The Inheritance of Metabolic Flux: Expressions for the ... - Europe PMC

Copyright 0 1990 by the Genetics Societyof America

The Inheritance of Metabolic Flux: Expressions for the Within-Sibship Mean and Variance Given the Parental Genotypes Patrick J. Ward Department of Biostatistics, Fred Hutchinson Cancer Research Center, Seattle, Washington 98104 Manuscript received September 1 1 , 1989 Accepted for publicationMarch 29, 1990 ABSTRACT Recent developments have related quantitative trait expression to metabolic flux. The present paper investigates some implications of this for statistical aspects of polygenic inheritance. Expressions are derived for the within-sibship genetic mean and genetic variance of metabolic flux given a pair of parental, diploid, n-locus genotypes. These are exact and hold for arbitrary numbers of gene loci, arbitrary allelic valuesat each locus, and for arbitrary recombination fractions between adjacent gene loci. The within-sibship, genetic variance is seen to be simply a measure of parental heterozygosity plus a measure of the degree of linkage coupling withinthe parental genotypes. Approximationsare given for the within-sibship phenotypic mean and variance of metabolic flux. These resultsare applied to the problem of attaining adequate statistical power in a testofassociationbetweenallozymic variation and inter-individual variation in metabolic flux. Simulations indicate that statistical power canbe greatlyincreased by augmentingthedata with predictionsandobservationsonprogeny statistics in relation to parental allozyme genotypes. Adequate power may thus be attainable atsmall sample sizes, and when allozymic variation is scored at a only small fraction of the total set of loci whose catalytic products determine theflux.

L

(1974, Chapter 1) discussed the dichotomy of “genotype” us. “phenotype,” and the apparently disparate, mathematical formulations of population us. quantitative genetics. This dichotomy partlyreflected the lack ofa set of mathematical, epigenetic rules whereby genotypes couldbe mapped to phenotypes. T h e recent realization that some phenotypic values are relatable to biochemical fluxes (KACSER and BURNS 1973, 1979, 1981; KEIGHTLEY and KACSER 1987) heralds promise for a solution of this disparity. According to this view, a phenotypic value is a function of the flux through a biochemical network underlying the expression of the trait. T h e phenotypic value may correspond fairly directly to a metabolic flux(e.g., as for active responses to environmental stimuli, such as those expressing poweroutput) but more generally would be a function of flux (such as the integralof flux over time,e.g., as in the amount of a pigment,or the weight of a keratinous structure, etc.). Metabolic fluxes are afundamental level of quantitative, polygenic expression and, in principle, are mapped to the genetic information carried on the chromosomes via the kinetic properties of the gene products catalysing the biochemical network. In the special case of a linear chain of unsaturated, monomolecular, reversible reactions at steady state, and in the absence of feedback inhibition and other non-linearities, metabolic flux has been shown to be inversely proportionaltothe sum of contributions across the n gene products catalyzing the pathway, EWONTIN

Genetics 1 2 5 655-667 (July, 1990)

according to the formula

J=”-

E

(1)

C gi i= 1 (KACSER and BURNS1973, 1981; HEINRICH and RA1974). Here] represents metabolic flux,g, = (M&-l)/Vi where Mi and Vi are, respectively, the Michaelis constant and the maximal velocity (V,,,,,) for the forward reaction stepcatalyzed by the product of the ith gene locus, K+1 is the chemical equilibrium POPORT

constant for the reaction up to andexcluding the ith step in the pathway, and E is the initial metabolite concentration minus the final metabolite concentration divided by the chemical equilibrium constant for the entire pathway (KACSER and BURNS 1973; KEIGHTLEYand KACSER 1987). This relatively simple mathematicalformulation (and its associated derivations) has shown a remarkable ability to encompass fundamental genetic properties. These include dominance (KACSER and BURNS 1981), epistasis (KACSER and BURNS 1979, 1981; DYKHUIZEN,DEAN and HARTL 1987; KEIGHTLEY 1989),pleiotropy (KEIGHTLEYand KACSER 1987),and the apparent selective neutrality of much allozymic variation (KACSER and BURNS1981; HARTL,DYKHUIZEN and DEAN1985; HARTL1989). Equation 1 also holds much of interest for the quantitativegeneticist, in helping to rationalize some basic, quantitative genetics concepts. Firstly, variations between individuals

656

P. J. Ward

in the g,reflect genetic variations in enzyme activity, measured as Vi/M,. The denominator of (1) thus defines a numerical value taken across the n gene loci which quantifies the direct contribution toJ from the genetic information carried at those loci. It thereby offers a rationale for the traditional, quantitative genetic concepts of genotypic, zygotic, gametic and allelic “value” (FALCONER 1960; GIMELFARB 1982), and also some justification for assuming additivity of such values, within and across loci, to produce a genotypic value, as often used in models of polygenic inheritance.This follows not only fromthe fact thatthe denominator of (1) is a summation across gene loci, but also from the empirical findingthat levels of catalytic activity in heterozygotes are usually midway between those of the two corresponding homozygotes (HARRIS1975; KACSERand BURNS1981; MIDDLETON and KACSER 1983; STANBURY et al. 1983), consistent with an additivity of allelic values within a locus. In contrast, the values assigned to polygenes in models of quantitativeinheritance, andthe assumption of their additivity across loci toproducea genotypic value, rarely have any obvious meaning at the molecular level. Secondly, inmany situations, Equation 1 will partition most of the genetic and nongenetic components of J across denominator and numerator terms respectively. The denominator encompasses the direct contribution to J from the information carried at the n gene loci, and hence must often contain a large part of the genetic component of J . Conversely, the numeratorencompasses the contribution toJ from factors external to that genetic information, and this would include most of the nongenetic component of J . This is consistent with the numerator being a function only of the difference between initial and final metabolite pool sizes. Environmental effects on J , as experienced within an individual’s lifetime, will often be filtered (WATT 1985) into metabolite pool sizes (KACSERand BURNS1981; KEIGHTLEY1989). E will typically also include background genetic effects, so that Equation 1 can be seen as paralleling the common, traditional practice of partitioning phenotypic value (metabolic flux in this case) into an additive genetic component versus an “all the rest” component (FALCONER1960),thelatterincorporating background genetic effects and environmental effects. The partitioning here, however, is in the form of a ratio E/G rather than the usual sum G + E . A study of the inheritance of metabolic flux might therefore help clarify the theoretical basis of quantitative genetics, placing it within a more precise, molecular framework. Despite its obvious merits, however, the significance of (1) for quantitative genetics theoryper se has been little appreciated [but see BEAUMONT (1988) and KEIGHTLEY (1989)], probably because the focus of development so far has not been

the relevance of the approach for the statistical properties of groups of individuals. Its relevance to this latter domain, however, may be great. As a simple illustration,Equation1 can be viewed as relating metabolic flux to a ratio of random variables. This indicates that the variance of flux, VJ,within a population should partition to a close order of approximation (i.e., ignoring the contributions from central moments of order higher than 2) as:

(MOOD,GRAYBILL and BOES1974). The subscripts E and G here refer to contributions from numerator and denominator terms respectively of (1); p E and pG being their mean values within the population, and V, and VC their variances. Equation 2 indicates clearly that, within a group of individuals, a change in the average size of the external contribution to J (corresponding to a change in p E ) should alter the extent to which VJ is dominated by the Vc (additive genetic) component. This prediction is indeed been borne out in numerous case studies where the genetic component of phenotypic variance has changed (usually increased) with the extremity of environmental conditions under which the trait is expressed (LANGRIDGE and GRIFFING1959; PANI and LASLEY1972; DERR 1980; BELYAEVand BORODIN1982; SCHNEEand THOMPSON 1984a; MARTINet al. 1985; WARD 1985; PARSONS1987). It may also explain why the phenotypic expression of some genetic variants depends on environmentalconditions (PANI and LASLEY1972; SUZUKIet al. 1976; DYKHUIZENand HARTL 1983; SCHNEEand THOMPSON 1984b). Expression (2) can be contrasted with the usual formula V p= VE Vc for the phenotypic variance (FALCONER1960), which is based on the supposition that phenotypic value is the sum of genetic and external components rather than their ratio. The present paper investigates statistical aspects of parenttooffspring transmission of metabolic flux, when the flux is as given by (1). The findings will be directly applicable to the inheritance of some traits, and may serve as a reasonable approximationin other cases. A common feature of the expressions derived for metabolic flux so far [ie., for the simple linear pathway (KACSERand BURNS 1981)andforthe branched pathway (KEIGHTLEY1989)] has been the appearance of “group enzyme activity” terms [as defined by KEIGHTLEY and KACSER (1987)l. A group enzyme activity is the sum of g values across a linear section of the pathway under consideration [e.g., as appears in the denominator of (l)].Exact expressions are derived here for two basic quantitative genetics statistics: the within-sibship mean and variance of g,(i.e., within-sibship p c and Vc) given a pair of

+

657

Inheritance of Metabolic Flux

parental, n-locus genotypes. Approximations are then given forthe within-sibship phenotypic mean and variance of J . It is shown that statistical power for detecting an association between allozymic variations and inter-individual variations in J , when J is as given by ( l ) , will be greatly improved if the study incorporates an analysis of progeny statistics resulting from matings amongthe individuals under study. Some single-locus associations with metabolic flux might therefore be detectable using this approach. WITHIN-SIBSHIP,GENETICMEANAND VARIANCEGIVENTHEPARENTALGENOTYPES

Consider the parent to offspring transmission of a group enzyme activity G = CY='=gi, l hereafter referred to simply as a genotypic value. Despite the relatively simple additive structure, no general, exactexpression yet exists for the within-sibship variance of such a quantity, given the parental genotypes, which is not unduly restrictive in terms of allelic values and/or recombination fractions [see GIMELFARB (1 982) for some approximations]. Expressions are obtained here for the within-sibship mean and variance of genotypic value, as generated through meiotic recombination of a given pair of diploid, parental, n-locus genotypes. These expressions are exact and hold for any number of gene loci, for arbitrary allelic values at any locus, and for arbitrary recombination fractions among the gene loci. In the following, a gametic value is taken to be the sum of allelic values carried by that gamete across the gene loci whose catalytic products contribute to the metabolic flux of interest. At fertilization, a pair of gametic values sum toproducea genotypic value. Note that this additivity of allelic values, within and across loci, todeterminean individual's genotypic value is not to be confused with any non-linear, relationship which may subsequently exist between that genotypic value and the individual's metabolic flux, e.g., as in the inverse relationship given by (1). Consider n polygene loci situatedalonga set of chromosomes. An individual carriesa homologous pair of each chromosome. At the ith chromosomal locus within this individual, the paternally derived strand carries an allele of value xi and the maternally derived strand carries anallele of value yi, with xi + yi = g,. The recombination fraction between the ith and the (i 1)th loci is r,, where 0 < ri < 0.5. This genotypic model will be referred to for convenience as a "polygenotype." Let G = parental genotypic value

+

irk:k = 1, . . ., n - 11, we seek information about the probability function of z, denoted q z ) . 9 ( z ) is a discrete distribution and, due to the requirement of arbitrariness in the allelic values, its domain is nonlattzce, i.e., the sample space of z is not confined to any regularly spaced or orderly set of points along the real number line. This need to allow for allelic values of arbitrary, real, positive value, as opposed to the usual assumption that all alleles have equal absolute value, and/or the restriction of such values to integers such as - 1, 0 , or 1, presents the major mathematical challenge since the lack of spatial regularity in the sample space of z renders it difficult to write down a closed form expression for qz). One way of circumventing this problem is to deal instead with the probability generating function of z, defined as

F(s) =

9(z)s2. z

This function is marginal to the sample space of z, being a function rather of the continuous variable s, but the entire sample space of z is nevertheless contained implicitly within it. The variable s is simply a dummy variable enabling manipulation of the generatingfunction toextractinformation on qz). In particular, the moments of z are contained in the derivatives of F(s) at the point s = 1 [see FELLER (1968, Chapter XI)]. Solutions are obtained here for the central moments of z by manipulating relations obeyed by F(s). These relations follow from basic principles of Mendelian transmission, which of course hold independently of the sample space of z. One such basic principle is that meiotic recombination and segregation lead to the production of gametes in complementary pairs. For example, if an individual's two-locus genotype is Ala B/b, and thatindividual has producedagamete Ab then it must also have produced the complementary gamete aB. Such complementary gametes share an equal probability. Thus, for each gamete of value z the value of its complementary gamete is G - z, and these share an equal probability. This symmetry within complementary pairs of gametes stems from the basic fact that, for any locus i, a member of such a pair carries either xi or yi with equal probability (=1/2), which is of course Mendel's first law. Anylocus could bechosen to define this latter effect, so consider the probability generating function of z conditional on the gamete carrying x, for some arbitraryj, denotedf(s). Then

n

=

C1

(xi + Yi)

(3)

I=

and let z = value of a gamete producedby that parent. Given n, and the sequences (xi,yi: i = 1, . . ., n ) and

Equation 4 expresses the production of gametes in complementary pairs. Differentiating (4) with respect to s, and putting s = 1 yields the expected gametic

658

P. J. Ward

value as

Rearranging yields

1 p ( r ) = F’(1) = -G. (5) 2 T h e form of (4)also indicates that the distribution of z is symmetrical about (1/2)G, and hencethat all higher order, odd, central moments of the gametic value distribution are zero. Taking the second derivative of (4) with respect to s and setting s = 1 yields 1 (6) 2 It is easily shown [e.g., FELLER(1968, Chapter XI)] that V(z) = F”(1) F’(1) - [F’(l)]’ which, by substitution of expressions (5) and (6),yields F”(1) =f”(l) - (C

1 - l)f’(l) + -G2

2

- -G.

+

V ( r ) =f”(l)

1 - (G - l)f’(l) + -G2. 4

C 9 ( z , e)sY z

e

where 9 ( z , e ) is conditional on the gamete carrying xi. [Note thatf(s) can be recovered by setting t = 1 in the solution for $(s, t ) ; i.e., f(s) = +(s, l).] That part of $,(s, t ) dueto even numbers of recombination events is (1/2)[$,(s, t ) + qi(s, -t)] and that part due to odd numbers of recombination events is (1/2)[$i(s, t ) - Icll(s, -t)]. Thus, assuming independence of recombination events between chromosomal regions (e.g., no cross-over interference between regions), the recursion of $ across consecutive loci is 1 $,+,(s, t ) =-[$i(s, t ) + ql(s, -t)][(l - T ~ ) s ~ ’ ++ ’ ritsYt+‘]

2

1 + -[&(s, t ) - $,(s, 2

+

+ ql(s, -t)(l

“t)][r,tsx~+1 (1 - ri)sYt+l].

+ ~ ~ t ) ( s ~+%sYt+~) +l

- ri - rrt)(sX1+l- s”+l).

(8)

Solutions forf”( 1) andf’( 1) are obtained from (8) by setting either t = 1 or t = - 1 , taking derivatives with respect to s, setting s = 1 , and solving the various difference equations thereby formulated.This partof the derivation is given in the Appendix. The resultant expressions forf”(1) andf’(1) are then inserted into (7), yielding V(Z) =

1 ” 41,,

-X

(Xi

- Ji)2

+ I-

1 ”

(7)

Solutions forf’( 1) andf”( 1) are obtainedby incorporating a surprisingly minimal amount of information concerningthe probability distribution of meiotic, recombination events. In what follows, a “recombination event” applies to chromosomally adjacent loci within the n-locus set, and has occurred if the paternal pair of genes move to opposite poles at thefirst meiotic division. (For linked loci, for example, this would be effected by theoccurrence of anoddnumber of chiasmata between the loci.) In general, if a gamete carries xj, it will also carry xi (as opposed to yi) only if there has been an even number of recombination events between thejth and the ithloci, and conversely it will carry y, (as opposed to xi) only if there has been an odd number of recombination events between the jth and theith loci. Recognition of this second Mendelian principle is sufficient to determine the variance of r (andindeed all its centralmoments). Let e be the number of recombination events occurring between the ith and thejth loci. Define the bivariate probability generating function

Gi(s, t ) =

2$i+l(s, t ) = $i(s, t)(l - r,

2i=2

(9)

i- 1

1

- ri)C(xj - r j )k=jn ( l j= 1

-

2Tk).

This equation for the gametic variance has an easy interpretation. Let II

H =

(X,

- yJ2

i= 1

and i- 1

n

C=

2 (X, - ~

i=2

i ) (xj j= 1

1-

I

- rj)n( 1 - 2rk) k=j

so that 1 V(Z) = -H 4

1 + -C. 2

(10)

The H component is a type of distance statistic between the maternally and paternally derived chromosomes, and can be broadly interpreted as the heterozygosity of that parent with respect to the polygenic trait under investigation. Thus, for a model in which each allelomorph has a value of either “0”or “1” (as inmany polygenic models), it is simply the number of heterozygous loci carried by that parent. The C component expresses the degree of linkage “coupling” of alleles of larger effect among these heterozygous loci. Thus, it has a maximum value when each difference ( x i - ri) has the same sign for all i, and when r, < 0.5 for all i ( i e . , when the allelomorphs of largest value at each locus are all on the same chromosomal strand), and is zero under free recombination (if?.,when ri = 0.5 for all i). Where the loci are spread across more than one homologous chromosome pair, those ri corresponding to the interchromosomal regions are simply equal to 0.5. This effectively splits the C expression into separate parts added together, consistent with the idea of statistically independent contributions to the gametic variance from the different homologouspairs. Equation 9 can equivalently be expressed in matrix

Inheritance of Metabolic Flux

659

tively, and the distribution of metabolic flux among their progeny is the distribution of a random ratio J = E / G for which the distributions of both numerator and denominator areconditional on the parental vectors. The following approximation holds forthe within-sibship, mean metabolic flux, provided higher order, central moments in the numerator and denominator variables are small:

notation as

with

and

where Pij

= Pj,i

= segregational correlationbetween ith andjth loci j- 1

= H(l

-2 4

k=i

Given expressions ( 5 ) and ( 1 0) for thegametic mean and variance, it is relatively easy tocomputethe within-sibship mean and variance of genotypic values resulting from a mating between two individuals of known polygenotype. Let values for G, H , and C be determined separately on each parent,and signify these by (GI, HI, C , ) and (G2, H Z , C2) respectively. Assume the gametes from these parents pair at random in the formation of progeny zygotes, and that each such pair of gametic values are additive in determining the progeny genotypic value. T h e expected genotypic value of the resultant sibship is then

and the within-sibship genetic variance is

v

- -(HI 1

+ Hz) + 1

$Cl

+ CP).

G-4 WITHIN-SIBSHIPMEANANDVARIANCE METABOLICFLUX

(MOOD, GRAYBILL and Bo= 1974), where PC and VC are as given by (12) and ( 1 3). The corresponding approximation for the within-sibship variance of metabolic flux has the same form as (2). Note that when applied to the within-sibship statistics in this way, PC and Vc in (2) and (14) are determined by genetic properties of the parents, as given by (5), (lo), (12) and (13), whilst and VE refer tothe mean and variance of E among the sibs. When considering the moments of a random ratio it is important to establish their existence and finiteness (MOOD, GRAYBILL and BOES 1974). It is thus noteworthy that distributional propertiesof G support the existence and finiteness of the moments of J , as approximated by (2) and (14). T h e number of segregating alleles per allozymic locus is typically small in natural populations [Le., in the order of five or less (SINGHand RHOMBERC1987)], so the sample space of G can be regarded as discrete and finite. Hence, all the moments of G will exist and be finite. Furthermore, G is a sum of positive g values and hence its domain is strictly positive. Therefore, J never has a zero in its denominator (except in the trivial case where an individual has null alleles at all of its loci). T h e existence and finiteness of the moments of J therefore depends solely on the corresponding existence andfiniteness of the moments of E . It is also noteworthy that approximations (2) and ( 1 4) are supported by the symmetry of the distribution of G (Equation 4), which renders all higher order, odd, central moments of G zero. These approximations may be weakened, however, by the occurrence of matings for which the within-sibship distribution of genotypic values has a large kurtosis.

OF

Consider two individuals which mate and generate a sibship. Assume J is as given by (1) and consider the distribution of J within the sibship. The form of this distribution is unknown, but measures of E , G, H a n d C (as defined above) taken on each parent suffice to approximate its mean and variance. Let these measures taken on the t w o parents be the vectors ( E l , G I , H I , C,) and (E2, G P , H z , C,) respectively. T h e two parental metabolic fluxes are EI/GI and E2/G2respec-

IMPROVINGSTATISTICAL POWER FOR DETECTINGSINGLELOCUSSELECTION

Laboratoryattempts to statistically relate singlelocus, allozyme variation to interindividual variations in metabolic flux have so far yielded either weak associations or no association, even where the locus is known to be biochemically involved in determining the flux, and where the allozymic variation reflects significant alteration to the catalytic properties of the enzyme (MIDDLETONand KACSER 1983; CLARKE et al.

660

P. J. Ward

1983; WATT 1985); BARNESand LAURIE-AHLBERG 1986;DYKHUIZEN, DEAN and HARTL1987;CARTER and WATT 1988). Theoreticalconsiderations based on (1)have shed light on this apparent lack of association (KACSERand BURNS1979, 198 1; MIDDLETON and KACSER1983; HARTL,DYKHUIZENand DEAN 1985;HARTL1989).T h e inverse relationship of flux to the g values renders flux relatively unaffected by allozymic substitution at any one step (relative, for example, to what would be the case if the relationship was linear), and this buffering increases in strength with the number of catalysed steps in the pathway. This has been related to thewider issue ofempirically identifying fitness differentials due to segregation at a single locus, which are of extremely small magnitude against sampling and experimental error (LEWONTIN 1974; KACSER and BURNS198 1; KOEHN, ZERA and HALL 1983; DYKHUIZEN,DEANand HARTL1987). Given the potential weakness of the associations being sought then, an important consideration is the statistical power of the experimental design and method of statistical analysis employed to detect such associations. The approach developed in this paper can be used to illustrate such low power for detecting theeffect of variation at a single locus on metabolic flux, and also suggests a way that this can be alleviated by progeny testing. The power of statistical tests of association between metabolic flux and allozyme genotype cannot be directly calculated, so simulation will be used. Assume that flux J through a biochemical pathway is as given by (l),and that an experimental population of individuals are each scored for J and for the g values at some subset of the n catalyzed pathway steps. N o information on the E values is obtained. The aim is to detect the statistical association between interindividual variation in J and theg values. A simple way of quantifying the association of J with the g value at the ith locus would be the sample, product-moment correlation between J and l.O/g,.(The exact numerator value is unimportant). Likewise, the sample correlation between J and 1 .O/CZtA gi quantifies the association between J and the multilocus genotype associated with a subset A of pathway steps, consisting of m of the n g values. Statistical power for detecting such an association can be equated to the probability that the sample correlation will be statistically significant, and is anticipated to be a function of the number of individuals in the sample, of the size of m relative to n, and of the true mean and variance of E within the experimental population. Table 1 presents power estimates obtained by computer simulation, for various combinations of these parameters. Clearly, statistical powerfordetecting such within-individual associations is likely to be very low ( i e . , less than ten percent) if interindividual vari-

ation in the unscored E component exists. It is noteworthy that the average value of E has a positive effect on statistical power, so that the association is more likely to bedetectedunderconditionseffectinga greater difference between initial and final pool sizes of the pathway. In practice, allozyme genotypes are usually treated as categorical variables i.e., information on the kinetics of the allozymes at a locus (the g values) is not directly utilized in the statistical analysis. The relative effect of this can only be to decrease statistical power from what is simulated here. Suppose this experimental procedure is augmented with the results of matings, so that four flux measurements are taken on each parental pair; two parental fluxes and a mean and variance of flux within their progeny. Each parent is also test-crossed to yield m locus, allozyme haplotypes underlying their g values. Given estimates of the recombination fractions within the m - 1 chromosomal regions, a score can then be obtained for G , H and C across the m gene loci on each parental individual, using (3)and (10). From Equations 2 and 14, predictors of the within-sibship mean and variance of metabolic flux are 1 .O/pc +VG/ p i and VG/&, respectively, with and VCcalculated from the two parental, m-locus polygenotypes using (1 2) and (1 3). Each mating thus generates eight data items; the fourflux measurements and theirrespective predictors. The statistical association between allozyme genotype and metabolic flux is quantified by the 8 X 8 matrix of correlations among these eight data items, calculated across the experimental population. A test of association between the four flux measurements and their fourcorresponding predictors can be carried outusing the matrix method outlinedin MORRISON (1967,Section 7.5). Table 2 presents power estimates for this latter test of association across various combinations of sample size, n, m, p E , and V E ,as obtained by computer simulation. These results illustrate that augmenting the analysis with data from progeny testing as described, renders statistical power relatively robust to VE.However, even when a sample size of 100 parental pairs are tested, adequate statistical power (of say 80% or more) still requires that g values at 30% or more of the pathway steps be scored,so the detection of singlelocus effects still requires large sample sizes. Nevertheless, the powerfordetecting the association is vastly improved by utilizing the expressions derived in this paper. A combination of progeny testing and a multilocus investigation is clearly the more effective means for detecting associations between metabolic flux and allozyme variation, particularly if larger sample sizes are used than simulated here. Implementing the progeny testing procedure requires prior knowledge of (i) the catalytic activity (gvalue) associated with allozymic

66 1

Inheritance of Metabolic Flux

TABLE I Statistical powerfor detecting a correlation between metabolic flux through an n step pathway and allozyme variation at m of the steps, as obtained by computer simulation

pj

V+

m

1

2

3

1

2

4

6

n

6

6

6

12

12

12

12

Sarnple si/r = 100 individuals

1.0

0.0 9.0 25.0

68.93 (2.98) 5.20 (0.28) 4.70 (0.40)

92.24 (1.94) 5.61 (0.32) 5.40 (0.62)

98.54 (0.67) 5.92 (0.31) 5.46 (0.61)

54.97 (2.86) 4.78 (0.20) 4.82 (0.43)

81.96 (2.42) 5.13 (0.24) 4.90 (0.21)

97.59 (0.87) 4.89 (0.29) 4.62 (0.19)

99.77 (0.16) 4.84 (0.41) 4.46 (0.28)

5.0

0.0 5.93 9.0 25.0

66.83 (3.64) (0.57) 7.15 5.33 (0.53) 72.77 (2.91)

90.41 (2.51) (0.68) 8.33 5.80 (0.58)

83.43 (1.79) (0.23) (0.35) 5.65 4.92 (0.18)

98.23 (0.46) 4.80 (0.19)

99.90 (0.05) 6.30 (0.45) 4.79 (0.30)

94.13 (1.54) (1.32) (0.56) 7.52

57.90 (1.97) 97.97 (1.01) (0.76) 5.05 (0.26) 5.13 5.20 (0.30) 6.18 (0.69) 99.26 (0.48) 54.99 (2.30) 15.40 (1.80) 6.48 (0.36) 6.71 (0.68) 5.15 (0.22)

80.19 (2.34) (0.47) (0.40) 5.51

97.01 (0.72) 8.60 (0.64) 6.19 (0.64)

99.73 (0.10) 10.50 (0.88) 6.97 (0.92)

97.16 (1.03) 5.71 (0.32) 5.43 (0.42)

99.78 (0.14) 5.94 (0.39) 5.44 (0.43)

67.15 (2.68) 4.98 (0.33) 4.68 (0.38)

89.90 (1.62) 5.01 (0.34) 4.99 (0.31)

99.36 (0.22) 4.95 (0.37) 5.24 (0.33)

99.99 (0.01) 4.97 (0.44) 5.20 (0.44)

99.49 (0.37)

65.66 (2.23) (0.28) 5.89 5.15 (0.32)

88.93 (1.56) (0.32) 5.29 (0.23) 88.72 (1.93) 8.05 (0.52) 6.23 (0.33)

99.27 (0.21) 99.99 (0.00) 6.38 (0.42) (0.64) 6.91 5.57 (0.32) 5.89 (0.44)

10.0

0.0 8.40 (0.88) 9.0 11.85 25.0

5.90 (0.57) 6.64

Sample size = 200 individuals 1.0

0.0 9.0 25.0

81.17 (3.68) 5.40 (0.40) 5.76 (0.46)

5.0

0.0 8.03 9.0 25.0

96.36 (1.73) 79.83 (4.12) (0.38) 9.65 (0.61) 11.28 (0.78) 5.57 6.04 (0.40) 5.37 (0.41) 75.30 (3.59) 94.96 (1.83) (1.53) 17.64 (2.81) 9.93 (0.99) 12.47

10.0

0.0 11.63 9.0 (0.64) 7.23 25.0

6.46 (0.49) 99.21 (0.50) 23.47 (3.70) (1.36) 5.48

65.88 (2.75) 6.90 (0.43) (0.32)

99.10 (0.35) 99.96 (0.03) 10.79 (0.99) 13.70 (1.51) 7.41 (0.46) 8.58 (0.62)

Figures are statistical power expressed as a percentage, with standard errors in parentheses. n = number of gene loci (= number of catalyzed steps in the pathway); m = number of gene loci used in prediction of flux; sample size = number of polygenotypes per simulated population; p+. = mean E value within each simulated population; V+,= variance of E values within each simulated population. To commence each simulation, two allelic values were generated per locus using a uniform ( I , 2) random number generator. T h e range ( I , 2) of the allelic value distribution is unimportant since the product-moment correlationis unaffected by scale. Simulated populations of polygenotypes were then constructed by generating pairs of integers using a uniform (0, 2“ - 1) random number generator, and using the pairwise bit sequences t o define polygenotypes; a binary digit “1” indicating one of the two possible allelic values is present at that locus, and “0” indicating the other allelic value is present. Successive simulated populations were generated by such random sampling ona common basis of allelic values. Linkage disequilibrium was not modeled, andso any lack of linkage equilibrium within a simulated population of genotypes is due tosampling error only. An E value was assigned to each genotype at randomusing an N(p+:,V+.)random number generator and an “observed”flux then calculated for each genotype using (1). Predictors of each flux were calculated using m-locus subsets of the n-locus genotype (see text), and a sample correlation between observed and predicted was calculated across the entire simulated population. For each m there are

(3

distinct

m-locus subsets of n loci. A satnple correlation between observed and predictedwas computed and tested forsignificance separately for each of these distinct m-locus subsets, across each simulated population of genotypes.T h e results were then pooled to give a single power estimate at that value of m. Each estimate of statistical power is obtained across 50 such replicate, simulated populations, as the frequency of sample correlations significant at the 0.05 level. A new cycle is then commenced with the generation of a new set of allelic values, and a second lot of 50 simulations performed, producing a second power estimate, andso on. Each power estimate given in the table is the mean of ten such replicate power estimates, and the standard errors are the sample standard errorscalculated across these ten replicates.

variants, (ii) parental haplotypes (e.g., as obtained from test crossing), and (iii) recombinationfractions between allozyme loci. Such information is readily obtainable for some laboratory organisms (e.g., Drosophila). Care would be required to randomize the progeny environment relative to the parental environment, thus minimizing correlations between parents and offspring in environmental effects (see FALCONER 1960). DISCUSSION

Equation 1 maps the information carried at single, gene loci to the metric value of a polygenic trait ( i e . , metabolic flux). Equations 2, 12, 13 and 14 map this information, carriedwithin parental genotypes, to the

mean and variance of this trait within a sibship. However, a numberof restrictive biochemical assumptions underlie the derivation of (l), and (2)and (14) are approximations only, ignoringany contributions from third and higher-order central moments in the numerator and denominator of (1). Some consideration must therefore be given to the validity of these assumptions, and to the extent to which these expressions apply to real traits. The assumption of unsaturated catalysis, underlying the derivation of (l), will hold in considerable generality. It is consistent with empirical findings (LOWRY and PASSONNEAU 1964; GARFINKEL and HESS 1964; held beliefs CLARKEet al. 1983)and withwidely concerning in vivo metaboliteconcentrations (CORNISH-BOWDEN 1976,1987; KACSER 1987).Natural

662

P. J. Ward TABLE 2

Statistical power for detecting a correlation between metabolic flux through ann step pathway and allozyme variationat m of the steps, as obtainedby computer simulation, when the testof association utilizesdata from progenytesting ~~

2 PP:

VP

m

1

n

6

3

1

2

4

6

6

12

12

12

~

6

Sample size = 50 mating pairs 1.0

0.0 9.0 25.0

51.80 (2.65) 38.97 (1.27) 36.30 (1.21)

80.61 (2.94) 72.31 (2.27) 64.60 (2.54)

93.81 (2.04) 90.13 (2.12) 83.81 (2.72)

32.57 (1.30) 21.27 (0.46) 21.85 (0.82)

56.19 (2.51) 40.14 (0.56) 39.34 (0.86)

85.92 (2.55) 73.11 (1.29) 74.15 (1.65)

96.59 (1.15) 91.88 (1.08) 92.49 (1.38)

5.0

0.0 9.0 25.0

48.13 (2.85) 41.80 (1.01) 38.00 (2.32)

76.65 (3.39) 73.59 (0.81) 63.71 (3.63)

91.75 (2.42) 91.85 (0.71) 82.42 (3.85)

32.38 (1.07) 22.38 (0.66) 21.23 (0.75)

55.42 (1.55) 39.43 (1.04) 39.77 (0.75)

85.09 (1.86) 71.10 (1.72) 69.20 (3.14)

96.57 (1.04) 90.36 (1.35) 87.94 (3.18)

10.0

0.0 9.0 25.0

48.93 (3.40) 38.37 (3.21) 39.03 (1.74)

77.47 (3.39) 62.45 (2.86) 65.01 (1.57)

91.79 (1.96) 81.38 (2.95) 83.94 (1.52)

33.22 (0.84) 21.81 (0.69) 2 1.43 (0.49)

59.97 ( I .49) 39.95 (0.64) 39.71 (0.71)

89.57 (1.58) 73.75 (1.12) 71.10 (1.56)

98.29 (0.63) 92.51 (0.85) 90.53 (1.33)

Sample size = 100 mating pairs

1.0

0.0 9.0 25.0

60.10 (3.38) 51.30 (3.37) 47.57 (2.20)

86.88 (2.82) 79.89 (3.41) 74.08 (2.22)

96.76 (1.23) 93.49 (1.85) 90.63 (1.50)

49.15 (1.03) 35.90 (1.02) 33.92(1.11)

76.94 (1.24) 62.03 (1.78) 59.33 (1.74)

97.17 (0.46) 90.47 (1.45) 88.56 (1.66)

99.86 (0.05) 98.50 (0.50) 97.81 (0.65)

5.0

0.0 9.0 25.0

67.97 (3.50) 53.57 (3.62) 54.07 (2.27)

91.08 (2.61) 81.55 (4.29) 83.67 (2.39)

98.30 (1.23) 93.70 (3.38) 95.97 (1.12)

44.68 (1.48) 38.07 (0.94) 34.13 (1.16)

72.87 (2.48) 63.72 (1.43) 58.94 (2.30)

94.97 (1.64) 91.71 (0.98) 87.61 (2.08)

99.24 (0.56) 98.99 (0.22) 97.46 (0.98)

10.0

0.0 9.0 25.0

64.53 (3.57) 52.40 (2.46) 5 1.27 (3.79)

90.57 (2.46) 80.71 (2.36) 80.45 (3.51)

97.90 (0.96) 93.93 (1.35) 93.33 (2.01)

49.83 (1.98) 35.94 (1.17) 36.95 (1.28)

77.00 (2.39) 62.59 (1.80) 62.40 (1.83)

96.48 (0.90) 91.49 (1.31) 90.63 (1.36)

99.69 (0.12) 98.84 (0.38) 98.59 (0.35)

~

~

~~

~~

~~

Figures are statistical power expressed as a percentage, with standard errors in parentheses. p E = mean E value for parental generation; V); = variance of E values for parental generation. The simulation procedure used to generate Table 1 was extended in the following manner: At the generation of the basic set of two allelic values per locus, n - 1 recombination fractions were also generated using a uniform (0, 0.5) random number generator. Within a population, the polygenotypes were associated at random into simulated parental pairs. For each pair, the “observed” parental metabolic fluxes were calculated as before, while the “observed” within-sibship mean and variance of metabolic flux generated by the mating were calculated using (12), (13), (14) and (2). For any particular sibship, the within-sibship mean value of E was assumed to be (1/2)(EI E 2 ) , where E1 and E2 are the particular E values for the parents, the latter values being obtained as before by random sampling on a N(p,.., VR:)distribution, and thewithin-sibship variance of E was set to (1/2)V#:.The latter assumes random mating and and CAVALLI-SFORZA 1977; RICE,CLONINCER and REICH 1978). Predictors of these an arbitrary but constant V, for all sibships (seeFELDMAN four flux measurements based on m-locus subsets of the parental, n-locus genotypes were also calculated (see text), and a test of statistical association between observed and predicted was then carried out using the 8 X 8 matrix of correlations among these data items, calculated across the entire simulated population of parental pairs, and the statistical method outlined in MORRISON (1967, Section 7.5). Power is estimated as the frequency of such associations which were significant at the 0.05 level, across 50 simulated populations. As in Table 1, each power estimate is the mean often, independent, power estimates, with a sample standard error calculated across these ten estimates.

+

selection must necessarily have evolved catalysts which effect high metabolic fluxes, thus holding intermediary substrate concentrations at very low levels. Were this not thecase, the thousands of known intermediary metabolites would easily exceed the solvent capacity of the cell (ATKINSON1969; CLARKet al. 1983). In addition, many such metabolites, being highly reactive and unstable compounds, would undergoreactions unrelated to metabolism and thus generate toxic effects if not kept at extremely low, steady-state concentrations (CLARKEet al. 1983). CORNISH-BOWDEN (1976) showed that, where natural selection favours high metabolic flux, the selective advantage goes to enzymes which hold their substrates well below saturation levels. The derivation of (1) (HEINRICHand RAPOPORT 1974; KACSERand BURNS1981; KEIGHTLEY and KACSER 1987) assumes a linear chain of monomolecular,

reversible reactions, with no feed-back inhibition or other nonlinearities. This limits its scope considerably. For example, if the hypothesized chain of reactions contains side branches,the flux across a branch point is not described by (1). Itis noteworthy, however, that the equations for flux across such a branch point share important features with (1). For the simple branched pathway analysed by KEIGHTLEYand KACSER(1987) and KEIGHTLEY(1 989), the equation for flux through the “reference” branch, as opposed to the“competing” branch and the “common” branch [see KEIGHTLEY (1989)l is still a ratio like (1) inwhich the internal contribution (group enzyme activity in the reference branch) is confined to the denominator and the external contribution (pool size differences) is confined to the numerator (KEIGHTLEY1989). It differs from (1) most significantly in having group enzyme activities of the competing and common branches in both numerator and denominator.

Inheritance of Metabolic Flux Most biochemical networks, however, incorporate the simple, linear segments assumed in the derivation of (I), and so analysing the mode of inheritance of flux through such a pathway uncovers a component of the inheritance of many phenotypic values. Seen in this context, the utility of the derivations presented here must be measured by their ability to shed light on empirical properties of quantitative inheritance. As noted in the Introduction, Equation 2 provides a simple illustration of how the genetic component of phenotypic variance can change when the trait is expressed under different environmental conditions, as has been empirically demonstrated across a wide variety of species, traits and conditions. In contrast, the traditional additive-dominance model (FALCONER 1960; MATHERand JINKS 1971), based on the supposition that genetic and non-genetic components are additive in their determination of a phenotypic value, is unable to easily explain this phenomenon, and cannot relate the effect to underlying biochemical processes. Equation 14 also accounts for a common result in standard crosses between inbred lines to produce an FI and an F2 generation. The F2 mean sometimes deviates more from the mean of the original, inbred, parental lines than does the F1 mean [e.g., see data in MATHERand JINKS (1971) and NYBERGand BOGAR (1986)]. Such findings are not easily explicable under the traditionaladditive-dominancemodel,but are consistent with Equation 14 which predicts a contribution to the F2 mean from the heterozygosity and the maximal degree of allelic coupling within the F1 polygenotypes. Such a contribution will be absent in the F1mean, reflectingthe homozygosity of the inbred parental lines. Equation 14 predicts further that this difference between the means of the first two filial generations should be more pronounced with smaller PC, e.g., with metabolic pathways manifesting high fluxes, or metabolic pathways of fewer steps. Equation 14 indicates that the within-sibship mean metabolic flux is not necessarily equal to the midparental metabolic flux (1/2)(E1/G1 E2/G2), reflecting the ability of Equation 1 to account for “dominance” as a biochemical system property without any need to explicitly specify parametersfor this effect in the model (KACSERand BURNS1981). Dominance arises directly from the inverse relationship between metabolic flux and its genetic basis (Equation l ) and its existence is thus entirely compatible with an additive model for thedeterminationofthat genetic basis across gene loci, as presented here. Equation 2 indicates that an increase in the genetic mean pc effects a decrease of the within-sibship variance in metabolic flux VJ, and this is also consistent with current theory on the dynamics of metabolic pathways (KACSERand BURNS 1981).The greater the number of catalysed steps there are in a pathway (which will usually cor-

+

663

respond to a greater pc) the greater is the degree of buffering of flux through that pathway to variability in the flux across each step (KACSER and BURNS198 l), corresponding to a decrease of VJ.Interestingly, Equation 2 indicates that parental homozygous loci affect within-sibship VJ via theircontribution to p ~ even , though they make no contribution to within-sibship genetic variance (Equation 9). As noted, Equation 1 partitions,across denominator and numerator respectively, the component of] due to the genetic information carried at the n gene loci from the component due to factors external to that information. The extent to which this corresponds to a partitioning of genetic and nongenetic effects across denominator and numerator of (1) will rarely be absolute. Most metabolic pathways are embedded within other pathways and this must generally effect some backgroundgeneticcomponent to initial and final pool sizes. Furthermore, even when the initial metabolite in the pathway is externally derived, or when the final metabolite is a waste or endproduct, such things asgenetic variation in feeding rate might effect a genetic component to pool size differences. Beyond this, any truly environmental component of intermediary, metabolite pool sizeshaspassed through the organism’s “environmentalfilters” [WATT’S(1985) terminology] and variation in the effectiveness of such filters can itself be partly genetic (WATT 1985). The denominator of ( 1 ) mayalso sometimes contain an environmental component. For example, changes in substrate pool sizes may be associated with changes in cytoplasmic temperature, pH, and osmolarity, which may in turn alter the kinetic properties of enzymes. In short, the view of (1) as partitioning an additive, genetic component in the denominator from “all the rest” in the numerator, the latter incorporating nongenetic effects with background genetic effects, must be seen as a reasonable, theoretical approximation. From a statistical viewpoint, such considerations as the above indicate that variation in the numerator of ( 1 ) cannot strictly be treated as independent of variation in its denominator. To accommodate this, Equations 2 and 14 could be rewritten as

(MOOD, GRAYBILL and BOES1974) [where COVEGsignifies the covariance of E and GI, thus allowing for contributions to V’ from statistical interactions between the numerator and denominator of (l), across members of the population. It remains a problem to expand this covariance in terms of variability in metabolite and enzyme concentrations, enzyme kinetics,

P. J. Ward

664

etc. However, within a sibship, this covariance term must typically be small in support of approximations (2) and (14). The within-sibship distribution of the denominator term in (1) is dominated by the effects of meiotic recombination and segregation of the parental polygenotypes, while the distribution of the numerator term is dominated by subsequent effects occurring within the lifetimes of the sibs, and these two causes of variation in J operate largely independently. Furthermore, within an individual, the denominator of (1) will be robust to variations in metabolite pool sizes (as occur within a sib's lifetime) when profiles of enzyme activity across cellular pH, temperature, osmolarity, etc. are fairly flat around the values encountered for these latter variables in vivo. Such a shape for anenzyme activity profile is typical for such profiles determined in vitro. Equation 1 1relates the gametic variance to a simple quadratic form in the allelic differences, in which the matrix R plays the role of a variance/covariance matrix. Interestingly, the assumption of no recombinational interference between chromosomalregions, which led to Equation 8, has reflected in the elements of R. The (i, j)th element of R is the segregational correlation pij between the ith and j t h loci, and is simply the product of the segregational correlations across successive chromosomal regions, as expected under the independence assumption. The fact that the elements of R turned out to be the segregational correlations suggests that other more general expressions for the gametic variance might be obtained by substituting more general expressions for p I J directly into R (e.g., ones incorporating parameters for recombinational interference). Following this hunch, it turns o u t that Equation 11 holds true also for arbitrary ~ i , j , a proof of which will be given in a subsequent publication. An expression for p i j can be easily extracted from any mapping function. Such functions take the general form r,j = (1/2)[ 1.0 - h(dij)], where rij is the recombination fraction between the ith and jth loci and h some function of map distance dij. Without placing any restrictions on pill, it is easily shown that r,,] is always related to p I j according to r i j = (1/2)( 1.O - pi;), i.e., p i j = h(dij).For example, wherethe recombination fractions depend not only on the distances between loci, but also on thedistances of the loci from the centromere, one might adaptOWEN'S (1950) mapping function and use p I j = e - 2 L ~ [ e ' ~cos

2(t1 - tj)

+ e-'$

sin 2(t, - ti)]

where ti is a measure of the distance of the ith locus from the centromere. Other mapping functions proposed in the literature (e.g., COBBS1978; STAM1979; SNOW 1979)might similarly be adapted to provide a choice of p t j suitable in a particular case. Equation 13 provides an exact, general expression for the within-sibship variance under additive inher-

itance, which is free of assumptions concerning allelic values and recombination fractions. The previous lack of such a simple, general expression for the withinsibship genetic variance has necessitated the use of some roughapproximations in models of additive, polygenic inheritance. One such practice has been to assign an identical variance to every sibship within a population, equal to one half the genetic variance of theentireparentalgeneration (see, e.g., CAVALLISFORZA and FELDMAN 1976; GIMELFARB 1983; RocERS 1983; ELSTON1986). Equation 13 has a simple structureandhence may prove useful in future models. Equation 10 clearly illustrates the susceptibility of genetic variance to evolutionary change via changes in degrees of allelic coupling amongst polygenic loci (MATHER1943; LEWONTIN 1974) as an alternative to changes in heterozygosity and/or levels of polymorphism. LEWONTIN (1974, Chapter 6) studied results of computer simulations of polygenic inheritance under a heterotic selection regime, and concluded that the number of parameters needed to account for the genetic variance of a polygenic trait approaches t v o as the number of gene loci increases, namely the maximum possible inbreeding depression andthe maplength. Consistent with this result, Equation 9 illustrates that genetic variance, as generated by meiotic recombination and segregation consists of two components, one reflecting heterozygosity and the other sensitive to the strength of linkage among the polygenic loci. The derivation presented here, however, does not depend on arguments about selection. The interpretation of phenotypic value in terms of metabolic flux (KACSERand BURNS1973,1979,1981) represents a significant conceptual step forward, the full potential ofwhich is yet to berealized. Based firmly on molecular considerations, the approach not only opens the way for a molecular account of quantitative inheritance, butalso provides the long awaited theoretical link between genotype andphenotype, thereby conceptually unifying the realms of population and quantitative genetics. Further exploration of its implications for statistical aspects of quantitative character dynamics is therefore warranted. Thanks to STEVESELF,ARY HOFFMANN and an anonymous referee for helpful discussions. This work was supported by National Institutes of Health grant GM24472.

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and metabolic structure. Genetics 117: 3 19-329. KOEHN, R.K., A. J. ZERA and J. G . HALL, 1983 Enzyme polymorphism and natural selection, pp. 1 15-1 36 in Evolution of Genes andProteins, edited by M. NEI and R. K. KOEHN.Sinauer Associates, Sunderland, Mass. J , , and B. GRIFFING, 1959 A study of high temperaLANGRIDCE, ture lesions in Arabidopsis thaliana. Aust. J. Biol. Sci. 12: I 17135. LEWONTIN,R. C., 1974 TheGeneticBasis of Evolutionary Change. Columbia University Press, New York. 1964 T h e relationships LOWRY,0. H.,and J. V. PASSONEAU, between substrates and enzymes of glycolysis in brain. J . Biol. Chem. 239: 31-42. J. B. GIBSON,G. A. STARMER, J. MARTIN,N. G., J. G. OAKESHOTT. PERL and A. V. WILKS, 1985 A twin study of psychomotor and physiological responses to an acute doseof alcohol. Behav. Genet. 15: 305-347. MATHER,K., 1943 Polygenic inheritanceandnatural selection. Biol. Rev. 18: 32-64. MATHER, K., andJ. L. JINKS, 1971 BiometricalGenetics, Ed. 2. Chapman & Hall, London. MIDDLETON, R. J., and H. KACSER,1983 Enzyme variation, metabolicflux and fitness: alcohol dehydrogenase in Drosophila melanogaster. Genetics 105: 633-650. MOOD,A. M., F. A. GRAYBILL and D. C. BOES, 1974 Introduction to the Theory of Statistics, Ed. 3. McGraw-Hill, New York. MORRISON,D. F., 1967 MultivariateStatisticalMethods, Ed. 2. McGraw-Hill, New York. NYBERG, D., and A. E. BOGAR, 1986 Genotypic and subgenotypic variation in heavy-metal tolerance in Paramecium. Am. Nat. 127: 6 15-628. OWEN, A. R. G.,1950 T h e theory ofgenetical recombination. Adv. Genet. 3: 117-157. PANI, S. N.,and J. F. LASLEY,1972Genotype x environment interactions in animals: Theoretical considerations and review findings. Research Bulletin 992, University of Missouri-Columbia, College of Agriculture, Agricultural Experiment Station, directed by ELMERR. KIEHL.University of Missouri-Columbia Press, Columbia, Mo. PARSONS,P. A., 1987 Evolutionary ratesunderenvironmental stress. Evol. Biol. 21: 31 1-347. RICE, J., C. R. CLONINCERand T . REICH,1978 Multifactorial inheritance with cultural transmission and assortative mating. 1. Description and basic properties of the unitary models. Am. J. Hum. Genet. 30: 618-643. ROGERS,A., 1983 Assortative matingandthesegregation variance. Theor. Popul. Biol. 23: 110-1 13. SCHNEE,F. B., and J. N. THOMPSON, JR.,1984aConditional polygenic effects in the sternopleural bristle system of Drosophila melanogaster. Genetics 108: 409-424. SCHNEE,F. B., and J. N. THOMPSON, JR.,1984bConditional neutrality of polygene effects. Evolution 38: 42-46. 1987 A comprehensive study SINGH,R. S., and L. R. RHOMBERG, of genic variation in natural populations of Drosophila melanogaster. 11. Estimatesofheterozygosity andpatterns ofgeographic differentiation. Genetics 117: 255-271. SNOW, R., 1979 Maximumlikelihoodestimation of linkage and interference from tetrad data. Genetics 92: 231-245. STAM,P., 1979 Interference in genetic crossing over and chromosome mapping. Genetics 92: 573-594. J. B., J. B. WYNGAARDEN, D. S. FREDRICKSON, J. L. STANBURY, of GOLDSTEINand M. S. BROWN,1983 TheMetabolicBasis Inherited Disease. McGraw-Hill, New York. SUZUKI, D. T . , T. KAUFMAN, D. FALKand the U. B. C. Drosophila Research Group, 1976 Conditionally expressed mutations in Drosophila melanogaster,pp. 207-263 in The Genetics andBiology of Drosophila, Vol. l a , edited by M. ASHBURNER and E. NovITSKI. Academic Press, London.

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With this set of four initial conditions, the difference equations listed above can besolved. Their general solutions are as follows: n- 1

Communicating editor: A. G . CLARK

$n(1,-1)=

n ( l -2rj) j=1

APPENDIX

Solutions forf'( 1) andf"( 1) are obtained by solving the following fourdifferenceequationsextracted from (8): 1. Putting s = 1 and t = -1 in (8) yields $,+1(1, -1) = h ( l , -1)(1

- 2ri).

2. Substituting t = -1in (S), differentiating with respect to s, setting s = 1 and substituting for $ i ( l , - 1) (see below) yields $:+l(l, -1) = $:(I, -1)(1

- 27-,)

3. Substituting t = 1 in (8), differentiating with respect to s, setting s = 1 and substituting for 1c/i(l, -1) yields

4. Substituting t = 1 in (8), takingthe second derivative with respect to s, putting s = 1 and substituting for $i( 1, - 1) yields f$1(1) =f:'(1) +f:(l)(xi+l

+ $((I,

+ ri+l)

- 2ri)(xi+l - yi.1) + (x?+] + ~21)- (xi+l + yi.1)

2

-1)(1

2

Initial conditions: For convenience, set j = 1 so that the gametes under consideration all carry x]. For i = 1, there is then only one possible outcome; i.e., (z = X ] , c = 0). Hence $l(S,t) = sX'

+ ql(l,t) = 1for

(15) all t.

Taking derivatives of (15 ) with respect to s and putting s = 1 yields ${(lJ)= x1

and $?(l,t) = xl(xI - 1)for

all t.

667

Inheritance of Metabolic Flux k= 3

The cancellations at this step in the derivation make

and

m= 1

j= I