Implementation of the Natural and Orthogonal InterAction (NOIA) model

Description

The NOIA model, as described extensively in Alvarez-Castro & Carlborg (2007), is a framework facilitating the estimation of geneticEffects and genotype-to-phenotype maps. This package provides the basic tools to perform linear and multilinear regressions from real populations, analyse pure genotype-to-phenotype (GP) maps in ideal populations, estimating the genetic effects from different reference points, the genotypic values, and the decomposition of genetic variances in a multi-locus, 2 alleles system. This package is extensively described in Le Rouzic & Alvarez-Castro (2008).

Details

Regression data set: The user must provide (i) The vector of phenotypes of all individuals measured in the population, and (ii) The matrix of the genotypes. There are two input formats for the genotype, see linearRegression.

Regression functions: linearRegression and multilinearRegression.

GP map data set: The user must provide (i) The 3^L (where L is the number of loci) vector of genotypic values (G in Alvarez-Castro & Carlborg (2007)) (ii) Allele or genotype frequencies in the reference population.

GP map analysis function: linearGPmapanalysis.

Change of reference: geneticEffects.

Genotype-to-phenotype map: GPmap.

Decomposition of genetic variance: varianceDecomposition.

Author(s)

Arnaud Le Rouzic, Arne B. Gjuvsland

Maintainer: Arnaud Le Rouzic

References

Alvarez-Castro JM, Carlborg O. (2007). A unified model for functional and statistical epistasis and its application in quantitative trait loci analysis. Genetics 176(2):1151-1167.

Alvarez-Castro JM, Le Rouzic A, Carlborg O. (2008). How to perform meaningful estimates of genetic effects. PLoS Genetics 4(5):e1000062.

Le Rouzic A, Alvarez-Castro JM. (2008). Estimation of genetic effects and genotype-phenotype maps. Evolutionary Bioinformatics 4.

Examples

set.seed(123456789)

map <- c(0.25, -0.75, -0.75, -0.75, 2.25, 2.25, -0.75, 2.25, 2.25)
names(map) <- genNames(2)
pop <- simulatePop(map, N=500, sigmaE=0.2, type="F2")

# Regressions

linear <- linearRegression(phen=pop$phen, gen=pop[2:3])

multilinear <- multilinearRegression(phen=pop$phen, gen=cbind(pop$Loc1, 
	pop$Loc2))

# Linear effects, associated variances and stderr
linear

# Multilinear effects
multilinear

# Genotype-to-phenotype map analysis
linearGP <- linearGPmapanalysis(map, reference="F2")

# Linear effects in ideal F2 population
linearGP

# Change of reference: geneticEffects in the "11" genotype (parental 1)
geneticEffects(linear, ref.genotype="P1")

# Variance decomposition
varianceDecomposition(linear)
varianceDecomposition(linearGP)

# GP maps
maps <- cbind(map, GPmap(linear)[,1], GPmap(multilinear)[,1])
colnames(maps) <- c("Actual", "Linear", "Multilinear")
maps

Names of Genetic Effects

Description

Provides and manipulates labels of genetic effects.

Usage

effectsNamesGeneral(nloc = 2, max.level=NULL, max.dom=NULL) 
effectsNamesMultilinear(nloc=2, max.level=2, max.dom=2)

Arguments

Details

The codes for genetic effects are stored into a vector of length 4, effectsNames. The first element of the vector is the code for the absence of effect (default: "."). The three other elements are respectively additive effects (default: "a") dominance effects (default: "d"), and multilinear epistatic effects (default: "e").

The names of genetic effects contains as many characters as the number of loci in the system. The additive effect of the first locus in a 3-locus system will be "a..", and the "Dominance by Dominance" between loci 2 and 4 in a 5-locus system will be ".d.d.". Directionality of epistasis between two (or more) loci is indicated by as many "e" as necessary (e.g. ".ee." for the interaction between loci 2 and 3 in a 4-locus case).

effectsNamesGeneral and effectsNamesMultilinear provide a list of the names of the genetic effects, in the correct order to be processed in the NOIA framework (Alvarez-Castro and Carlborg 2007).

References

Alvarez-Castro JM, Carlborg O. (2007). A unified model for functional and statistical epistasis and its application in quantitative trait loci analysis. Genetics 176(2):1151-1167.

Le Rouzic A, Alvarez-Castro JM. (2008). Estimation of genetic effects and genotype-phenotype maps. Evolutionary Bioinformatics, 4.

See Also

geneticEffects, genNames, linearRegression, multilinearRegression.

Examples

effectsNamesGeneral(3)
  

Noia analysis of genotype-to-phenotype (GP) maps in ideal populations

Description

Functions for doing a NOIA analysis of a GP map for L loci in a population where the loci are in complete linkage equilibrium.

Usage

linearGPmapanalysis(gmap, reference="F2", freqmat=NULL, 
                    max.level=NULL , S_full=NULL)

Arguments

Details

The algebraic framework is described extensively in Alvarez-Castro & Carlborg 2007. When analysing GP maps in ideal populations we can work directly with the S matrix and do not have to consider the X and Z matrices used in linearRegression. When it comes to the S_full argument keeping the multilocus S matrix in memory is generally fastest for computing all 3^L genetic effects. However it does not allow for computing only a subset of the effects and also runs out of memory for L>8 on a typical desktop machine. For S_full=NULL in linearGPmapanalysis a full S matrix is used if L<=8 and max.level=NULL, while L single locus S matrices are used otherwise.

Value

linearGPmapanalysis returns an object of class "noia.linear.gpmap" , with its own print method: print.noia.linear.gpmap.

Author(s)

Arne B. Gjuvsland

References

Alvarez-Castro JM, Carlborg O. (2007). A unified model for functional and statistical epistasis and its application in quantitative trait loci analysis. Genetics 176(2):1151-1167.

Cheverud JM, Routman, EJ. (1995). Epistasis and its contribution to genetic variance components. Genetics 139:1455-1461.

Le Rouzic A, Alvarez-Castro JM. (2008). Estimation of genetic effects and genotype-phenotype maps. Evolutionary Bioinformatics 4.

Zeng ZB, Wang T, Zou W. (2005). Modelling quantitative trait loci and interpretation of models. Genetics 169: 1711-1725.

See Also

varianceDecomposition

Examples

map <- c(0.25, -0.75, -0.75, -0.75, 2.25, 2.25, -0.75, 2.25, 2.25)

# Genotype-to-phenotype map analysis
linearGP <- linearGPmapanalysis(map, reference="F2")

# Linear effects in ideal F2 population
linearGP

Genetic Effects

Description

geneticEffects displays the genetic effects (and their standard errors) from the result of linearRegression. If a new reference point is provided, a "change of reference" operation is performed (Alvarez-Castro and Carlborg 2007).

Usage

geneticEffects(obj, reference="P1", ref.genotype = NULL)

Arguments

Details

Variance decomposition and change of reference operation are not possible from the result of a multilinear regression.

Author(s)

Arnaud Le Rouzic

References

Alvarez-Castro JM, Carlborg O. (2007). A unified model for functional and statistical epistasis and its application in quantitative trait loci analysis. Genetics 176(2):1151-1167.

Le Rouzic A, Alvarez-Castro JM. (2008). Estimation of genetic effects and genotype-phenotype maps. Evolutionary Bioinformatics, 4.

See Also

linearRegression, multilinearRegression.

Examples

map <- c(0.25, -0.75, -0.75, -0.75, 2.25, 2.25, -0.75, 2.25, 2.25)
pop <- simulatePop(map, N=500, sigmaE=0.2, type="F2")

# Regressions

linear <- linearRegression(phen=pop$phen, gen=cbind(pop$Loc1, pop$Loc2))

geneticEffects(linear, "P1")

Linear and Multilinear Genetic Regressions

Description

The regression aims at estimating genetic effects from a population in which the genotypes and phenotypes are known.

Usage

linearRegression(phen, gen=NULL, genZ=NULL, 
    reference="noia", max.level=NULL, max.dom=NULL, fast=FALSE)
multilinearRegression(phen, gen=NULL, genZ=NULL, 
    reference="noia", max.level=NULL, max.dom=NULL, fast=FALSE, 
    e.unique=FALSE, start.algo = "linear", start.values=NULL, 
    robust=FALSE, bilinear.steps=1, ...)

Arguments

Details

If a gen data set is provided, it will be turned into a genZ. Missing data (unknown genotypes) are considered as loci for which genotypic probabilities are identical to the genotypic frequencies in the population.

The algebraic framework is described extensively in Alvarez-Castro & Carlborg 2007. The default reference point ("noia") provides an orthogonal decomposition of genetic effects in the 1-locus case, whatever the genotypic frequencies. It remains a good approximation of orthogonality in the multi-locus case if linkage disequilibrium is small. Other optional reference points are those of the "G2A" model (Zeng et al. 2005), and the unweighted regression model "UWR" (Cheverud & Routman, 1995). Several key populations can be taken as reference as well: "F2", "F1", "Finf" (F infinity), and the two "parental" homozygous populations "P1" and "P2".

The multilinear model for genetic interactions is an alternative way to model epistatic interactions between at least two loci (see Hansen & Wagner 2001). The computation of multilinear estimates requires a non-linear regression step that relies on the nls function. Providing good starting values for the non-linear regression is a key to ensure convergence, and different algorithms are provided, that can be specified by the "start.algo" option. "linear" performs a linear regression and approximates the genetic effects from it, while "multilinear" performs a simpler multilinear regression (without dominance) to initialize the genetic effects. "subset" estimate all genetic effects from a random subset (50%) of the population, and "bilinear" estimate alternatively marginal and epistatic effects.

Value

linearRegression and multilinearRegression return an object of class "noia.linear" or "noia.multilinear", both having their own print methods: print.noia.linear and print.noia.multilinear.

Author(s)

Arnaud Le Rouzic

References

Alvarez-Castro JM, Carlborg O. (2007). A unified model for functional and statistical epistasis and its application in quantitative trait loci analysis. Genetics 176(2):1151-1167.

Alvarez-Castro JM, Le Rouzic A, Carlborg O. (2008). How to perform meaningful estimates of genetic effects. PLoS Genetics 4(5):e1000062.

Cheverud JM, Routman, EJ. (1995). Epistasis and its contribution to genetic variance components. Genetics 139:1455-1461.

Hansen TF, Wagner G. (2001) Modeling genetic architecture: A multilinear theory of gene interactions. Theoretical Population Biology 59:61-86.

Le Rouzic A, Alvarez-Castro JM. (2008). Estimation of genetic effects and genotype-phenotype maps. Evolutionary Bioinformatics 4.

Zeng ZB, Wang T, Zou W. (2005). Modelling quantitative trait loci and interpretation of models. Genetics 169: 1711-1725.

See Also

geneticEffects, GPmap, varianceDecomposition.

Examples

set.seed(123456789)

map <- c(0.25, -0.75, -0.75, -0.75, 2.25, 2.25, -0.75, 2.25, 2.25)
pop <- simulatePop(map, N=500, sigmaE=0.2, type="F2")

# Regressions

linear <- linearRegression(phen=pop$phen, gen=cbind(pop$Loc1, pop$Loc2))

multilinear <- multilinearRegression(phen=pop$phen, 
    gen=cbind(pop$Loc1, pop$Loc2))

# Linear effects, associated variances and stderr
linear

# Multilinear effects
multilinear

Names of Genotypes

Description

genNames provides the names of all possible genotypes in the order required by the NOIA model (Alvarez-Castro and Carlborg 2007). The codes for the genotypes are stored in the vector genotypesNames.

Usage

genNames(nloc = 2)

Arguments

Details

The names of the genotypes are stored in the vector genotypesNames. By default, they are "1", "2", and "3", the heterozygotes being "2". The genotypes at different loci are then put together, such as "123" for 3 loci.

Author(s)

Arnaud Le Rouzic

References

Alvarez-Castro JM, Carlborg O. (2007). A unified model for functional and statistical epistasis and its application in quantitative trait loci analysis. Genetics 176(2):1151-1167.

Le Rouzic A, Alvarez-Castro JM. (2008). Estimation of genetic effects and genotype-phenotype maps. Evolutionary Bioinformatics, 4.

Examples

    genNames(3)

Genotype-to-Phenotype Mapping

Description

The Genotype-to-Phenotype map is a vector providing the estimate of the genotypic value for any multi-locus genotype. The estimates may be computed from linearRegression or multilinearRegression.

Usage

GPmap(obj)

Arguments

Value

Returns a matrix with two columns: the first one is the estimate of genotypic effects, the second one the standard error of this estimate.

Author(s)

Arnaud Le Rouzic

References

Le Rouzic A, Alvarez-Castro JM. (2008). Estimation of genetic effects and genotype-phenotype maps. Evolutionary Bioinformatics, 4.

See Also

linearRegression, multilinearRegression, genNames.

Examples

set.seed(123456789)

map <- c(0.25, -0.75, -0.75, -0.75, 2.25, 2.25, -0.75, 2.25, 2.25)
pop <- simulatePop(map, N=500, sigmaE=0.2, type="F2")

# Regression
linear <- linearRegression(phen=pop$phen, gen=cbind(pop$Loc1, pop$Loc2))

# GP map
GPmap(linear)

Estimation of parameters for specific allele frequencies

Description

This function computes some parameters of interest (mean phenotype, genetic variance, additive variance, and evolutionary change in additive variance) for a combination of allele frequencies, based on a genotype-phenotype map.

Usage

marginallocus(gmap, freq=NULL, what="mean", definition=11, mc.cores=1, ...)
## S3 method for class 'noia.marloc'
plot(x, xlab=NULL, ylim=NULL, ylab=attr(x, "what"), ...)
## S3 method for class 'noia.marloc'
image(x, xlab=NULL, ylab=NULL, zlim=NULL, 
      main=attr(x, "what"), col.max="red", col.min="blue", col.zero="white", 
      n.cols=1000, zeropart=0.01, contour.levels=10, contour.options=list(), ...) 

Arguments

.

Details

marginallocus computes a population parameter for a series of allele frequencies. The loci under investigation are provided through the freq vector, which need to have as many elements as loci in the system. Values of the freq vector indicate fixed allele frequencies, while NA indicate loci under investigation. For instance, freq=c(NA, 1, NA, 0.5), will investigate the effect of varying loci 1 and 3, while keeping loci 2 and 4 at constant allele frequencies. The population is assumed to be at Hardy-Weinberg frequencies. If freq is not provided, all loci will be investigated.

Value

marginallocus returns an array with as many dimensions as loci under investigation. This array is an object of class "noia.marloc" which can be graphically illustrated through the provided plot (for 1-dimensional data) and image (for 2-dimensional data). Arrays of higher dimensionality cannot be represented graphically.

Author(s)

Arnaud Le Rouzic

See Also

linearGPmapanalysis

Examples

map <- c(0.25, -0.75, -0.75, -0.75, 2.25, 2.25, -0.75, 2.25, 2.25)

mrg2D <- marginallocus(map)
mrg1D <- marginallocus(map, freq=c(NA, 0)) # the second locus is fixed for allele 1

image(mrg2D)
plot(mrg1D)

Simulates a Population from a Genotype-Phenotype Map

Description

The simulatePop function takes a Genotype-to-Phenotype map (i.e. a vector defining the genotypic value of all possible genotypes) and returns a data frame containing the simulated population.

Usage

simulatePop(gmap, N = 100, sigmaE = 1, type = "F2", freqmat=NULL)

Arguments

Details

The type of population refers to the expected allelic and genotypic frequences:

• "F1"First generation of an intercross between two parental populations fixed for alleles A and B respectively; expected genotypic frequencies are: AA: 0, AB: 1, BB: 0.

• "F2"Second generation of an intercross between two parental populations fixed for alleles A and B respectively; expected genotypic frequencies are AA: 0.25, AB: 0.5, BB: 0.25.

• "Finf"Theoretical population from an infinite number of generations after an intercross between two parental populations fixed for alleles A and B respectively; expected genotypic frequencies are AA: 0.5, AB: 0, BB: 0.5.

• "UWR"Theoretical population corresponding to ideal (but experimentally unrealistic) equal genotypic frequencies; expected genotypic frequencies are AA: 0.333, AB: 0.333, BB: 0.333. In such a population, the "UnWeighted Regression model" (UWR) by Cheverud and Routman 1995 provides orthogonal estimates.

• "G2A"Population at Hardy-Weinberg frequencies; expected genotypic frequencies are: AA: p*p, AB: 2p(1-p), BB: (1-p)(1-p), the frequency of allele A (p) at locus i being provided by the i-th element of vector freqmat. "G2A" is the name of the statistical model by Zeng et al. (2005) in which genetic effects estimated from such a population are orthogonal.

• "noia"Population in which genotypic frequencies are arbitrary; expected genotypic frequencies are: AA: pAA, AB: pAB, BB: pBB, frequences pAA, pAB, and pBB at locus i being provided by the i-th line of matrix freqmat. "noia" is the name of the statistical model by Alvarez-Castro and Carlborg (2007) in which genetic effects estimated from such a population are orthogonal. In all populations, loci are considered as independent and are at linkage equilibrium.

Value

Returns a data frame, in which the first column ($phen) contains the phenotypes, and the following ones ($Loc1, $loc2, etc) the genotypes of all individuals.

Author(s)

Arnaud Le Rouzic, Arne B. Gjuvsland

References

Alvarez-Castro JM, Carlborg O. (2007). A unified model for functional and statistical epistasis and its application in quantitative trait loci analysis. Genetics 176(2):1151-1167.

Cheverud JM, Routman, EJ. (1995). Epistasis and its contribution to genetic variance components. Genetics 139:1455-1461.

Le Rouzic A, Alvarez-Castro JM. (2008). Estimation of genetic effects and genotype-phenotype maps. Evolutionary Bioinformatics, 4.

Zeng ZB, Wang T, Zou W. (2005). Modelling quantitative trait loci and interpretation of models. Genetics 169: 1711-1725.

See Also

GPmap, genNames

Examples

set.seed(123456789)

map <- c(0.25, -0.75, -0.75, -0.75, 2.25, 2.25, -0.75, 2.25, 2.25)
pop <- simulatePop(map, N=500, sigmaE=0.2, type="F2")
str(pop)

## Create a "noia" population with genotype frequencies 1/3,1/3,1/3 for locus 1 
## and 0.2,0.6,0.2 for locus 2
pop = simulatePop(map, N=1000, sigma=1, type='noia', 
  freqmat=matrix(c(1/3,1/3,1/3,0.2,0.6,0.2),nrow=2, byrow=TRUE))

Decomposition of Genetic Variance

Description

Variance decomposition in a classical operation in quantitative genetics (e.g. Fisher 1918, Lynch and Walsh 1998). The genetic variance, i.e. the part of phenotypic variance that can be identify as due to genetic factors, can be decomposed into several orthogonal components (generally, the part due to additive factors Var(A), to dominance factors Var(D), and to genetic interactions Var(I)).

Usage

varianceDecomposition(obj)
## S3 method for class 'noia.vardec'
print(x, ...)

Arguments

Details

The details of the variance decomposition are provided for all levels of interaction: Var(A) and Var(D) for marginal effects, Var(AA), Var(AD) and Var(DD) for 2nd order interactions, etc.

Value

varianceDecomposition returns a list of vectors. Each element of the list corresponds to an order of interactions, and the vectors detail the variance decomposition within each level. print.noia.vardec prints the previous list in a nice way, and computed the percentage of genetic variance explained by each variance component.

Author(s)

Arnaud Le Rouzic

References

Alvarez-Castro JM, Carlborg O. (2007). A unified model for functional and statistical epistasis and its application in quantitative trait loci analysis. Genetics 176(2):1151-1167.

Fisher RA. (1918). The correlation between relatives on the supposition of Mendelian inheritance. Thans. Roy. Soc. Edinburgh 52:339-433.

Le Rouzic A, Alvarez-Castro JM. (2008). Estimation of genetic effects and genotype-phenotype maps. Evolutionary Bioinformatics, 4.

Lynch M, Walsh B (1998) Genetics and Analysis of Quantitative Traits. Sunderland, MA; Sinauer Associates.

See Also

linearRegression

Examples

map <- c(0.25, -0.75, -0.75, -0.75, 2.25, 2.25, -0.75, 2.25, 2.25)
pop <- simulatePop(map, N=500, sigmaE=0.2, type="F2")

# Regression

linear <- linearRegression(phen=pop$phen, gen=cbind(pop$Loc1, pop$Loc2))

# Variance decomposition
varianceDecomposition(linear)

Graphical display of genetic regressions and genotype-phenotype maps

Description

These functions allow a graphic representation of the result of genetic regressions from linearRegression and GPmap.

Usage

## S3 method for class 'noia.linear'
plot(x, loc = 1:x$nloc, effect=TRUE, epistasis = TRUE, 
ylim=range(GPmap(x)[,1]) + c(-1,1)*max(GPmap(x)[,2]), ...)
## S3 method for class 'noia.gpmap'
barplot(height, GPcol = c("indianred", "palegreen", "royalblue"), 
arrowscol = "purple", stderr = TRUE , main=NA, ylab=NA, ...)

Arguments

Author(s)

Olivier Ariste, Arnaud Le Rouzic

Printing Genetic Regressions and GP map analyses

Description

Display the output of functions linearRegression, multilinearRegression and linearGPmapanalysis

Usage

## S3 method for class 'noia.linear'
print(x, ...)
## S3 method for class 'noia.multilinear'
print(x, ...)
## S3 method for class 'noia.common'
print(x, ...)
## S3 method for class 'noia.linear.gpmap'
print(x, ...)

Arguments

Details

The print method being actually very similar for the linear and multilinear regressions, both call the common method print.noia.common.

Author(s)

Arnaud Le Rouzic, Arne B. Gjuvsland

References

Le Rouzic A, Alvarez-Castro JM. (2008). Estimation of genetic effects and genotype-phenotype maps. Evolutionary Bioinformatics, 4.