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Keywords:

  • epistasis;
  • gene regulatory networks;
  • genetic variance;
  • genotype–phenotype map;
  • monotonicity

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models and methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix
  10. Supporting Information

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.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models and methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix
  10. Supporting Information

In the ‘Generation of Animals’ (c. 340 bc), 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 reproduction (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 recombination and the subsequent billions of highly complex and nonlinear processes setting up the genotype–phenotype (GP) map, should on average resemble its parents more than a randomly drawn couple from the population. 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 proportion of the phenotypic variance VP and (in the absence of strong negative genetic-environmental covariance) 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, overdominance 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 considerable 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 subsequent 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 quantitative 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

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models and methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix
  10. Supporting Information

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 Γi = {11, 12, 22} and the multilocus genotype space Γ contains 3N genotypes constructed by concatenating elements from the single-locus genotype spaces

  • image

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 inline image. 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,

  • image(1)

When modelling and analysing the genetic effects and variance arising from G, we take advantage of the natural and orthogonal interactions (NOIA) framework presented 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 sensuCockerham (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 frequencies inline image, inline image and inline image, the NOIA statistical model is G = SiEi where the vector of genotypic values

  • image

the one-locus genetic-effect design matrix

  • image

and the vector of genetic effects

  • image

A model involving all N loci is given by the equation

  • image(2)

Here, inline image 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 ⊗ S− 1 ⊗ ⋯ ⊗ S2 ⊗ S1 and ES is a vector containing all genetic effects (average effects, dominance deviations and up to N-way interactions among these). The genetic-effect design matrix SS together with the diagonal matrix F = diag (p1111⋯11, p1211⋯11. . .p2222⋯22) of multilocus genotype frequencies (assuming linkage equilibrium) has the property that inline image 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,

  • image(3)

(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,

  • image(4)

with covariance matrix inline image, 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 r 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 3− 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 × 3− 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

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models and methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix
  10. Supporting Information

Genetic variance components of random GP maps

Analytical results

Here, we derive analytical expressions for the distributions of ratios of genetic variance components for different populations with intermediate allele frequencies. The results also provide a basis for assessing the numerical work below.

F2populations: For a single locus in an ideal F2 population, the relevant matrices are

  • image(5)

and the distribution of genetic effects for the random GP map (1) is specified by the covariance matrix

  • image

so that ∼ N(0, 0.52) and ∼ N(0, 1.5) are independent. Inserting this and elements from inline image into eqn (3), the proportion of genetic variance explained by the additive genetic effect is given by

  • image(6)

where inline image (except for the genetic effects being random variables, eqn (6) is equivalent to eqn 8.7 in Falconer & Mackay (1996)). The expected value of the proportion is

  • image

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 wherep11 = p12 = p22 = 1/3: In a hypothetical population where inline image for all loci, the matrix SS itself becomes orthogonal, and the distributions of genetic effects under a random GP map are given by

  • image(7)

The matrix of genotype frequencies is inline imageand eqn (3) simplifies to

  • image(8)

i.e. the scaled genetic effects ɛi are independent and identically distributed. The proportion of VG explained by a single genetic effect ej is given by

  • image(9)

In the case of a subset S of genetic effects, the variance explained by the nS effects in this subset is

  • image(10)

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 = 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 inline image, 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; Balasubramanian et al., 2009), we get a similar situation as above. As can be seen from the one-locus matrix

  • image(11)

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

  • image(12)

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 inline image, respectively).

Numerical results

Simulations 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 proposed 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 populations, 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.

image

Figure 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.

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Order-breaking: a distinctive feature of random GP 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 underdominance 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 Γk by 11 < 12 < 22, and for each genetic background (g1g2. . .g− 1 gk + 1. . .gN) for locus k, this gives an ordering of three elements in Γ,

  • image(13)

This defines a strict partial order on Γ with 3N inequalities ordering pairs of genotypes (for = 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

  • image(14)

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 population). 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).

image

Figure 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.

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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 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 relationships.

Monotonicity of the gene regulation function constrains the genotype to expression phenotype map

Motivated by the aforementioned results, we investigated 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 αij be the maximal production rate of the allele and γij 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(yθp) = yp/(θp + yp), where parameter θ 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:
    • image(15)
  • The variables aj and bj are used to code genotype at genes 1 and 2, respectively; for genotype klmn, we set a1 = ka2 = lb1 = m and b2 = n.

  • 2
     The nonmonotone unimodal function inline image, 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:
    • image(16)
  • 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, γij was kept nonpolymorphic at 10, whereas allelic values of the other parameters were either sampled uniformly in the intervals αij:(100, 200), θ2j:(20, 40), p2j:(1, 9), μ2j:(25, 35), σ2j:(2, 4), or the mid-value of the respective intervals were used as nonpolymorphic values. The steady-state expression level of gene 2 (y2 = x21 + x22) was used as the phenotype. We did 5000 Monte Carlo simulations for seven different parameter scenarios (cf. rows in Table 1) of polymorphic and nonpolymorphic parameters.

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 functionNonmonotone gene regulation function
Polymorphic parametersFrequency of order-breaking with respect toPolymorphic parametersFrequency of order-breaking with respect to
Gene 1Gene 2Gene 1Gene 2
α1α200α1α20.3380
α1θ200α1μ20.4880.476
α1p200.507α1σ20.3360
α1α2θ200.062α1α2σ20.5130.452
α1α2p200.256α1α2μ20.3320.249
α1θ2p200.130α1μ2σ20.4980.455
α1α2θ2p200.137α1α2μ2σ20.5010.434

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 α2j and the threshold θ2j are polymorphic. 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 monotonicity of the same nullclines makes order-breaking with respect to gene 1 impossible.

image

Figure 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.

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image

Figure 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 α11 = 182.04, α12 = 159.72, α21 = 190.58, α22 = 135.08, θ21 = 32.06, θ22 = 20.86, p21 = 8.84, p22 = 7.91, and for the nonmonotonic, α11 = 139.44, α12 = 159.20, α21 = 144.15, α22 = 124.87, μ21 = 26.36, μ22 = 32.33, σ21 = 3.02, σ22 = 2.27. Variance ratios (VA/VGVD/VGVI/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.

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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 μ2j or both the maximal production rate α2j and the width parameter σ2j 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

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models and methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix
  10. Supporting Information

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 ubiquitously present in gene regulatory systems as well as metabolic systems lead to GP maps where at least some 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 (Hallgrimsdottir & Yuster, 2008; Livnat et al., 2008). Our use of such maps was key to disclose that a major principle underlying a predictable parent–offspring relationship is monotonicity 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 transcriptional dose-response is that the main step for regulating 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 transcriptional machinery such as synergy and cooperativity (Veitia, 2003) have been used as arguments for sigmoidal dose–response relationships. Furthermore, detailed modelling 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 sigmoidal transcription responses for many complex cis-regulatory set-ups. When two transcription factors regulate the same gene, the cis-regulatory input function must integrate both inputs into one output, and it has been shown both theoretically (Buchler et al., 2003) and experimentally (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 overdominance) 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. However, 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 genotype-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 parameter. 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 regulation 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 inline image. 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 regulation, 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

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models and methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix
  10. Supporting Information

We thank Thomas Hansen for useful comments on the manuscript. This study was supported by the Norwegian eScience program (eVITA) (RCN grant no. NFR178901/V30).

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models and methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix
  10. Supporting Information

Appendix

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models and methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix
  10. Supporting Information

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

Consider a random GP map inline image 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

  • image

The effect of this sorting of three and three genotypic values is that G1 will consist of 3− 1 triplets of genotypes {g1. . . g− 111gk + 1. . . gNg1. . . g− 112gk + 1. . .gNg1. . . g− 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

  • image

Now let N [RIGHTWARDS ARROW] ∞ and observe that for a population with equal genotype frequencies p11 = p12 = p22 the population mean μ [RIGHTWARDS ARROW] E(G1) = 0 and the total genetic variance VG [RIGHTWARDS ARROW] 1. Furthermore, the variance explained by an additive effect of locus k will approach

  • image

As N [RIGHTWARDS ARROW] ∞, the additive variance explained by other loci than k will tend towards zero (see main paper) and so

  • image

Thus, as the number of loci becomes large, the introduction 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

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models and methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix
  10. Supporting Information

Figure S1VA/VG in partially order-preserving GP maps.

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