Fluxes and Metabolic Pools as Model Traits for ... - Genetics

Copyright © 1999 by the Genetics Society of AmericaFluxes and Metabolic Pools as Model Traits for Quantitative Genetics.I. The L-Shaped Distribution of Gene EffectsBruno Bost,* Christine Dillmann* ,† and Dominique de Vienne**Station de Génétique Végétale, INRA/UPS/INAPG, Ferme du Moulon, 91190 Gif-sur-Yvette, Franceand † Institut National Agronomique Paris-Grignon, 75005 Paris, FranceManuscript received June 7, 1999Accepted for publication August 25, 1999ABSTRACTThe fluxes through metabolic pathways can be considered as model quantitative traits, whose QTL arethe polymorphic loci controlling the activity or quantity of the enzymes. Relying on metabolic controltheory, we investigated the relationships between the variations of enzyme activity along metabolic pathwaysand the variations of the flux in a population with biallelic QTL. Two kinds of variations were taken intoaccount, the variation of the average enzyme activity across the loci, and the variation of the activity of eachenzyme of the pathway among the individuals of the population. We proposed analytical approximations forthe flux mean and variance in the population as well as for the additive and dominance variances ofthe individual QTL. Monte Carlo simulations based on these approximations showed that an L-shapeddistribution of the contributions of individual QTL to the flux variance (R 2 ) is consistently expected inan F 2 progeny. This result could partly account for the classically observed L-shaped distribution of QTLeffects for quantitative traits. The high correlation we found between R 2 value and flux control coefficientsvariance suggests that such a distribution is an intrinsic property of metabolic pathways due to thesummation property of control coefficients.THE pioneering work of Kacser and Burns (1981)illustrated the power of the metabolic control theory(MCT) in accounting for fundamental genetic phenomenasuch as recessivity of deleterious alleles, epistasis,or selection of selective neutrality. The MCT describeshow the properties of individual enzymes of a pathwayinfluence the flux through the pathway, and thus providesa biochemical link between the genetically determinedenzyme activities/concentrations and the flux,which is a global property of the pathway. This mecha-nistic model of a quantitative phenotype has been suc-cessfully used in quantitative and population genetics.The variability of the flux was theoretically analyzed asa function of the effect and frequency of mutations inpopulations (Keightley 1989), or within sibship whenparental genotypic values are known (Ward 1990). De-velopments of this model shed light on the variability ofenzyme activities in populations under mutation-selectionbalance (Clark 1991; Hastings 1992). The relation-ship between metabolic flux and fitness was exploredin Escherichia coli (Dykhuizen et al. 1987), leading tothe concept of natural selection of selective neutrality(Hartl et al. 1985). Beaumont (1988) pointed outthat stabilizing selection arises as a consequence of thestructure of metabolic pathways; and Keightley (1996)showed how dominance and directional epistasis, whichare automatically generated in metabolic pathways, leadCorresponding author: Bruno Bost, Station de Génétique Végétale(INRA-UPS-INAPG), Ferme du Moulon, 91190 Gif-sur-Yvette, France.E-mail: bost@moulon.inra.frto an asymmetrical pattern of response to directionalselection. Finally Szathmary (1993) showed that epistasisbetween deleterious mutations for enzyme activity is synergisticin most kinds of selection, except for selection formaximizing the flux, where epistasis is antagonistic.In the terminology of modern quantitative genetics,the enzymatic loci can be regarded as putative quantitativetrait loci (QTL) of the flux, characterized by theircontribution to the flux variance in a population. Assumingthat macroscopic and quantitative traits are proportionalto metabolic fluxes in the cell, we considered thefluxes as model traits to analyze the quantitative geneticvariation. In this work, the MCT was used to predict theshape of the distribution of flux QTL effects in a segregat-ing population derived from the cross between two individ-uals drawn at random in a species. We considered boththe variation of the average enzyme activities across themetabolic pathway and the variation of activity of singleenzymes between individuals of the population. Usinganalytical developments and simulations, we showed thatan L-shaped distribution of flux QTL effects is consis-tently observed. This distribution is related to the L-shapeddistribution of flux control coefficients, which is a consequenceof the summation theorem (Kacser and Burns1973), and is also observed experimentally.THEORETICAL BACKGROUNDMetabolic flux as a function of enzyme activity: Theflux through a linear metabolic pathway is described asa hyperbolic function of the activity of each enzymeinvolved in the pathway (Kacser and Burns 1973). Con-Genetics 153: 2001–2012 ( December 1999)

2002 B. Bost, C. Dillmann and D. de Viennesider a linear metabolic pathway, with n enzymes (E 1 ,E 2 ...,E i ,...,E n ), converting a substrate (S 1 ) into afinal product (S n1 ),S 1 → E 1S 2 → ...→ S i → E iS i1 → ... → S E nn → S n1and define E i , which for simplicity will be called theactivity of enzyme i, asE i V iM iK 1,i ,To study the flux QTL distribution, we considered thepopulations resulting from a cross between two diploidparents drawn at random. In those populations, we variedboth the average activities among loci and the extentof the genetic variability of activity of the enzymes thatcontrol the metabolic pathway. The resulting variationsobserved at the flux level were analytically studied, anda set of relevant variables to describe QTL effects andmetabolic control was defined. Then, we used MonteCarlo simulations to analyze the distribution of thesevariables and their relationships.Variation at the enzyme level: We defined a given individualk of the species by the vector E k {E k1 , E k2 ,...,E ki , . . .} of enzyme activities of the biallelic loci governingthe metabolic pathway. Without any knowledge of thedistribution of enzyme activities among the loci of actualmetabolic pathways, we supposed that the E ki are randomvariables, independently and identically distributed accordingto a given law L( k ), where k is the vector ofthe parameters of L for individual k. We consideredthe population resulting from the cross between twoindividuals k and h and supposed that the loci governingenzyme activities are independent, and without linkagedisequilibrium in the population. In this case, the distributionof the flux is determined by the enzyme activitiesE k and E h for each parent, and by the matrix of allelicfrequencies {p ij }, where p ij is the frequency of allele j forenzyme i in the resulting population.In particular, we considered the F 1 hybrid resultingfrom the cross between two inbred lines and the F 2population obtained by selfing the F 1 hybrid. In case ofindependent loci and with no dominance at the enzymelevel, enzyme activity at locus i is defined by the averageactivity m i (E ki E hi )/2 and the additive allelic effecta i (E ki E hi )/2. Note that m i and a i are not independent,because for all i, |a i | m i . In an F 2 population, it iseasily shown (appendix a) that the coefficient of variation(cv i ) of the activity of enzyme i iswhere V i is the maximum velocity of enzyme E i , M i isits Michaelis constant, and K 1,i j1K i1j,j1 is the prod-uct of equilibrium constants of reactions 1, 2, . . . , i.At the steady state, and assuming that all enzymes arefar from saturation, the flux through the pathway isJ [S 1] [S n1 ]/K 1,n1, (1)i1(1/E n i )where [S 1 ] and [S n1 ] are the concentrations of the sub-strate S 1 and product S n1 , respectively. [S 1 ] and [S n1 ] arefixed parameters of the system, while the intermediatemetabolite concentrations ([S i ] for i 2ton) are variables.Control coefficient of the flux: To quantify how theflux reacts when an infinitesimal change occurs in theactivity of a given enzyme, Kacser and Burns (1973)defined the control coefficient C J i of the flux J, withrespect to activity E i of enzyme E i , as the ratio of aninfinitesimal relative variation of the flux to an infinites-imal relative variation of an enzyme:C J i JJ E i.E iUnder the assumptions mentioned above, we have1/EC J i ij1(1/E n j )and hence i1C n J i 1.This summation theorem (Kacser and Burns 1973)applies more generally than to linear pathways, for examplein branched pathways, pathways with feedbackregulation (Kacser and Burns 1973), pathways wheresome metabolites are involved in a moiety-conservedcycle (Hofmeyr et al. 1986), or pathways with two stepscatalyzed by the same protein molecule (Cascante etal. 1990). The most important consequence of the theoremis that the control of the metabolic system may beshared among all the enzymes, a view quite differentfrom that of “rate-limiting” or “bottleneck” concepts. Inthe context of the MCT, the rate-limiting steps are stepswith control coefficients close to one, and they arehighly dependent on the genetic background. Experimentaldata are rather consistent with a distribution ofthe control across the metabolic pathway (see results).METHODScv i a im i √2 . (2)Hence, the F 2 population can alternatively be de-scribed by the distribution law L( m )ofthem i and thedistribution law L( cv ) of the cv i . The former describesthe distribution of the average enzyme activity across theloci, while the latter describes the distribution of thedifferences between the parents k and h, because for agiven m i value, the a i value can be deduced from the cv ivalue.Variation at the flux level: In a segregating population,each enzyme whose activity is genetically variableexplains a part of the genetic variance of the flux. Inother words, the polymorphic loci responsible for en-zyme variations are QTL of the flux and of any traitproportional to the flux. However, there is no simple

MCT and Distribution of QTL Effects2003tween the flux of the F 1 hybrid and the mean flux ofthe parents. Figure 1B shows that the control coefficientincreases as the enzyme activity decreases, and is closeto zero for values of activity corresponding to the plateauin the flux curve. Due to this hyperbolic relationship betweenflux and activity, a relative change in activity leadsto a smaller relative change of the flux, and the low-effectalleles are recessive (Kacser and Burns 1981). If bothparents exhibit low control coefficients (i.e., high valuesof activity), the flux values will be close to each other,while if both parents have high control values, the heterozygotewill exhibit additivity, a prediction consistentwith known cases of heterozygotes between sets of“lower” alleles in a series, as for pigmentation in guineapig (Wright 1960) or mouse (Grünenberg 1952).In the general case, all enzyme activities may differbetween the parents. Hence, the flux of each individualdepends on its genotype at the different enzymatic loci.The average flux J in a population can be approximatedby developing the function (1) expressing theflux into a second-order Taylor series. We chose thesecond order as a good compromise between precisionand heaviness of the calculations. Provided there is nolinkage disequilibrium, and taking the derivatives of J with respect to allelic frequencies, the additive anddominance effects ( i and i ) of QTL i, and the epistatic(additive additive) effect ( ij ) of a pair (i, j) of QTL,could be calculated (appendix b; Kojima 1959), as wellas the contributions of the QTL to the components ofFigure 1.—Relationship between the parameters that detheflux variance: additive ( A 2 i) and dominance ( D 2 i)scribe the variability at the flux level, and the correspondingparameters at the enzyme level, in a simple situation where variances at QTL i, and epistatic variance ( AAij) 2 for theall enzymes of the pathway, except one (enzyme i), have the pair of QTL (i, j). The QTL i contributes for a fractionsame activity for the cross between two inbred lines k and h. R 2 i of the total variance of the flux,(A) E ki , E hi , and E F1 , are the enzyme activities of parent k, parenth and F 1 hybrid, respectively. m i is the average activity and a iis the additive allelic effect of the locus i. J ki , J hi , and J F1 are the R i 2 A 2 iD 2 i, (3)flux values of parent k, parent h, and F A 2 D 2 AA2 1 hybrid, respectively. J is the average flux in an F 2 population resulting from theselfing of the F 1 hybrid. i and i are, respectively, the additive where A, 2 D, 2 and AA, 2 are the additive, dominance, andand dominance effect of the locus i: 2 i J hi J ki and 2 i epistatic (additive additive) variances of the flux inJ F1 (J hi J ki )/2. (B) C kiJ , C hiJ , and CF J1, are the flux controlthe population, respectively. The total genetic variancecoefficients of the enzyme i, in parent k, parent h, and the F 1hybrid, respectively. Parent k has a control coefficient higher of the flux also comprises other components, like thethan parent h, which is nearly on the plateau (C Jki is five times (additive dominance) and (dominance dominance)higher than C hiJ ). This situation leads to a high additive effect epistatic variances, as well as higher-order variance comwithan intermediary degree of positive dominance.ponents, which were neglected here. In this article, the“additive allelic effect” of enzyme locus i refers to a i ,relationship between the variation of enzyme activity at while the “flux QTL effect” of QTL i refers to R i 2 .a locus i and its additive and dominance effect ( i and The sharing out of the control between the enzymes i , respectively) upon the flux.of the metabolic pathway is different for each individualThe simplest case of an F 2 population where all enwecomputed, for each enzyme i, the average flux con-of the population. To characterize each F 2 population,zymes but one have the same activity in both parentsis represented in Figure 1. The hyperbolic relationship trol coefficient, and its variance, Var[C J i], and we de-between flux and enzyme activity leads to a saturation in fined the concept of “populational control coefficient”the flux when the activity increases (Figure 1A; Equation for enzyme i as1). The additive effect i of QTL i upon the flux is equalto half of the difference between the parents, and thedegree of dominance i is one-half the difference be-C J i 1/E i n j1(1/E j ) , (4)

2004 B. Bost, C. Dillmann and D. de Viennewhere E i (respectively E j ) is the average activity of en- tions of those parameters were obtained by pooling thezyme i (respectively j) in the population.5000 resulting values. The populational control coeffi-The populational control coefficient of enzyme E i is cient and R 2 i distributions were characterized by the followingnot equal to its average control coefficient, but correspondsparameters: mean, skewness (Sokal and Rohlfto the control coefficient of an “average” individ- 1995), i1R n 2 i averaged over populations, and percentageual, i.e., an individual displaying the average activities of values 0.02, which is the value expected for 50 equivaforall enzymes. It does not depend on the additive lent QTL (1/50).allelic effect a i of the QTL, unlike the R i 2 .InF 2 populationswithout dominance at the enzyme level, E i m iso that the populational flux control coefficient is alsoRESULTSthe control coefficient of the F 1 hybrid for enzyme i. Relationships between enzymatic allelic effects andSimulation of flux QTL effects and control coeffi- flux QTL effects: As the relationship between flux andcient distributions: To simulate the distributions of flux enzyme activities is nonlinear, the average flux J in aQTL effects or control coefficients we considered 50 population depends not only on the average enzymeindependent enzymatic loci in F 2 populations. A four- activities, E i , but also on the variances, E 2 i, of enzymestep procedure was used:activities, with a negative relationship between average1. Draw of the m i values. We considered several distributionflux and activity variances. As shown in appendix b,laws for m i , corresponding to different degreesji(1/E j )( n j1(1/E j )) 3. (5)of dispersion of the average enzyme activity acrossthe metabolic pathway. Those distributions were constant(m i 10, ∀i reference case), uniform (in the J f(E 1 ,...,E n ) i1 n 2 E i 2KE 3 iFor the same reason, the flux variance is related notrange [0, 30]), normal (10, 2.5), or exponentialonly to the variances of enzyme activity at each QTL( 16.2, 1.2). The value of was chosen so but also to their average activities. These features clearlythat all the distributions have roughly the same range differ from the classical additive models used in quanti-of variation, and the probability density function of the tative genetics. Moreover, the formulas show that it isexponential law is f(x) (1/)exp(( x)/). not the average activity of the enzymatic locus that di-2. Draw of the cv i values. We have chosen to consider rectly influences the flux variance, but the relativethe distribution of cv i rather than the distribution of weight of the enzyme in the pathway, expressed as thea i for two reasons. First, it made it easier to take “populational control coefficient” (C i), J or control coefficientthe constraint |a i | m i , ∀i into account. Second, weof the “average” individual (Equations B13–B15).observed that our approximations for the average flux The additive contribution of QTL i to the flux varianceand its variance were better for cv i values 0.3. Three (A 2 i) is also affected by the other QTL through theircontrasted distributions were considered: (i) cv i variability (contribution of a QTL is reduced by an in-0.2, ∀i, i.e., there is a strict positive relationship between crease of the variability of the other enzymes) andmean and additive effect of enzyme activity; (ii) nor- through their populational control coefficients, whichmal, with an average of 0.35 (middle of the range are related by the summation property,for the possible values of cv i , given m i ; see appendixa) and standard deviation fitted to get all values A 2 i E 2 i(C i J ) 4 K 2 [1 3cv i2 (C i J 1) 2within the range of possible cv values; (iii) gamma, [cvj C j(3C Jj J 2)]] 2 ,fitted to get 95% of the values between 0 and 0.3.ji(6)3. Computation of the flux QTL effects. For each pairwhere cv i a i /m i √2, with m i the average activity of enof{m i } and {cv i } vectors, we used our approximationszyme i and a i its additive allelic effect.to compute the flux of the F 1 hybrid and the parame-It is worth noting that the relationship between thoseters of the F 2 population: populational flux controlfactors is not tight: an enzymatic locus with a large addicoefficientC J i, average flux J , total genetic variance,tive allelic effect may have a small effect upon the fluxand the flux QTL effects R 2 i (see appendix b).variance if its control on the pathway is weak in both4. Distribution of the flux control coefficients. For anparents (Figure 1).F 2 population, 10,000 individuals were randomly gen-The flux QTL effects are L-shaped distributed: Figureerated, according to the parental genotypes at each2 compares the distributions of the average enzymelocus. For each individual and each locus, we comactivity,m i , across the loci, to the corresponding distribuputedthe flux control coefficient and inferred thetions of flux QTL effects, R i2 , for a gamma distributioncorresponding variance Var[C J i] for each enzyme.of cv i . When all enzymes have identical m i and a i , theThose steps were iterated 100 times to simulate 100 QTL have the same R i2 (Figure 2A). Our simulationsdifferent F 2 populations. Hence, we computed 100 J show that, as soon as there is any difference in theirvalues and a total of 100 50 5000 different values average activity or variability, the distribution of R i2 ex-for m i ,cv i , C J i, R i 2 , and Var[C J i]. The expected distribu- hibits an L shape: few steps have high R i2 values, more

MCT and Distribution of QTL Effects2005Figure 2.—Distributionsof flux QTL R 2 (percentages)for a 50-enzyme pathwayin 100 F 2 populations,computed for four distributionsof average activity (m i ):(A) Reference (same m i forall the QTL); (B) normal;(C) exponential; and (D)uniform. The coefficients ofgenetic variation of theQTL (cv i ) are randomlydrawn in a gamma distribution.have moderate values, and a large number have small of the m i values, it is about the same whatever the distributionor very small values. This shape is more pronouncedof the cv i .with the uniform distribution of m i , which leads to more Another feature of flux QTL is that they behave as ifcontrasted m i values, but remains with the exponential, they are nearly additive. Without epistasis, the R i2 shouldwhich still displays a J-shaped distribution of m i , or with sum up to 100% in a given population (Equation 3).the normal distribution (Figure 2). Numerical characteristicsWith the parameters chosen in our simulations, theof the R i2 distributions confirm these observa- average R i2 value is just below 1/50, and their averagetions: all skewness values are significantly positive (P sum over the flux QTL ranges from 95.1 to 99.9%. Of0.001). As shown Table 1, 61.4–91.5% of the QTL have course we did not consider all the epistatic terms in thean R i2 value below 1/50, depending on the m i and cv i denominator of Equation 3. However, we checked, bydistributions, and do not really contribute to the vari- calculating the difference between the total genetic varianceance of the flux in the population. However, this skewnessand the denominator, that the epistatic terms ofis more affected by unequal m i values across loci higher order are negligible so this approximation doesrather than by unequal cv i values: for a given distribution not significantly modify the results. The more pronounced

2006 B. Bost, C. Dillmann and D. de VienneTABLE 1Distributions of flux QTL R 2 for a 50-enzyme pathway in an F 2 populationQTL R 2 distributionAverage Sum Percentagecv i and m i distributions (%) (%) Skewness under 2%∀i, cv i 0.2∀i, m i m (reference) 1.99 99.5 — 100.0m i : normal 1.99 99.3 4.86 67.4m i : exponential 1.99 99.3 21.02 76.2m i : uniform 1.97 98.4 7.12 89.0cv i : normal distribution∀i, m i m (reference) 1.96 98.2 3.46 68.4m i : normal 1.96 98.1 9.25 71.8m i : exponential 1.96 98.2 13.66 74.1m i : uniform 1.90 95.1 6.90 90.6cv i : gamma distribution∀i, m i m (reference) 1.99 99.7 6.95 76.2m i : normal 1.99 99.7 8.30 78.0m i : exponential 1.99 99.7 9.21 77.4m i : uniform 1.99 99.4 7.07 90.9The R 2 distributions were computed for a 50-enzyme pathway in 100 F 2 populations, with four distributionsof average activity (m i ), reference (same m i for all the QTL), normal, exponential, and uniform; and threedistributions of coefficients of genetic variation (cv i ), same cv i for all the QTL, normal, and gamma. Mean R 2value of the distribution and average sum of the 50 R 2 are in percentage of total flux variance.the L-shaped distribution across loci, the higher the vari- control coefficients and a few steps with a large control;ance that results from additive additive interactions: it i.e., the distribution of control coefficients across loci isis equal to 0.6% with constant m i and cv i , and rises up expected to exhibit an L shape.to 4.9% with uniform m i and normal cv i . Moreover, when Experimental data: We analyzed three experimental orthe number of QTL decreases, the additive additive modeling studies by pooling for each one all the controlepistasis increases—for example, from 0.6 to 2.8% for 50 coefficients estimated under various conditions (Groenand 10 QTL, respectively (constant m i and cv i , not shown). et al. 1986; Albe and Wright 1992; Hill et al. 1993).As seen in Figure 1 and methods, the additive and Groen et al. (1986) have estimated the control coeffidominanceeffects of an enzymatic locus upon the flux cients of eight steps of gluconeogenesis in isolated ratare related to the difference in the control coefficients liver cells under various experimental conditions (eachbetween the parental genotypes. On the other hand the “step” actually included several reactions, the wholecontrol coefficients are linked through the summation pathway being considered). About 70% of the controlproperty. Thus, looking for a possible intrinsic relation- coefficients were under the average value (0.125), withship between flux QTL R 2 and summation property, we 50% under 0.05. The shape of the distribution of theanalyzed the sharing out of the control in the parents, coefficients was skewed to the right, with a skewnessand we studied the relationship between the R i 2 and the value of 20.42 (significant with P 0.001; Figure 3A).variance of the control coefficients in the population. Similar results were obtained for five steps of succinateThe parental flux control coefficients are L-shaped oxidation in cucumber cotyledon mitochondria underdistributed: In linear pathways of unsaturated Michae- various experimental conditions (Hill et al. 1993; skewnesslian enzymes, the flux control coefficients are all positive.27.00, P 0.001, Figure 3B). Finally, a steadylianWith an n-enzymes pathway, the summation theo- state model for the tricarboxylic acid cycle has beenrem implies that, for a given individual, when one or a established from experimental data in Dictyostelium dis-few steps have control coefficients greater than 1/n, the coideum (Albe and Wright 1992). The control coefficientsother steps will necessarily have coefficients below 1/n.for the CO 2 flux produced by the cycle and theHence, the average value of the control coefficients is total CO 2 production were estimated for each of the 26expected to be 1/n. If a mutation decreases one enzyme steps, with six different ranges of variation of enzymeactivity close to 0, its control coefficient will rise up to activities. The distribution of the coefficients was skeweda value close to 1, and the other coefficients will become toward weak control (skewness 12.29, P 0.001, Figurenegligible. Thus, as soon as there is some variation for 3C), and 66% of the values were below the expectedenzyme activity across loci in large metabolic systems, average of 1/26. These data are consistent with numer-there would be many steps exhibiting small or very small ous partial characterizations of other metabolic path-

MCT and Distribution of QTL Effects2007Figure 3.—(A) Distribution of flux control coefficients for8 steps of the gluconeogenesis pathway, from lactate to glucose,in isolated rat liver cells in various experimental conditions.Data from Groen et al. (1986). (B) Distribution of fluxcontrol coefficients for 5 steps of the succinate oxidation pathwayin isolated cucumber cotyledon mitochondria. The fluxis the O Figure 4.—Relationship between flux QTL R 2 2 consumption. Data from Hill et al. (1993). (C)and popula-Distribution of CO 2 flux control coefficients for 26 steps of a tional control coefficients (A and B) or variance of the controlcomputer model of the tricarboxylic acid cycle in Dictyostelium coefficient (C) for a 50-enzyme pathway in 100 F 2 populations.discoideum. Data from Albe and Wright (1992).Average activity (m i ) is normally distributed (see Figure 2B).(A) The coefficient of genetic variation of each QTL (cv i )israndomly drawn in a gamma distribution. (B) As for A, butonly 1 of the 100 populations is represented. For the threeways, which show that the control is not equally shared marked QTL, see results. (C) Relationship between R 2 andbetween the different steps of metabolic chains.control coefficient variances in the same population as for B.Simulations: The same kind of L-shaped distributionwas found in our simulations, considering the 2 100parents of the F 2 populations and a 50-step metabolic the enzyme i. However, the relationship between thepathway (results not shown). Whatever the distribution of R i2 and the populational control coefficient in a givenenzyme activity, skewness values are positive and highly population is complex and appears to be very loosesignificant (P 0.001) and there are much more control (Figure 4A). QTL with similar populational control coefficientsmay have quite different R i2 , depending on a ivalues below 1/50 (51–81% of the values) than over.Relationship between control coefficients and flux values, while QTL may exhibit low R i2 even though theQTL effects: From Equations 5 and B20, it appears that control coefficient is high, if a i is low (compare QTL 1,the contributions of QTL i to the flux additive and 2, and 3 in Figure 4B).dominance variances are related to the populational On one hand, a given allelic additive effect (a i )iscontrol coefficient and to the variance of the activity of expected to affect the flux all the more if the difference

MCT and Distribution of QTL Effects2009repulsion), or heritability, may produce L-shaped distri- 1995) in humans. Actually, the proportion of polymorphicgenes encoding or controlling enzymes is notbutions for traits following the classical additive modelof quantitative genetics, even though the skewness is known, and the extent to which our analysis applies toconsistently higher with the metabolic model (B. Bost, gene products involved in the timing or tissue specificityC. Dillmann and D. de Vienne, unpublished results). of gene expression, or to hormone/receptor systems,However, it is highly likely that the L-shaped distribution is difficult to appreciate.also has biological bases. First, there is no genetic reason In conclusion, the hyperbolic relation between enzymeactivity and metabolic flux accounts for thefor a discontinuity between all-or-null (wild-type/mutant)variation and quantitative variation. Thus in pea, a major L-shaped distribution of control coefficients and hencegene for Ascochyta blight resistance was mapped on could be one of the factors explaining the L-shapedchromosome 4 using both a QTL detection approach distribution of gene effects in quantitative genetics.and a Mendelian analysis after partitioning the distribu- Thanks go to F. Hospital for useful discussions and reading thetion of the resistance in the progeny into two classes manuscript, and also to C. Damerval and A. Leonardi for reading the(Dirlewanger et al. 1994). Second, a survey of the manuscript. We thank G. de Jong, who brought to our attention theliterature shows that for various traits, the same major Kojima approach on gene effect decomposition. B. Bost was supportedby a Ph.D. grant from the French Ministry of National Education,QTL may be found in different populations or environ-Research and Technology (MENRT).ments, which seems quite unlikely in case of erroneousestimates of the QTL effects. For example, in two differentF 2 populations derived from crosses between maizeLITERATURE CITEDand teosinte, Doebley and Stec (1993) found similarAlbe, K. R., and B. E. Wright, 1992 Systems analysis of the tricarboxsuitesof QTL for architectural traits, with the same ylic acid cycle in Dictyostelium discoideum. II. Control analysis. J.order of QTL effects for several of them. In three tomatoBiol. Chem. 267: 3106–3114.populations derived from the same parents (one F Beaumont, M. A., 1988 Stabilizing selection and metabolism. He-2 andredity 61: 433–438.two F 2/3 ) and grown in different environments, Pater- Beavis, W. D., 1994 The power and deceit of QTL experiments:son et al. (1991) found QTL for fruit traits common to lessons from comparative QTL studies, pp. 250–266 in 49th An-two or even three conditions. Also significant are the QTL nual Corn and Sorghum Research Conference. ASTA, Washington,DC.apparently commonly found in different species of Byrne, P. F., M. D. McMullen, M. E. Snook, T. A. Musket, J. M.Poaceae (Paterson et al. 1995) or Fabaceae (FatokunTheuri et al., 1996 Quantitative trait loci and metabolic pathetal. 1992; Maughan et al. 1996; Timmerman-Vaughanways: genetic control of the concentration of maysin, a cornearworm resistance factor, in maize silks. Proc. Natl. Acad. Sci.et al. 1996). Finally, in a noninfinitesimal model, equal USA 93: 8820–8825.QTL effects would imply that for any given trait, there Carbonell, E. A., T. M. Gerig, E. Balansard and M. J. Asins, 1992would be a given, limited number of detectable QTL,Interval mapping in the analysis of nonadditive quantitative traitloci. Biometrics 48: 305–315.which seems to be biologically irrelevant. Carbonell, E. A., M. J. Asins, M. Balsega, E. Balansard and T. 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MCT and Distribution of QTL Effects2011Combining (B2), (B3), and these simplifications, an0 cv i 1 √2 . (A3) approximation of the population average flux isi1 2 fE 2 iE i2 J f(E 1 ,...,E n ) 1 (E2 n1 ,...,E n ) . (B4)APPENDIX BQTL contributions to the flux variance components: The second partial derivative of the function f with re-Shown are calculation of approximations of the additive spect to E i isand dominance effects ( i and i ) and additive and 2 f(EE i2 1 ,...,E n ) 2K ji1/E jdominance variance ( 2 E i3 ( j11/E n j ) 3. (B5)A iand D 2 i) at QTL i, and approximationsof epistasis (additive additive) effect andvariance (Thus, introducing (B5) into (B4), we have an approxi-ij and AA 2 ij) for a pair of QTL, i and j.mation of the population average flux,The QTL are controlling the flux through a linearpathway of n enzymes, in a segregating population, as-K J i11/E n i 1 i1 n E 2 isuming that the QTL are not linked. Each enzyme ofC i(1 J C JE ) i2 i the pathway is controlled by one biallelic QTL i, with afrequency p i for the upwardly acting allele.The flux in the pathway for an individual j in thepopulation isJ(j) K ni11E i (j) f(E 1 (j),...,E i (j),...,E n (j)),(B1)where K [S 1 ] [S n1 ]/K 1,n1 (see Equation 1).Expanding B1 into a second-order Taylor series, wehavef(E 1 (j),...,E i (j),...,E n (j)) f(E 1 ,...,E n ) ni1 (E i E i ) fE i(E 1 ,...,E n ) 1 2 n ni1 (E i E i ) 2 2 fE i2i1j i (E i E i )(E j E j )(E 1 ,...,E n ) i 1 22 fE i E j(E 1 ,...,E n ) ,(B2)where E i is the population average activity of the enzymei.The population average flux is J E[f(E 1 (j),...,E i (j),...,E n (j))].(B3)K n i11/E i 1 ni1[cv 2i C Ji(1 C i)] J ,(B6)where E 2 iand cv i are, respectively, the variance and thegenetic coefficient of variation of activity of enzyme iin the population, and C iJ is the “populational” fluxcontrol coefficient of enzyme i:C J i 1/E i.j11/E n jKojima (1959) showed that the additive and dominanceeffects of a gene i upon a trait are related, respec-tively, to first and second partial derivatives of the populationmean of the trait with respect to allelic frequency(p i ). The (additive additive) epistasis effect of a pairof QTL (i and j) is related to the mixed partial derivativeof the mean with respect to both allelic frequencies (p iand p j ). Applying these formulas to the flux, we get Jp i(B7)⇒ 2 A i 2p i (1 p i )( i ) 2 (B8) i 1 2 2 Jp 2 i⇒ 2 D i p 2 i (1 p i ) 2 ( i ) 2 2 J(B9)(B10) ij 1 (B11)4 p i p j⇒ AA 2 ij 4p i p j (1 p i )(1 p j )( ij ) 2 . (B12)So following Equation B7, we have, for the flux addi-Some simplification occurs: tive effect of the locus i,E[f(E 1 ,...,E n )] f(E 1 ,...,E n )E[E i E i ] 0.And as there is no linkage disequilibrium in the population,⎡⎢⎢⎢⎢⎣ E i i K p i 1 n[cvj 2 C J (3C Jj j 2)] cvi 2 (3 4C J ) i 2 (C Jj1i )2E 2 ip i 1 C JiE i⎤⎥⎥⎥⎥⎦.(B13)E[(E i E i )(E j E j )] 0From Equation B9 we get the flux dominance effectE[(E i E i ) 2 ] E 2 i. of the locus i,

2012 B. Bost, C. Dillmann and D. de Vienne i K 2 (C J⎡⎢⎢⎢⎢i )2 E ip i 2 ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡⎢3(C Ji )2 n⎢⎢ C Ji n⎢⎢⎣j1 cv2 jj1 cv j2 2 cv2 iE i[6(C J 2 2 E ip 2 i 1 C J iE i E i 2 E ip i p i 6 1 C J iE i 2 2 E ip 2 i [1 n⎤⎤(2C Jj 1)E j⎥⎥ ⎥⎥ ⎥(2 3C JjE ) j⎥ ⎥⎥i )2 8C Ji 3] 1 C ⎥Ji ⎥ ⎥E i ⎦ ⎥⎥.⎥⎥⎥⎥⎥[cv 2 j C J(3C Jj j 2)] cv 2 i(3 4C J)] ⎥i⎥j1⎦(B14)And from Equation B11 we get the epistatic effect ofthe pair of loci i, j, ji K 4 (C Ji C J⎡⎢j )2 E iE 2 jp i p j 3C Jj 1C J (E j j) 2⎢⎢⎢⎢⎢⎢⎢⎢⎣ E ip iE jp j nk1 E jE 2 ip j p i 3C Ji 1C J (E i i) 2i1E6cv 2 (1 2C J )i 2 6 n[cvkC 2 k J (2Ck J 1)]k k1 6cvj 2 (1 2C Jj )⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(B15)In an F 2 population with additive enzymatic loci, thereare some simplifications in Equations B13 to B15:E i m i 2 E i 1 2 a2 iE ip i 2a i 2 E ip i 0 2 E ip 2 i 0 2 E 2 i4ai2p 2 i⎥⎥⎥⎥⎥C Jicv i 1/m i n j1[1/m j ]a im i √2 .Hence an approximation of the average flux in an F 2population is J K n i11/m i 1 ni1[cvi 2 C i(1 J C i)] J .(B16)An approximation of the additive effect of the QTL iupon the flux is 2 A i 2 E i(C Ji) 4 K 2 1 3cv2 i (C J i 1) 2 [cvj 2 C j(3C J Jj 2)]ji2. (B18)The dominance effect of the QTL i is⎡⎢⎢⎢⎢⎢⎢⎣ 3(C J i 2Kai 2 (C Ji )2i) 2 j (2C Jji cv2 j 1)m j C J ji (2 3C Jji cv2 m ) jj 1 [6cvi 2 (C Ji 1) 3 C Ji 1]m i⎤⎥⎥⎥⎥⎥⎥⎦(B19)and the contribution of the QTL i to the flux dominancevariance is 2 D i ( 2 E i) 2 (C J⎡⎢⎢⎢i) 4 K 2 ⎢⎢⎢⎣3(Ci J ) 2 j (2Cji cv2 jJ 1)m j C i J j (2 3Cji cv2 j J )m j 1 [6cv 2 i (C iJ 1) 3 C i J 1]m i⎤⎥⎥⎥⎥⎥⎥⎦2.(B20)The epistatic effect of the pair of loci i, j (i j) is ji Ka i a j (C iC J Jj ) 2 n i Ka i (C i) J 2 1 3cv2 i (C i J 1) 2 with ∀i, AA 2 ii 0.k11m k⎡⎢⎢⎣3cv 2 i (1 2C Ji)(1 C Ji) 2 3 [cv 2 kC k(2C J J 1)]kki,j⎤⎥⎥⎦ 3cv 2 j (1 2C J )(1 C J )j j(B21)and the contribution of the QTL i and j (i j) totheflux epistatic variance is⎡3cv ⎢⎢ 2 i (1 2C J )(1 C J ) ⎤i i2⎥⎥ 2 AA ij 2 E i 2 E j(C i J C Jj )4 K 2 2 3 [cv 2 kC J k⎢(2C J 1)]kki,j⎥⎣ 3cv 2 j (1 2C J )(1 C J ) ⎦j j(B22) [cvj 2 C j(3C JjJ 2)]ji (B17)and the contribution of the QTL i to the flux additivevariance isThe R 2 of the QTL i is calculated as described inEquation 3 (methods), withA 2 nA 2 i, D 2 nD 2 i, AA 2 ni1i1i1j i 2 AA ij.