning a matrix with two columns: the maternal and paternal alleles for each individual. Continuing with the simulation in the previous section (an AIL with 30 breeding pairs), here’s how to grab the genotype at 30 cM:

g30 <- get_geno(xodat, 30)
tail(g30)
##     mat pat
## 509   1   2
## 510   1   1
## 511   1   1
## 512   1   2
## 513   1   2
## 514   1   2

The get_geno function could be useful, for example, for grabbing QTL genotypes for use in constructing a phenotype.

The convert2geno function takes the detailed crossover information plus a vector of marker locations and returns a matrix with marker genotypes.

First, construct a vector with the marker locations.

map <- seq(0, 100, by=10)
names(map) <- paste0("m", map)
map
##   m0  m10  m20  m30  m40  m50  m60  m70  m80  m90 m100
##    0   10   20   30   40   50   60   70   80   90  100

Then, pass the crossover information and map to convert2geno. I print the data for the last five individuals.

geno <- convert2geno(xodat, map)
geno[nrow(geno)-(4:0), ]
##     m0 m10 m20 m30 m40 m50 m60 m70 m80 m90 m100
## 510  1   2   2   1   1   2   2   2   2   2    2
## 511  2   3   2   1   1   2   2   1   2   2    2
## 512  2   1   2   2   1   2   2   3   2   2    2
## 513  2   2   2   2   1   2   2   3   3   3    3
## 514  2   2   2   2   1   2   2   2   3   3    3

Here the genotypes get recoded as 1 / 2 / 3 for 11 / 12 / 22. That is, genotypes 1 and 3 are the homozygotes for the allele from founders 1 and 2, respectively, and genotype 2 is the heterozygote.

For crosses with more than two founders, the output of convert2geno is a three-dimensional array, individuals × markers × alleles (maternal and paternal). For example, here are genotypes for the sixth generations of inbreeding of an eight-way RIL.

rilped <- sim_ril_pedigree(ngen=6, parents=1:8)
dat <- sim_from_pedigree(rilped, L=100)
geno <- convert2geno(dat, map)
geno[nrow(geno)-(1:0),,]
## , , mat
##
##    m0 m10 m20 m30 m40 m50 m60 m70 m80 m90 m100
## 27  7   5   5   5   5   3   3   3   3   3    7
## 28  7   5   5   5   5   3   3   5   5   6    7
##
## , , pat
##
##    m0 m10 m20 m30 m40 m50 m60 m70 m80 m90 m100
## 27  7   5   5   5   5   3   3   3   3   3    7
## 28  7   5   5   5   5   3   3   3   3   3    7

More commonly, one may be interested in individuals’ SNP genotypes. This can also be obtained with convert2geno; one just needs to provide a matrix of SNP alleles for the founder lines, with the argument founder_geno. This should be a matrix of 1s and 2s, of dimension n_founders × n_markers.

Let’s simulate SNP alleles for eight founder lines.

fg <- matrix(sample(1:2, 8*length(map), replace=TRUE), nrow=8)

And then here are the SNP genotypes for the sixth generation of inbreeding an eight-way RIL.

snpgeno <- convert2geno(dat, map, fg)
snpgeno[nrow(snpgeno)-(1:0),]
##    m0 m10 m20 m30 m40 m50 m60 m70 m80 m90 m100
## 27  3   3   1   1   1   3   3   3   3   1    3
## 28  3   3   1   1   1   3   3   2   2   1    3

The genotypes are again coded as 1 / 2 / 3 for 11 / 12 / 22.

## Simulating multiple chromosomes

One can use sim_from_pedigree and convert2geno to simulate multiple chromosomes at once.

To simulate multiple chromosomes with sim_from_pedigree, provide a vector of chromosome lengths. In this case xchr should be either a logical vector, of the same length as L, or a character string with the name of the chromosome in L that correspond to the X chromosome.

ailped3 <- sim_ail_pedigree(ngen=12, npairs=30, nkids_per=3)
xodat3 <- sim_from_pedigree(ailped3, c("1"=100, "2"=75, "X"=100), "X")
xodat3alt <- sim_from_pedigree(ailped3, c("1"=100, "2"=75, "X"=100),
c(FALSE, FALSE, TRUE))

To indicate that all chromosomes are autosomes, you can use xchr="", xchr=FALSE, or xchr=NULL. For example:

xodat3_noX <- sim_from_pedigree(ailped3, c("1"=100, "2"=75, "3"=100), "")

Having simulated multiple chromosomes with sim_from_pedigree, you may wish to use convert2geno to convert those results to marker genotypes. This is done by providing the output of sim_from_pedigree as well as a genetic marker map that is a list of vectors of marker locations.

Let’s first construct the marker map, assuming three chromosomes with lengths 100, 75, and 100.

map <- list("1"=seq(0, 100, by=5),
"2"=seq(0, 75, by=5),
"X"=seq(0, 100, by=5))
for(i in seq(along=map))
names(map[[i]]) <- paste0("m", names(map)[i], "_", map[[i]])

We then use convert2geno, ensuring that the inputs are both lists with the same length.

geno <- convert2geno(xodat3, map)

For crosses like the DO, in which founder genotypes are needed, the input founder_geno must be a list of matrices (one matrix per chromosome).

Here are some simulated founder genotypes:

fg <- vector("list", length(map))
for(i in seq(along=map))
fg[[i]] <- matrix(sample(1:2, 8*length(map[[i]]), replace=TRUE), nrow=8)

And now here is the simulation of a small DO population for multiple chromosomes. When simulating the pedigree, we need to have 16 founders (the 8 female founders and then the 8 male founders). After simulating the genotypes with sim_from_pedigree(), we use collapse_do_alleles() to collapse the alleles 9-16 (for the male founders) into 1-8.

doped <- sim_do_pedigree(ngen=4, nkids_per=1)
xodat <- sim_from_pedigree(doped, L=c("1"=100, "2"=75, "X"=100), "X")
xodat <- collapse_do_alleles(xodat)
geno <- convert2geno(xodat, map, fg)

## References

Broman KW (2012) Genotype probabilities at intermediate generations in the construction of recombinant inbred lines. Genetics 190:403-412 doi: 10.1534/genetics.111.132647

Broman KW, Rowe LB, Churchill GA, Paigen K (2002) Crossover interference in the mouse. Genetics 160:1123-1131

Copenhaver GP, Housworth EA, Stahl FW (2002) Crossover interference in arabidopsis. Genetics 160:1631-1639

Foss E, Lande R, Stahl FW, Steinberg CM (1993) Chiasma interference as a function of genetic distance. Genetics 133:681-691

Kover PX, Valdar W, Trakalo J, Scarcelli N, Ehrenreich IM, Purugganan MD, Durrant C, Mott R (2009) A Multiparent Advanced Generation Inter-Cross to fine-map quantitative traits in Arabidopsis thaliana. PLoS Genetics 5:e1000551

McPeek MS, Speed TP (1995) Modeling interference in genetic recombination. Genetics 139:1031-1044

Mott R, Flint J (2002) Simultaneous detection and fine mapping of quantitative trait loci in mice using heterogeneous stocks. Genetics 160:1609-1618

Svenson KL, Gatti DM, Valdar W, Welsh CE, Cheng R, Chesler EJ, Palmer AA, McMillan L, Churchill GA (2012) High-resolution genetic mapping using the mouse Diversity Outbred population. Genetics 190:437-447

Threadgill DW, Churchill GA (2012) Ten years of the Collaborative Cross. Genetics 190:291-294

Zhao H, Speed TP, McPeek MS (1995) Statistical analysis of crossover interference using the chi-square model. Genetics 139:1045-1056