Order‐preserving principles underlying genotype–phenotype maps ensure high additive proportions of genetic variance

Order-preserving principles underlying genotype–phenotype mapsensure high additive proportions of genetic variance

A. B. GJUVSLAND*, J. O. VIK*, J. A. WOOLLIAMS�� & S. W. OMHOLT�*Department of Mathematical Sciences and Technology, Centre for Integrative Genetics (CIGENE), Norwegian University of Life Sciences, As, Norway

�Department of Animal and Aquacultural Sciences, Centre for Integrative Genetics (CIGENE), Norwegian University of Life Sciences, As, Norway

�The Roslin Institute and Royal (Dick) School of Veterinary Studies, University of Edinburgh, Roslin, Midlothian, UK

Introduction

In the ‘Generation of Animals’ (c. 340 BCBC), Aristotle

observed that ‘[offspring] take after their parents more

than after their earlier ancestors, and after their ancestors

more than after any casual person’ (translation by Peck

(1942)). More than 2000 years later, this phenomenon

that like begets like became one of the key premises of

Darwinism, based on the rationale that if offspring on

average did not resemble their parents more than other

couples in the population, there would be no natural

selection and no adaptation.

Charles Darwin considered the resemblance between

parents and offspring to be almost implied by reproduc-

tion (Darwin, 1859: 489–90), an assumption largely

unquestioned to this day. However, this resemblance is

really quite enigmatic from a causal point of view. There

is no a priori reason why an offspring, arising from the

random sorting of chromosome pairs plus genetic recom-

bination and the subsequent billions of highly complex

and nonlinear processes setting up the genotype–pheno-

type (GP) map, should on average resemble its parents

more than a randomly drawn couple from the popula-

tion. It could equally well be that this would give rise to a

quite unpredictable parent–offspring relationship. We

know that it does not. But by taking this evolutionary

phenomenon for granted, we are asserting the existence

of generic features of the GP map that currently have no

scientific explanation in terms of proximate principles

and mechanisms.

The framework of quantitative genetics offers concepts

and models for population-level description of the degree

of resemblance between parents and offspring. The

regression of offspring on single-parent phenotypes gives

an estimate of h2 ⁄ 2, where h2 is the narrow-sense

heritability. The phenotypic variance can be decomposed

using models of quantitative trait loci (QTLs) in terms of

allele frequencies and gene action, assuming Hardy–

Weinberg and linkage equilibria, leading to the equation

h2 = VA ⁄ VP. Fractions of additive-by-additive interactions

also contribute to parent–offspring regression (Jacquard,

1983), but in practice, the effect is small (Falconer &

Mackay, 1996). In quantitative genetic terms, a strong

resemblance between parent and offspring suggests that

the additive genetic variance VA makes up a considerable

Correspondence: Arne B. Gjuvsland, Centre for Integrative Genetics

(CIGENE), Norwegian University of Life Sciences, P.O. Box 5003,

N-1432 As, Norway.

Tel.: +47 64 96 52 92; fax: +47 64 96 54 01;

e-mail: [email protected]

ª 2 0 1 1 T H E A U T H O R S . J . E V O L . B I O L . 2 4 ( 2 0 1 1 ) 2 2 6 9 – 2 2 7 9

J O U R N A L O F E V O L U T I O N A R Y B I O L O G Y ª 2 0 1 1 E U R O P E A N S O C I E T Y F O R E V O L U T I O N A R Y B I O L O G Y 2269

Keywords:

epistasis;

gene regulatory networks;

genetic variance;

genotype–phenotype map;

monotonicity.

Abstract

In quantitative genetics, the degree of resemblance between parents and

offspring is described in terms of the additive variance (VA) relative to genetic

(VG) and phenotypic (VP) variance. For populations with extreme allele

frequencies, high VA ⁄ VG can be explained without considering properties of

the genotype–phenotype (GP) map. We show that randomly generated GP

maps in populations with intermediate allele frequencies generate far lower

VA ⁄ VG values than empirically observed. The main reason is that order-

breaking behaviour is ubiquitous in random GP maps. Rearrangement of

genotypic values to introduce order-preservation for one or more loci causes a

dramatic increase in VA ⁄ VG. This suggests the existence of order-preserving

design principles in the regulatory machinery underlying GP maps. We

illustrate this feature by showing how the ubiquitously observed monotonicity

of dose–response relationships gives much higher VA ⁄ VG values than a

unimodal dose–response relationship in simple gene network models.

doi: 10.1111/j.1420-9101.2011.02358.x

proportion of the phenotypic variance VP and (in the

absence of strong negative genetic-environmental covari-

ance) an even larger proportion of the genetic variance

VG. Hill et al. (2008) recently showed that VA ⁄ VG will be

high when most alleles at underlying QTLs are at

extreme frequencies (close to 0 or 1), and in this case,

nonadditive gene actions like dominance, overdomi-

nance and epistasis contribute little to VG and hence to

the phenotypic variance. A main conclusion drawn from

their explanation is that the specific structure of

the genotype-to-phenotype map has little explanatory

significance in natural populations.

However, the VA ⁄ VG ratios are high also in populations

with intermediate allele frequencies (e.g. F2 crosses and

collections of recombinant inbred lines [RILs]). Hill et al.

(2008) themselves report an average VA ⁄ VG of 0.5 for

maize yield traits. Moreover, the ratio is on average 0.77

for plant architectural traits and fruit yield in a melon

cross (Zalapa et al., 2006), 0.46 for leaf morphological

traits in two line crosses of upland cotton (Hao et al.,

2008) and 0.75 for 22 quantitative traits (including

developmental rates and sizes) in a cross between two

strains of Arabidopsis thaliana (Kearsey et al., 2003). Based

on this, we hypothesized that resemblance between

parents and offspring and the underlying high VA ⁄ VG

ratios cannot be fully accounted for without considering

properties of the mapping of genotypes to phenotypes.

This motivated us to study random GP maps, made by

randomly assigning genotypic values and monitoring in

F2 populations how the ratio between additive variance

and total genotypic variance changes as a function of

the number of loci contributing to a trait. We show

that the VA ⁄ VG ratio drops to unrealistic levels very fast as

the number of underlying loci increases. This implies that

high VA ⁄ VG values are indeed not fully accounted for by

an allele-frequency explanation and that we need to

include constraints on the GP map.

Despite the huge number of nonlinear regulatory

processes underlying a GP map, the parent–offspring

relationship remains predictable. The underlying causal

machinery thus appears to behave linearly to a consid-

erable degree. This suggests that the type of nonlinearity

matters, that some nonlinear relationships preserve the

predictability much better than others and that these

should be over-represented in regulatory systems. We do

indeed find that a distinctive feature of random GP maps

is that they are nonmonotone (or order-breaking) with

respect to the partial genotype order of a given locus. We

then move on to rearrange genotypic values to introduce

monotonicity (or order-preservation) for one or more

loci and see a sharp increase in VA ⁄ VG values. Guided by

this, we focus on gene regulatory networks, asking

whether monotone and nonmonotone dose–response

relationships differ in their capacity to generate additive

variance. By use of mathematical modelling and sub-

sequent quantitative genetic analysis of the simulation

data, we find that monotonic and saturating dose–

response relationships ubiquitously present in gene

regulatory systems as well as in metabolic systems result

in GP maps with more order-preservation and higher

VA ⁄ VG than the more infrequently observed unimodal

dose–response relationships.

Our results point to the need for disclosing how

various observed regulatory designs and combination of

designs influence the parent–offspring relationship and

whether they exist because of systemic necessity or have

been picked from a pool of alternative designs through

natural selection. In this way, they also add a new

dimension to the existing body of research integrating

mathematical models of biological systems with quanti-

tative genetics (Kacser & Burns, 1981; Keightley, 1989;

Omholt et al., 2000; Bagheri & Wagner, 2004; Peccoud

et al., 2004; Welch et al., 2005; Gjuvsland et al., 2007).

Models and methods

Random GP maps

Consider a diploid genetic model with N biallelic loci

underlying a quantitative trait. Using the indexes 1 and 2

for the two alleles, the genotype space for locus i is

Gi = {11, 12, 22} and the multilocus genotype space Gcontains 3N genotypes constructed by concatenating

elements from the single-locus genotype spaces

C ¼YNi¼1

Ci ¼ fg1g2 . . . gN jgi 2 f11; 12; 22gg:

By a GP map, we mean a mapping that maps genotypes

into real-valued genotypic values, and in the matrix

notation used here, these genotypic values are laid out as

a 3N · 1 vector G ¼ G1111��11 G1211��11 � � � G2222��22½ �T .

By a random GP map, we mean that the genotypic values

are sampled independently from some distribution. Here,

to ensure analytic tractability, we use the normal

distribution with unit variance,

G � Nð0; I3N Þ: ð1ÞWhen modelling and analysing the genetic effects and

variance arising from G, we take advantage of the natural

and orthogonal interactions (NOIA) framework pre-

sented in Alvarez-Castro & Carlborg, 2007 and make

use of the same notation [which was adapted from Zeng

et al. (2005)]. We use only a part of the framework, the

orthogonal model, which is a matrix formulation based

on orthogonal scales sensu Cockerham (1954) and where

the matrices for multilocus models are conveniently built

by Kronecker products of single-locus matrices. In the

following, we describe details on the relevant parts of the

NOIA model and its use in this study to partition genetic

variance (Le Rouzic & Alvarez-Castro, 2008).

Based on the assumption that the N loci are in linkage

equilibrium in the study population, NOIA offers an

orthogonal model for the genetic effects underlying the

genotypic values. For a single locus i with genotype

2270 A. B. GJUVSLAND ET AL.


J O U R N A L O F E V O L U T I O N A R Y B I O L O G Y ª 2 0 1 1 E U R O P E A N S O C I E T Y F O R E V O L U T I O N A R Y B I O L O G Y

frequencies p11, p12 and p22, the NOIA statistical model is

G = SiEi where the vector of genotypic values

G ¼G11

G12

G22

24

35;

the one-locus genetic-effect design matrix

Si ¼1 �p12 � 2p22 � 2p12p22

p11þp22�ðp11�p22Þ2

1 1� p12 � 2p224p12p22

p11þp22�ðp11�p22Þ2

1 2� p12 � 2p22 � 2p12p22

p11þp22�ðp11�p22Þ2

2664

3775

and the vector of genetic effects

Ei ¼li

ai

di

24

35:

A model involving all N loci is given by the equation

G ¼ SSES: ð2ÞHere, G ¼ G1111...11 G1211��11 . . . G2222...22½ �T is the

vector of all 3N genotypic values, the 3N · 3N genetic-effect

design matrix SS is given by the Kronecker product of the

single-locus matrices SS = SN � SN ) 1 � . . . � S2 � S1

and ES is a vector containing all genetic effects (average

effects, dominance deviations and up to N-way interac-

tions among these). The genetic-effect design matrix SS

together with the diagonal matrix F = diag (p1111. . .11,

p1211. . .11, . . ., p2222. . .22) of multilocus genotype frequen-

cies (assuming linkage equilibrium) has the property that

STSFSS is a diagonal matrix. Owing to this orthogonality,

the genetic variance in the population can be split into a

sum of contributions from each genetic effect,

VG ¼X3N

i¼2

e2SiFiis

Ti si; ð3Þ

(this equation is equivalent to eqn (6) in Le Rouzic &

Alvarez-Castro (2008)) and the contribution to total

genetic variance from a specific type of genetic effect (e.g.

additive, dominance and two-way interactions) can be

obtained by summing over the relevant indexes only. By

combining eqns (1) and (2), we see that with a random

GP map, the genetic effects will be normally distributed,

ES � Nð0;RÞ; ð4Þwith covariance matrix R ¼ S�1

S ðS�1S Þ

T, and so the genetic

variance (eqn 3) or any component of it will be a linear

combination of chi-square distributions. In cases where

these distributions are independent, we derive analytic

results for VA ⁄ VG; otherwise, we resort to Monte Carlo

studies with numeric NOIA analysis on sampled GP

maps. For the numeric analyses, we extended the RR

package noia (Le Rouzic & Alvarez-Castro, 2008) with

a high-level function linearGPmapanalysis as well as

a number of low-level functions. We submitted the

modifications to the package maintainer, and they are

available as of version 0.94 of the package (http://cran.

r-project.org/web/packages/noia/).

Introduction of order-preservation

To introduce order-preservation for a single locus, we

went through all 3N ) 1 background genotypes in

random order, and for each of them, we checked

whether eqn (14) was fulfilled. If not, we reassigned

the three genotypic values to the three genotypes so as to

fulfil the inequalities in eqn (14). To introduce order-

preservation for M loci, we did the same operation

processing the M · 3N ) 1 versions of eqn (14) in random

order.

Gene regulatory simulations

Parameters were sampled in Python and the differential

equations solved with the SUNDIALS library (https://

computation.llnl.gov/casc/sundials/). To avoid artefacts

arising from the error tolerance in the ODE solver, data

sets were omitted from analysis if all genotypic values

were lower than 10)6 (this happened for at most 6 of

5000 data sets per parameter scenario).

Results

Genetic variance components of random GP maps

Analytical resultsHere, we derive analytical expressions for the distribu-

tions of ratios of genetic variance components for

different populations with intermediate allele frequen-

cies. The results also provide a basis for assessing the

numerical work below.

F2 populations: For a single locus in an ideal F2 population,

the relevant matrices are

ES ¼la

d

2435; SS ¼

1 �1 �0:51 0 0:51 1 �0:5

24

35; F¼

0:25 0 0

0 0:5 0

0 0 0:25

24

35;ð5Þ

and the distribution of genetic effects for the random GP

map (1) is specified by the covariance matrix

R ¼0:375 0 0:25

0 0:5 0

0:25 0 1:5

24

35;

so that a � N(0, 0.52) and d � N(0, 1.5) are indepen-

dent. Inserting this and elements from STSFSS=

diagð1; 0:5; 0:25Þ into eqn (3), the proportion of genetic

variance explained by the additive genetic effect is given

by

VA

VG

¼ 0:5a2

0:5a2 þ 0:25d2¼ 1

1þ 0:5 da

� �2¼ 1

1þ 0:5u2ð6Þ

where u � Cauchyð0;ffiffiffi3pÞ (except for the genetic effects

being random variables, eqn (6) is equivalent to eqn 8.7

Order-preserving genotype–phenotype maps 2271



in Falconer & Mackay (1996)). The expected value of the

proportion is

EVA

VG

� �¼Z1

�1

1

1þ 12

u2f ðuÞdu ¼ 1ffiffi

32

qþ 1� 0:449

As seen from the covariance matrix, the columns of SS

are not orthogonal in the F2 case, and for two or more

loci, this introduces covariance between the genetic

effects, making further analysis a daunting task. We

therefore move on to analyse other cases and continue

the treatment of F2 populations in the Numerical Results

section.

Populations where p11 = p12 = p22 = 1 ⁄ 3: In a hypothetical

population where p11 ¼ p12 ¼ p22 ¼ 1=3 for all loci, the

matrix SS itself becomes orthogonal, and the distribu-

tions of genetic effects under a random GP map are given

by

ei � Nð0;RiiÞ; i ¼ 2; 3; . . . ; 3N : ð7Þ

The matrix of genotype frequencies is F ¼ ð1=3NÞI3N and eqn (3) simplifies to

VG ¼X3N

i¼2

e2SiFiis

Ti si ¼

X3N

i¼2

e2Si

1

3N

1

Rii

¼X3N

i¼2

e2Si;

eSi ¼ eSi

ffiffiffiffiffiffiffiffiffiffiffi1

3NRii

r� N 0;

1

3N

� � ð8Þ

i.e. the scaled genetic effects ei are independent and

identically distributed. The proportion of VG explained by

a single genetic effect ej is given by

Vej

VG

¼ e2j =X3N

i¼2

e2i : ð9Þ

In the case of a subset S of genetic effects, the variance

explained by the nS effects in this subset is

VS

VG

� bnS

2;3N � nS

2

� �: ð10Þ

From the expected values of this distribution, some

interesting observations can be made: (i) The expected

value of the proportion of genetic variance explained by

any single effect is 1 ⁄ (3N ) 1); (ii) The expected

proportion of genetic variance being additive is

N ⁄ (3N ) 1), and this holds also for dominance genetic

variation. For n = 1, this means half the variance is

expected to be additive, but the expectation tends

quickly to zero as n increases (for 2, 3 and 4 loci

E(VA ⁄ VG) is 1=4; 3=26 and 1=20, respectively); and iii.

With more than a few loci in a random GP map,

practically all genetic variance is epistatic.

Comparing results for the single-locus case, we see that

the proportion of additive variance relative to total

genotypic variance is slightly smaller in an F2 population

than the corresponding one half found for

p11 = p12 = p22 = 1 ⁄ 3. This is intuitive because an F2

population has the same allele frequency, but fewer

homozygotes, and thus, additive effects explain less of

the variance.

Collections of RILs: In the case where p11 = p22 and p12 = 0,

which corresponds to the genotype frequencies in a

collection of RILs (e.g. Hrbek et al., 2006; Balasubrama-

nian et al., 2009), we get a similar situation as above. As

can be seen from the one-locus matrix

SS;F1 ¼1 �1 0

1 0 1

1 1 0

24

35; ð11Þ

the SS matrix is not orthogonal. But because there are no

heterozygotes in RIL populations and no dominance

effects, we can remove the second row and the last

column and obtain a reduced 2 · 2 matrix

SS;F1 ¼1 �1

1 1

� �; ð12Þ

which is orthogonal. The same argument as above can be

used to show that eqns (7)–(10) with 3N replaced by 2N

(to account for the removal of all dominance-related

effects) hold also for this population. Increasing N, the

expected proportion of variance being additive tends to

approach zero more slowly than the above case (for 1, 2,

3 and 4 loci, E(VA ⁄ VG) = N ⁄ (2N ) 1) is 1; 1=2; 3=7and 4=15, respectively).

Numerical resultsSimulations of 10 000 random GP maps with 1–13 loci

were performed for F2 populations as well as for

populations with allele frequencies sampled from the

uniform distribution and the U-shaped distribution pro-

posed by Hill et al. (2008). For all populations, the mean

value of VA ⁄ VG across simulations decreases considerably

as the number of loci increases (Fig. 1). For F2 popula-

tions, the drop is very fast, and the mean values are close

to the expected values for the case p11 = p12 = p22. With a

U-shaped distribution, the overall level of VA ⁄ VG is higher

and it decreases slower as the number of loci increases.

Order-breaking: a distinctive feature of randomGP maps

In the one-locus case, it is straightforward to understand

how the proportion of additive genetic variance is

determined by the properties of the GP map in terms of

additive gene action, dominance and overdominance. In

particular, the majority of the genetic variance will be

nonadditive when the locus shows over- or underdom-

inance characterized by |G11| < |G12| > |G22|, i.e. the

genotypic values are not ordered according to the allele

content. In the following, we generalize this broken

ordering, building on order theory and the definitions of

genotype spaces and GP maps stated at the beginning of

the Results section.




For a particular locus k, we order the genotypes in

Gk by 11 < 12 < 22, and for each genetic background

(g1g2 . . . gk ) 1 gk + 1 . . . gN) for locus k, this gives an

ordering of three elements in G,

g1 . . . gk�111gkþ1 . . . gN < g1 . . . gk�112gkþ1 . . . gN

< g1 . . . gk�122gkþ1 . . . gN

ð13Þ

This defines a strict partial order on G with 3N

inequalities ordering pairs of genotypes (for N = 1, the

three pairs are 11 < 12, 11 < 22 and 12 < 22), and we

call it the partial genotype order relative to locus k.

Without loss of generality, we assume that the allele

indexes at each locus have been chosen such that

G1111. . .11 £ Gg1g2. . .gNfor all homozygote genotypes. We

call a GP map monotone or order-preserving with respect to

locus k if it preserves the partial genotype order relative

to locus k, i.e. if

Gg1 ...gk�111gkþ1 ...gN�Gg1...gk�112gkþ1...gN

�Gg1...gk�122gkþ1 ...gN

ð14Þ

for all genetic backgrounds for locus k. By allowing

nonstrict inequalities, we include GP maps showing

complete dominance and complete magnitude epistasis

in the class of order-preserving GP maps. Conversely, we

call a GP map nonmonotone or order-breaking with respect

to locus k if it does not preserve the partial genotype

order relative to locus k. These definitions are easily

applied to the classical one- and two-locus GP maps. In

the one-locus case, a nonmonotone GP map is equivalent

to overdominance, whereas for two or more loci, this

property of the GP map arises from conditional (on one

or more background genotypes) overdominance as well

as from sign epistasis (Weinreich et al., 2005).

Order-breaking is a characteristic of random GP maps

with multiple loci. Only 4 of the 10 000 Monte Carlo

simulations for two loci resulted in GP maps that were

order-preserving with respect to both loci, whereas 181

GP maps were order-preserving with respect to one locus.

For three or more loci, every random GP map was order-

breaking with respect to all loci.

Starting out with the sampled random GP maps, we

constructed order-preservation for any number of loci by

rearranging genotypic values (see Models and Methods).

The effect on VA ⁄ VG of this manipulation is dramatic

(Fig. 2). Introduction of order-preservation with respect

to one locus results in an increase in the mean value of

VA ⁄ VG of around 0.4 across the range of loci in the

original random GP map. Also, the variation in VA ⁄ VG

between the sampled GP maps becomes smaller and

smaller as the number of loci increases and the trend is

that VA ⁄ VG converges to just above 0.4 (see Appendix for

an analytic treatment of this in a p11 = p12 = p22 popu-

lation). Introducing order-preservation for more loci

gives further increase in the proportion of genetic

variance being additive, but with diminishing gains (up

to a mean of 0.89 for a GP map that is order-preserving

for all of its eight loci). The transition from random to

fully order-preserved GP maps involves several sorting

operations. In the resulting series of partially order-

preserving GP maps, we observe that after the first few

sorting operations, the VA ⁄ VG ratio increases steadily as a

function of the number of sortings (see Fig. S1).

Introduction of order-preservation with respect to

even a single gene imposes a strong constraint on the

GP map; considering random GP maps with three or

more loci, none of the 10 000 sampled GP maps fulfilled

the constraint without any reordering of genotypic

Number of loci in random GP map

Mea

n(V

A/V

G)

0.0

0.2

0.4

0.6

0.8

1.0

2 4 6 8 10 12

Frequency distribution

F2

Uniform

U N = 100

U N = 1000

U N = 10 000

Fig. 1 The effect of number of loci and allele frequencies on VA ⁄ VG. Mean values of VA ⁄ VG in random GP maps with 1–13 loci in F2 populations

and populations with allele frequencies sampled from the uniform distribution and the U-shaped distribution proposed by Hill et al. (2008);

10 000 random GP maps were sampled for each case.




values. However, as random GP maps are incapable of

producing the amounts of additive variance observed in

populations with intermediate allele frequencies, we

argue that such constraints do indeed exist through the

design of proximate mechanisms or regulatory principles

of biological systems. In the next section, we illustrate

this point by showing how the characteristic monotonic

dose–response relationships in regulatory networks result

in more order-preserving GP maps than do the less

frequently observed nonmonotone dose–response rela-

tionships.

Monotonicity of the gene regulation functionconstrains the genotype to expression phenotypemap

Motivated by the aforementioned results, we investi-

gated the effect of varying the shape of the gene

regulation function in a diploid gene regulatory model

on order-preservation and additive genetic variance. To

this end, we studied a very simple two-gene regulatory

system where gene 1 is constitutively expressed and is

regulating the expression of gene 2. Following the

sigmoid modelling formalism (Mestl et al., 1995; Plahte

et al., 1998) for diploid organisms (Omholt et al., 2000),

we set up a system of ordinary differential equations

where the state variable xij describes the expression level

of the j-th allele of gene i. We let aij be the maximal

production rate of the allele and cij the relative decay rate

of the expression product. We then compared two

different gene regulation functions (GRFs) (Rosenfeld

et al., 2005) or cis-regulatory input functions (Setty et al.,

2003) describing the relative production rate of gene 2 as

a function of the expression level of gene l:

1. The monotone Hill function H(y, h, p) = yp ⁄ (hp + yp),

where parameter h gives the amount of regulator y

needed to obtain 50% of maximal production rate and p

determines the steepness of the response. For simplicity,

we assumed that the allele products of gene 1 were

equally efficient as regulators and use just their sum

(y1 = x11 + x12) in the regulatory function. Then, the

two-gene system is described by four ordinary differential

equations:

_x1j ¼ a1aj� c1aj

x1j;

_x2j ¼ a2bjHðy1; h2bj

; p2bjÞ � c2bj

x2j; j ¼ 1; 2:ð15Þ

The variables aj and bj are used to code genotype at genes

1 and 2, respectively; for genotype klmn, we set a1 = k,

a2 = l, b1 = m and b2 = n.

2. The nonmonotone unimodal function Rðy; l; rÞ ¼e�ððy�uÞ2=2r2Þ, which is the probability density function of

the normal distribution scaled such that the maximum

function value is 1. The equations for this system are

given by:

_x1j ¼ a1aj� c1aj

x1j;

_x2j ¼ a2bjRðy1; l2bj

; r2bjÞ � c2bj

x2j:ð16Þ

We did a series of simulations for both systems

studying the amount of order-breaking and additive

genetic variance arising from polymorphisms affecting

one or more parameters. For both alleles of both genes in

both systems, cij was kept nonpolymorphic at 10,

whereas allelic values of the other parameters were

either sampled uniformly in the intervals aij:(100, 200),

h2j:(20, 40), p2j:(1, 9), l2j:(25, 35), r2j:(2, 4), or the

mid-value of the respective intervals were used as

Fig. 2 The effect of introducing order-preservation on VA ⁄ VG. The effect of introducing order-preservation in random GP maps on the

proportion of genetic variance explained by additive effects in F2 populations (y-axis). The x-axis shows the total number of loci per GP map,

whereas the colours indicate the number of loci (ranging from 0 to the total number of loci in the GP map) for which order-preservation has

been introduced. Summary statistics for VA ⁄ VG in ideal F2 populations for 10 000 randomly sampled GP maps are shown as follows: Boxplots

display the median and the first and third quartile, whereas the lines show the mean.




nonpolymorphic values. The steady-state expression

level of gene 2 (y2 = x21 + x22) was used as the pheno-

type. We did 5000 Monte Carlo simulations for seven

different parameter scenarios (cf. rows in Table 1) of

polymorphic and nonpolymorphic parameters.

For system (1), the genetic variance is highly additive

across all simulations (Fig. 3 left panel), the smallest

observed VA ⁄ VG fraction ranges from 0.722 to 0.956 and

the mean fractions range from 0.966 to 0.992. The

resulting GP maps are always order-preserving with

respect to gene 1 (Table 1), and order-breaking with

respect to gene 2 is only seen when either the steepness

parameter p2j is polymorphic or both the maximal

production rate a2j and the threshold h2j are polymor-

phic. These results can be understood by examining the

nullclines of eqn (15) (the curves obtained by setting the

equations equal to zero one at a time); the intersection of

all the nullclines gives the steady state of the system.

Figure 4a,b shows genotypic values and the homozygote

nullclines (the sum of the two identical allelic nullclines)

of eqn (15) for one of the statistically most epistatic data

sets. The crossing of the two monotonically increasing

nullclines for gene 2 opens for order-breaking with

respect to gene 2 (and such crossing of two Hill functions

can only occur for the types of polymorphisms seen to

give order-breaking in Table 1), whereas the monoto-

nicity of the same nullclines makes order-breaking with

respect to gene 1 impossible.

For system (2), order-breaking with respect to gene 1 is

observed for all combinations of polymorphic parameters

(Table 1). The same is true for gene 2 when the peak

parameter l2j or both the maximal production rate a2j and

the width parameter r2j are polymorphic. This results in a

much broader spectrum of genetic variance components

(Fig. 3, right panel). For all combinations of polymorphic

parameters, data sets with essentially no additive variance

are observed, as well as data sets with only additive

variance, and the mean value of VA ⁄ VG is between 0.646

and 0.689. Major features of one of the most epistatic data

sets are shown in Fig. 4c,d. For the parameters of this

system, the crossing of the two nullclines for gene 2

enables order-breaking with respect to gene 2, whereas

the nonmonotonicity of the same nullclines makes order-

breaking with respect to gene 1 possible.

Discussion

The main results of this study can be summed up as

follows: (i) The high VA ⁄ VG ratios observed in F2 crosses

are not accounted for by an allele-frequency explanation;

(ii) With a random GP map, the U-shaped allele-

frequency distribution used by Hill et al. (2008) does

not ensure high levels of additive variance (Fig. 1); (iii)

Introduction of order-preservation for just a few loci in a

multilocus random GP map is a sufficient constraint to

ensure high VA ⁄ VG ratios also in populations with

intermediate allele frequencies; (iv) The monotonic (i.e.

order-preserving) dose–response relationships ubiqui-

tously present in gene regulatory systems as well as

metabolic systems lead to GP maps where at least some

Table 1 Order-breaking in genotype–phenotype maps for gene

regulatory motifs with monotone and nonmonotone gene regulation

functions. Each row reports averages over 5000 simulations for a

given set of polymorphic parameters.

Monotone gene regulation function

Nonmonotone gene regulation

function

Polymorphic

parameters

Frequency of

order-breaking

with respect toPolymorphic

parameters

Frequency of

order-breaking

with respect to

Gene 1 Gene 2 Gene 1 Gene 2

a1, a2 0 0 a1, a2 0.338 0

a1, h2 0 0 a1, l2 0.488 0.476

a1, p2 0 0.507 a1, r2 0.336 0

a1, a2, h2 0 0.062 a1, a2, r2 0.513 0.452

a1, a2, p2 0 0.256 a1, a2, l2 0.332 0.249

a1, h2, p2 0 0.130 a1, l2, r2 0.498 0.455

a1, a2, h2, p2 0 0.137 a1, a2, l2, r2 0.501 0.434

Fig. 3 Relationship between shape of gene regulation function and VA ⁄ VG. Additive variance in F2 populations for the gene regulatory system

(15) with a monotone gene regulation function (left panel) and the system (16) with a nonmonotone gene regulation function (right panel).

Each boxplot summarizes VA ⁄ VG for 5000 Monte Carlo simulations, and parameter sets are numbered 1–7 corresponding to the row numbers in

Table 1.




loci show order-preservation. Our results suggest that

strong additivity-enhancing effects of constraints (such as

order-preservation) on the GP map is an important

complement to the allele-frequency explanation of high

VA ⁄ VG ratios.

Taking advantage of the empirically highly unrealistic

scenario of random GP maps as a strategy for gaining

biological insight is not unique to this study (Hallgrims-

dottir & Yuster, 2008; Livnat et al., 2008). Our use of such

maps was key to disclose that a major principle under-

lying a predictable parent–offspring relationship is mono-

tonicity in the mapping from genotypes to phenotypes.

But it should be emphasized that our work also shows

the need for developing new concepts around the

properties of multilocus GP maps. For the diallelic one-

locus case, the situation is simple, with additivity and

various degrees of dominance describing the whole range

of possible gene actions. The two-locus case is well

described by classical concepts from the Mendelian (e.g.

duplicate dominant genes) and Fisherian (e.g. additive-

by-additive) schools, see (Phillips, 1998) for a review.

There is also an innovative recent attempt to unify these

two by using a geometric approach (Hallgrimsdottir &

Yuster, 2008) to identify 69 symmetry classes of

the shapes of two-locus GP maps. However, when the

number of loci increases, the need for describing the

main aspects of the GP map with lower-dimensional

descriptors increases. Here, we have focused on the

order-preservation with respect to partial genotype

orders and shown that it is a defining property of random

GP maps as well as a key determinant of VA ⁄ VG in

populations with intermediate allele frequencies. Based

on these preliminary concepts, we think there is much to

learn by using available tools from function theory and

multivariate analysis to find descriptors that separate

biologically constrained GP maps from the random ones.

Nijhout (2008) asserts that an important reason for the

inability of quantitative genetics to predict long-term

evolution is that the relationship between genetic and

phenotypic variation is nonlinear. More specifically, he

claims that a general reason for this nonlinearity is that

the relationships between cause and effect, such as

transcriptional activator concentration and transcription

rate, are saturating and have a hyperbolic or sigmoid

form. Our results suggest that this conception may need

to be qualified to some degree, as they show that

monotonic and saturating dose–response curves do in

fact preserve the features of a linear GP map much more

than, for example, unimodal dose–response curves. That

is, the type of nonlinearity appears to be essential.

The rationale for our focus on the shape of transcrip-

tional dose-response is that the main step for regulating

(a) (b)

(c) (d)

Fig. 4 Genotypic values and nullclines for highly epistatic data sets. Lineplots with genotypic values (a ⁄ c) and nullclines (curves) and

equilibrium points (blue circles) for all homozygotes (b ⁄ d) for the statistically most epistatic data sets from simulations with monotonic (a ⁄ b)

and nonmonotonic (c ⁄ d) gene regulation functions (GRFs). Parameter values for the monotonic case are a11 = 182.04, a12 = 159.72,

a21 = 190.58, a22 = 135.08, h21 = 32.06, h22 = 20.86, p21 = 8.84, p22 = 7.91, and for the nonmonotonic, a11 = 139.44, a12 = 159.20,

a21 = 144.15, a22 = 124.87, l21 = 26.36, l22 = 32.33, r21 = 3.02, r22 = 2.27. Variance ratios (VA ⁄ VG, VD ⁄ VG, VI ⁄ VG) in F2 populations are

(0.748, 0.004, 0.248) for the monotonic case and (0.007, 0.002, 0.991) for the nonmonotonic case.




gene expression is at the initiation of transcription (Carey

& Smale, 2000), and the shape of the GRF determines

key features of cellular behaviour, including regulatory

switches such as the lysogeny–lysis switch in phage

lambda or gene networks exhibiting sustained

oscillations of mRNA or protein levels (Rosenfeld et al.,

2005). The commonly used classification of cis-regulatory

elements into enhancers and silencers (Davidson, 2006)

shows that current molecular biology subscribes to a

conceptual model where basic gene regulation functions

are monotonic. More formally, properties of the tran-

scriptional machinery such as synergy and cooperativity

(Veitia, 2003) have been used as arguments for sigmoidal

dose–response relationships. Furthermore, detailed mod-

elling of a number of promoter regions using statistical

mechanics (Buchler et al., 2003; Bintu et al., 2005a,b)

and reaction kinetics (Verma et al., 2006) together with

experimental data (Kringstein et al., 1998; Hooshangi

et al., 2005; Rosenfeld et al., 2005) also indicates sigmoi-

dal transcription responses for many complex cis-regula-

tory set-ups. When two transcription factors regulate the

same gene, the cis-regulatory input function must inte-

grate both inputs into one output, and it has been shown

both theoretically (Buchler et al., 2003) and experimen-

tally (Yuh et al., 2001; Setty et al., 2003; Istrail &

Davidson, 2005; Mayo et al., 2006) that different Boolean

functions can be obtained by small variations in the

regulatory sequence. The literature also contains some

examples of nonmonotone gene regulation functions

that can be achieved, for instance, by multiple enhancer

sequences where one overlaps the core promoter

(Ptashne et al., 1976; Wang & Warner, 1998) or as a

result of incoherent feedforward motifs, which are quite

common in eukaryotes (Kaplan et al., 2008).

Our Monte Carlo simulations suggest that both order-

breaking with respect to all loci (like we find in the

random GP maps) and order-preservation with respect to

all loci (which is implied by the traditional quantitative

genetics models with interlocus additivity and no over-

dominance) are hard to realize even in very simple

dynamic gene regulatory models generating GP maps.

The gene regulatory model with monotone GRF (eqn 15)

illustrates this point. From the two-first rows of Table 1,

we see that if we restrict polymorphisms to maximal

production rates (or thresholds), while allowing only one

polymorphic parameter per gene, this creates fully order-

preserving GP maps throughout parameter space. How-

ever, when we introduce genetic variation in more than

one parameter per gene, we see that this enables order-

breaking behaviour at locus 1. Characterizing the geno-

type-to-parameter map of models at different abstraction

levels is a large and important research programme in

itself, but available theory (Bintu et al., 2005a) and

empirical data (Rosenfeld et al., 2005; Mayo et al., 2006)

on this map for gene regulation functions indicate that

point mutations can easily affect more than one param-

eter. This leads to the empirically testable prediction that

the GP maps arising from genetic variation in typical

gene regulatory networks will show a high degree of

order-preservation as a result of monotonic gene regu-

lation functions, but that order-breaking for a few loci is

still a ubiquitous phenomenon. The latter property

contrasts with the series of GP maps arising from linear

metabolic pathways as studied by Hill et al. (2008, cf.

table 3). These models are derived from the work by

Kacser & Burns (1981) and Keightley (1989). In this

framework, the genotype at locus i is assigned an enzyme

activity Ei and intralocus additivity for this activity is

assumed. Steady-state flux J is used as a phenotype and

under simplifying assumptions (see Bagheri & Wagner,

2004) about enzyme kinetics J /P

1=Ei½ ��1. As J

increases as a function of Ei independently of all other

enzyme activities, it is clear that this class of GP maps are

order-preserving with respect to all loci. This suggests

that simple metabolic systems lead to even more order-

preserving GP maps than those arising from simple gene

expression networks, and also helps explain the high

VA ⁄ VG ratios reported (Keightley, 1989; Hill et al., 2008)

for metabolic models in F2 populations.

Our results suggest that GP maps are kept order-

preserving by the action of regulatory principles or

mechanisms in sexually reproducing organisms. This

makes the parent–offspring relationship more predictable

across a very wide range of evolutionary settings. Bas

Kooijman elegantly alluded to this phenomenon a

decade ago: ‘Neither the cell nor the modeller needs to

know the exact number of intermediate steps to relate

the production rate to the original substrate density, if

and only if the functional responses of the subsequent

intermediate steps are of the hyperbolic type. If, during

evolution an extra step is inserted in a metabolic pathway

the performance of the whole chain does not change its

functional form.’ (Kooijman, 2000, pp. 75). The basic

features of these additivity-enhancing principles must

have appeared very early in the history of life, and a

crucial question is of course whether their appearance is

caused by some sort of systemic necessity concerning

how complex biological structures can be built at all or

whether natural selection has been responsible for it.

Starting out with random GP maps, Livnat et al. (2008,

2010) showed that sexual reproduction favours alleles

with high mixability, meaning that they perform well

across genetic background and suggest that sex selects for

alleles with an additive effect that rises above the forest of

epistasis effects. If a GP map is order-preserving with

respect to a given locus, the high-performing allele per

definition does well across all genetic backgrounds. Our

results therefore suggest that if biological systems have

much monotonic regulatory behaviour out of systemic

necessity, high mixability could very well be ensured

long before the appearance of sexual reproduction and

thus even facilitate its emergence.

A natural extension of our work on simple gene

regulatory motifs is to look for design principles




promoting a predictable parent–offspring relationship in

large-scale biological networks involved in gene regula-

tion, signalling and metabolism. Although this will be a

much more demanding exercise both analytically and

numerically than the current study, it is encouraged by

recent findings pointing to a high degree of monotonicity

in cellular networks (Baldazzi et al., 2010; Iacono &

Altafini, 2010).

Acknowledgments

We thank Thomas Hansen for useful comments on the

manuscript. This study was supported by the Norwegian

eScience program (eVITA) (RCN grant no. NFR17

8901 ⁄ V30).

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Appendix

The effect of introducing order-preservation for a single locus in apopulation where p11 = p12 = p22

Consider a random GP map G � Nð0; I3N Þ in a population

with p11 = p12 = p22.

We study the effect of introducing order-preservation

for a single locus k by creating a new GP map G1, which is

a permutation of G such that

G1g1 ... gk�111gkþ1 ...gN

� G1g1...gk�112gkþ1 ...gN

� G1g1 ...gk�122gkþ1 ...gN

:

The effect of this sorting of three and three genotypic

values is that G1 will consist of 3N ) 1 triplets of

genotypes {g1. . . gk ) 111gk + 1. . . gN, g1. . . gk ) 112gk + 1. . .

gN, g1. . . gk ) 122gk + 1. . . gN}, which follow the order

statistics of samples of 3 from the standard normal

distribution. Following (Jones, 1948) the expected values

for such a triplet are

� 3

2ffiffiffipp ; 0;

3

2ffiffiffipp

:

Now let N fi ¥ and observe that for a population

with equal genotype frequencies p11 = p12 = p22 the

population mean l fi E(G1) = 0 and the total genetic

variance VG fi 1. Furthermore, the variance explained

by an additive effect of locus k will approach

� 32ffiffipp

h i2

þ02 þ 32ffiffipp

h i2

3¼ 3

2p:

As N fi ¥, the additive variance explained by other

loci than k will tend towards zero (see main paper) and so

VA

VG

! 3

2p� 0:477:

Thus, as the number of loci becomes large, the introduc-

tion of order-preservation for a single locus will create a

single purely additive locus, which explains just below

50% of the genetic variance.

Supporting information

Additional Supporting Information may be found in the

online version of this article:

Figure S1 VA ⁄ VG in partially order-preserving GP maps.

As a service to our authors and readers, this journal

provides supporting information supplied by the

authors. Such materials are peer-reviewed and may be

re-organized for online delivery, but are not copy-edited

or typeset. Technical support issues arising from sup-

porting information (other than missing files) should be

addressed to the authors.

Received 11 October 2010; revised 14 June 2011; accepted 17 June 2011




Order‐preserving principles underlying genotype–phenotype maps ensure high additive proportions of genetic variance

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