Chapter 2




Wim E. Crusio




Génétique, Neurogénétique et Comportement

UPR 9074 CNRS, Institut de Transgénose

3b, rue de la Férollerie

45071 Orléans Cedex 02, France

Send correspondence and proofs to Dr. Wim E. Crusio at the above address.

Tel: (33) 2 38 25 79 74; Fax: (33) 2 38 25 79 79


It is kindly requested that no changes or "corrections" be made in the manuscript without prior consent of the author. Thank you.



This chapter provides a brief overview of quantitative-genetic theory. Quantitative-genetics provides important tools to help elucidate the genetic underpinnings of behavioral and neural phenotypes. This information can then provide substantial insights into the previous evolutionary history of a phenotype, as well as into brain-behavior relationships.

The most often employed crossbreeding designs are the classical Mendelian cross and the diallel cross. The information rendered by the former is limited to the two parental strains used and cannot be broadly generalized. The principal usefulness of this design is for testing whether a given phenotype is influenced by either one gene or by more genes. The diallel cross renders more generalizable information, the more so if many different strains are used, such as estimates of genetic correlations. To estimate the latter, correlations between inbred strain means may provide a helpful shortcut.

Some commonly encountered mistakes in the interpretation of the results of quantitative-genetic studies are presented and explained.


1.  Introduction

Behavior is an animal's way of interacting with its environment and is therefore a prime target for natural selection. Furthermore, as behavior is the output of an animal's nervous system, this indirectly leads to selection pressures on neuronal structures. In consequence, each species' behavior and nervous system have co-evolved in the context of its natural habitat and can be properly comprehended only when their interrelationships are regarded against that background [4]. To arrive at a profound understanding of neurobehavioral traits, one will therefore have to consider problems of causation. Van Abeelen [37] distinguished between the phenogenetic and the phylogenetic aspects of causation. Both aspects deal with the genetic correlates of neurobehavioral traits, the first in a gene-physiological, the second in an evolutionary sense. In other words, neurobehavioral geneticists try to uncover the physiological pathways underlying the expression of a trait and to evaluate its adaptive value for the organism. As I have shown before [9], quantitative-genetic methods may be employed with profit to address problems related to both aspects of causation.


2.  Phylogenetic Aspects of Causation


Selection pressures mold a population's genetic make-up, which subsequently will show traces of this past selection. Therefore, information about the genetic architecture of neurobehavioral traits might permit us to deduce the probable evolutionary history of these traits [2]. With genetic architecture we mean all information pertaining to the effects of genes influencing a particular phenotype in a given population at a given time, including information concerning the presence and size of certain genetic effects, the number of genetic units involved, etc. Generally, however, information about the presence and nature of dominance suffices [7].

With very few exceptions, natural, non-pathological variation in neurobehavioral phenotypes is polygenically regulated. If dominance is present, we may envisage two different situations: either (uni)directional or ambidirectional dominance. In the first case, dominance acts in the same direction for all genes involved (e.g., for high expression of the trait), whereas in the latter case it acts in one direction for some genes and in the opposite one for others. In its most extreme form, ambidirectional dominance may lead to situations where an F1 hybrid is exactly intermediate between its parents, despite the presence of strong dominance effects.

Mather [29] distinguished between three kinds of selection: stabilizing, directional, and disruptive. Stabilizing selection favors intermediate expression of the phenotype, directional selection favors either high or low expression, whereas with disruptive selection more than one phenotypic optimum exists. Disruptive selection will lead to di- or polymorphisms, which may be in stable equilibrium or may even lead to breeding isolation and incipient speciation [35]. The commonest example of a dimorphism is the existence of two sexes, whereas a possible example of speciation as a consequence of disruptive selection is the explosive adaptive radiation and speciation found among fish species belonging to the family Cichlidae in the great East-African lakes [20].

Selection acts in favor of those genotypes that not only produce the phenotype selected for, but are also capable of producing progeny that differs little from this phenotype. In the end, this results in a population whose mean practically coincides with the optimum. Stabilizing and directional selection have therefore predictably different consequences for the genetic architecture of a trait. Genes for which a dominant allele produces a phenotypic expression opposite to the favored direction will become fixed very rapidly for the recessive allele under directional selection. A similar rapid fixation will then occur for genes for which dominance is absent, i.e., where the heterozygote is intermediate between the two homozygotes. In contrast, selection against recessive alleles is much slower. The result will be that after only a relatively short period of directional selection the first two types of genes will not contribute to the genetic variation within the population anymore. Those genes where the dominant allele produces the favored phenotypical expression will remain genetically polymorphic for a much longer time, conserving genetic variance. Thus, directional selection leads to situations where dominance is directional, in the same direction as the selection. Stabilizing selection leads to situations where dominance is either absent or ambidirectional. Furthermore, directional selection generally results in lower levels of genetic variation than ambidirectional selection does [29]. The genetic architecture of a trait may be uncovered by using appropriate quantitative genetic methods.

2.2.  Theoretical Background

At this point, a brief excursion into the field of quantitative genetics is necessary. For the sake of simplicity, considerations of possible interactions between genes (epistatic interactions) will be omitted from the present treatment. Similarly, we will assume that sex-linked genes as well as pre-and postnatal effects are absent. Pertinent references and more technical details for cases where these simplifying assumptions do not hold true may be found in [7].

Classical Mendelian analysis studies characters influenced by a limited number of genes, two or three at the most, that are easy to separate into discrete phenotypic classes. Many characters, however, show continuous gradations of expression and are not separable into discrete classes (so-called quantitative traits). The simultaneous actions of a large number of genes (polygenes), combined with phenotypic deviations caused by variations in the environment (environmental variance, E), may explain such patterns of phenotypic variation [19]. The genetic contribution to a phenotype can then be divided into two main sources: additive-genetic effects and dominance deviations. In the case of one single gene, with alleles A and a, we may denote the phenotypical values of the three possible genotypes as follows:

AA = m + da                   Aa = m + ha                            aa = m - da                                              (1)
The parameter da is used to represent half the difference between the homozygotes, ha designates the deviation of the heterozygote from the midparental point m. Note that in quantitative genetics capital letters are used to indicate increaser alleles, which are not necessarily also the dominant ones. Hence, da is positive by definition, whereas ha may attain all possible values. If we now consider two inbred strains A and B in a situation where many genes affect the phenotype, we may denote the average phenotype of strain A by
m + S(d+) + S(d-)                                                                                                            (2)
(shortened to m + [d] for ease of representation), where S(d+) indicates the summed effects of those genes that are represented by their increaser alleles and S(d-) indicates the same for decreaser alleles. Parameter m is a constant, reflecting the average environmental effects both strains have in common as well as genetic effects at loci where the strains are fixed for the same alleles. The average phenotype of strain B will then equal m - [d]. Similarly, the phenotypic value of an F1 hybrid between A and B may be written as
m + S(h+) + S(h-)                                                                                                            (3)
(shortened to m + [h]). It must be noted that [h] is the sum of the dominance deviations of many genes. If these effects are balanced in opposite directions, [h] can be low or zero, even with dominance present. The same applies to [d], of course.

Because variations in a phenotype can be thought of as the summed effects of variations in genotype and environment, plus the interaction and covariation between these two factors, we may express the phenotypic variance P of a population as:

P = G + E + G*E + 2cov(g,e)                                                                                                 (4)
In the controlled situation of animal experiments in the laboratory (but not in the field) the covariance between genotype (g) and environment (e) can be minimized. Further, the absence of genotype-environment interaction means that all individuals, regardless of phenotype, will have similar sensitivities to environmental variations. As a result, genetically homogeneous groups (for example, inbred strains and their F1 hybrids) should exhibit similar variances. If such is not the case, the effects of (G*E) may often be removed by choosing an appropriate measurement scale [6], leaving
P = G + E                                                                                                                                (5)
The genetic component of the variance (G) can, of course, be divided into components due to additive-genetic variation (D) and dominance deviations (H).

We may demonstrate the partitioning of genetic variance into its additive-genetic and dominance components by the example of an F2 cross between two inbred strains. When only one gene with two alleles influences the phenotype, the expected genetic composition of the F2 population will be AA, 25%; Aa, 50%; aa, 25%. From the foregoing, this leads to a phenotypic mean of

¼ da + ½ ha - ¼ da = ½ ha                                                                                                                           (6)
(m is set at zero by a simple shift of the measurement scale). The sum of squares of deviations from the mean then equals
½ da2 + ¼ ha2                                                                                                                               (7)
In the absence of epistasis and linkage, the contribution of a number of genes (k) to the F2 variance becomes
½ di2 + ¼ hi2                                                                                                                 (8)
shortened to
½ D + ¼ H                                                                                                                               (9)
for ease of representation. The total phenotypical variance of an F2 is thus
VP = ½ D + ¼ H + E                                                                                                                         (10)
By using groups with different genetic compositions it will be possible to obtain estimates for the three parameters D, H, and E. From these parameters we may then estimate the proportion of the phenotypical variance due to additive-genetic effects, the heritability in the narrow sense
h2n = (½ D)/VP                                                                                                                     (11)
and the proportion of the phenotypical variance due to all genetic effects, the heritability in the broad sense
h2b = (½ D + ¼ H)/VP                                                                                                                                              (12)
When investigating populations other than an F2 between two inbred strains (such as a diallel cross), allele frequencies need not be identical. Using u to indicate the frequency of the increaser allele and v as the frequency of the decreaser allele (with u = 1 - v) we may amend the definitions of D and H as follows:
D = ½ 4uividi2                                                                                                             (13)
H = ¼ 4uivihi2                                                                                                            (14)
(Formally, this definition of H should be called H1, to distinguish it from H2, the other one of the two diallel forms of H; see [30]).

It should be noted that because D and H represent summations of the squared effects of single genes, they can only be zero if additive-genetic effects or dominance, respectively, are absent. This is in obvious contrast to [d] and [h].

The crossbreeding designs employed most often in neurobehavioral genetic studies are the classical Mendelian cross and the diallel cross (for other possible designs, see chapter 10 and [7]). The former consists of two inbred strains and, at least, their F1 and F2, often supplemented with backcrosses of the F1 with both parentals. The latter consists of a number of inbred strains (at least 3), that are crossed in all possible combinations. The two designs render different types of information on the genetic architecture of a trait. The classical cross permits very detailed genetic analyses and the detection of very small genetic effects, but on a very restricted sample of two inbred strains, only. Furthermore, it is very hard and almost always outright impossible, to distinguish between directional and ambidirectional dominance when using this design, because a significant parameter [h] only indicates that dominance is present and, at least, not completely balanced. In fact, a classical Mendelian cross is, generally speaking, only useful if one wants to establish whether the difference between two inbred strains is determined by either one gene or by more genes. In contrast, the analysis of a diallel cross renders information on a larger genetic sample and is therefore much more generalizable, but this carries a price in that the information obtained is less detailed. A great advantage, however, is the possibility to distinguish between ambi- and unidirectional dominance. To uncover the genetic architecture of a trait the diallel cross will be nearly always the design of choice.

When results from different crosses are available, it should be realized that comparing them is not very informative. Obviously, different results may be obtained depending on the genetic make-up of the parental strains used, especially if the crossbreeding design employed has a low generalizability (e.g., the classical cross). However, such results may be combined in order to provide a more complete and generalized picture of the genetic architecture of a trait. For instance, if one crossbreeding experiment indicates dominance in the direction of, say, high expression of the trait, but another cross indicates dominance in the opposite direction, then this constitutes prima-facie evidence for ambidirectional dominance. In fact, the presence of directional dominance may only be inferred if all available evidence indicates that dominance is acting in the same direction.


Crusio and van Abeelen [14] addressed the question of what exactly is the adaptive value of various mouse exploratory behaviors carried out in novel surroundings. As one result of exploration is the collection of new, or the updating of previously acquired, information, we argued that, if an animal enters a completely novel environment, it is obviously of prime interest to collect as much information as possible in a short time. On the other hand, high exploration levels will render the animal more vulnerable to predation. Taken together, we hypothesized an evolutionary history of stabilizing selection for exploration. This hypothesis was subsequently confirmed by the results of several crossbreeding experiments [11, 14] that revealed genetic architectures comprising additive genetic variation and/or ambidirectional dominance for most behaviors displayed in an open-field. The above reasoning is, of course, not specific for the species mouse. Indeed, Gerlai et al. [21] found similar genetic architectures for exploration in an open field in a diallel cross between inbred strains of Paradise Fish, Macropodus operculatus.

3.  Phenogenetic Aspects of Causation


When a behavioral neuroscientist wants to investigate the function of some brain system, he will often do so by manipulating the system in question. Brain lesions or pharmacological interventions to impair the functioning of the structure of interest are among the most often used techniques. Recently, Farah [18] reviewed the problems connected with the use of the so-called locality assumption, that more or less equalizes the function of an impaired structure with the defects exhibited by the damaged brain and which is almost always invoked to interpret the results of interventionist studies. In an elegant way, Farah [18] provided evidence that this reasoning may lead to false conclusions. An additional disadvantage of interventionist studies is related to the fact that large interindividual differences in brain structure exist. This heritable variation of the brain is an aspect that many neuroscientists tend to ignore, most likely at their own peril. For example, widely divergent behavioral effects of septal [15, 16] or limbic-system lesions [1], or of pharmacological interventions in the hippocampus [38] have been reported in mice, depending on which particular inbred strain was being used.

It appears that, in the field of neurobehavioral genetics, an alternative approach that does not suffer from these drawbacks exists: using genetic methods exploiting naturally occurring individual differences as a tool for understanding brain function. No brain is like another and every individual behaves differently. The assumption that there is a link between the variability of the brain and individual talents and propensities seems quite plausible. This approach differs from the usual one in neuropsychology in two important aspects. First, no subjects are studied that, by accident or by design, have impaired or damaged brains. Rather, all subjects fall within the range of normal, non-pathological variation (provided animals carrying deleterious neurological mutations are excluded). Second, instead of comparing a damaged group with normal controls, we study a whole range of subjects and try to correlate variation at the behavioral level with that at the neuronal level.

This non-invasive strategy is reminiscent of the phrenological approach propagated by Franz Josef Gall (1758-1828); Lipp has coined the name "microphrenology" for it [28]. It appears that, as long as variation in one neuronal structure is independent of that in another, there will be no need for a locality assumption to interpret results of experiments carried out along these lines. Especially when used in combination with methods permitting the estimation of genetic correlations [7, 8], this strategy yields a very powerful approach.

3.2.  Genetic Correlations

A weakness inherent in correlational studies is that a phenotypical correlation between characters does not necessarily reflect a functional relationship. On the other hand, if two independent processes, one causing a positive relationship, the other causing a negative relationship, act simultaneously upon two characters, the effects may cancel each other so that no detectable correlation can emerge.

These problems can largely be avoided by looking at the genetic correlations, that is, at correlations between the genetic effects that influence certain characters. Such correlations are caused either by genes with pleiotropic effects or by a linkage disequilibrium. With linkage disequilibrium we mean situations where certain allele combinations at closely linked genes are more frequent than might be expected based on chance. This will occur, for instance, in F2 crosses between two inbred strains (or populations derived therefrom, such as Recombinant Inbred or Recombinant Congenic Strains).

By using inbred strains that are only distantly related, the probability that a linkage disequilibrium occurs may be minimized so that a possible genetic correlation will most probably be caused by pleiotropy, that is, there exist one or more genes that influence both characters simultaneously. Thus, for these characters, at least part of the physiological pathways leading from genotype to phenotype must be shared and a causal, perhaps also functional, relationship must exist. It is this special property that makes the genetic-correlational approach such a uniquely valuable addition to the behavioral neuroscientists' toolbox. It should perhaps be noted at this point (as stated by Carey [5] and to be seen easily from the equations below) that the inverse need not be true: that is, the genetical correlation can still be low or even zero although pleiotropic genes are present.

3.3.  Theoretical Background

In the univariate analysis, we may partition the phenotypic variation into its components E, D, and H, the environmental, additive-genetic, and dominance contributions. In the bivariate analysis, the covariation between two traits x and y is partitioned into its equivalent components Exy, Dxy, and Hxy. We may define the latter two parameters as:
Dxy = ½ 4uividxidyi                                                                                            (15)
Hxy = ¼ 4uivihxihyi                                                                                             (16)
where dxi and dyi are the additive-genetic effects of the ith gene on characters x and y, respectively, and hxi and hyi are the respective dominance deviations due to the ith gene. Evidently, only genes that have effects on both of the characters x and y contribute to the genetic covariance terms, whereas all genes that affect either x or y contribute only to the respective genetic variance terms. Combining these components of the covariance with the components of the variance obtained in the univariate analyses we may estimate genetic correlations as follows:
rD = Dxy/                                                                                             (17)
rH = Hxy/                                                                                              (18)
As is the case with normal correlations, genetic correlations are bound by -1 and 1. If a genetic correlation equals unity, then all genes affecting character x also affect character y with gene effects on both characters being completely proportionally. A genetic correlation will become zero only in case no gene at all affects both characters simultaneously or in some balanced cases. For instance, if the effects of genes on character x are uncorrelated to the effects on character y (so that some genes influence both characters in the same direction whereas others do so in opposite directions)[5]. Obviously, similar observations can be made about rE, the correlation between environmental effects on two phenotypes x and y.

In principle, every breeding design allowing the partitioning of variation also enables one to partition covariation. However, in practice some designs turn out to be not very well suited to estimating genetical correlations. Especially the classical cross is very problematic in this respect and although many examples exist in the literature in which authors claim to have analyzed genetic correlations with this design, none really have done so. The problem appears to be mainly due to the frequent occurrence of the phenomenon, first observed by Tryon [36], that the variance of a segregating F2 population is not significantly larger than those of non-segregating populations (in extreme cases, it will even be smaller). Several possible explanations have been brought forward. Firstly, Hall [23] attributed the Tryon effect to an insufficient degree of inbreeding of Tryon's selected (but not inbred) strains. Of course, this would enlarge the genetic variation within the parental and F1 generations, but one would still expect the F2 to have a somewhat larger variance. In addition, the Tryon effect has since also been observed in crosses between highly inbred strains. A second explanation was presented by Hirsch [26]. He argued that most phenotypes are influenced by more than one gene. If we take the rat, with a karyotype of 21 chromosome pairs, as an example and, for simplicity, treat these chromosomes as major indivisible genes, one can see that this organism can produce 221 different kinds of gametes, leading to 321 (= 1.05 x 1010) different possible genotypes. In reality, this number will be even larger because chromosomes are not indivisible. Obviously, no experiment can take from an F2 generation a sample large enough to have all these genotypes represented and Hirsch [26] assumed this sampling effect to lower the observed F2 variances below expected levels. Tellegen [34] quite correctly countered that as long as the sampling from the F2 is random, an unbiased estimate of the population variance should be obtained. In addition, it can easily be seen that even if Hirsch's reasoning were correct, the F2 variance would still be expected to significantly exceed that of non-segregating generations, being the sum of environmentally-induced variation and, in his reasoning, at least some genetic variation.

Bruell [3] had observed that the amount of variance caused by segregation in the F2 increases if gene effects are larger and decreases if more genes influence the phenotype studied (cf. equations 13 and 14). If environmental influences on the phenotype are large, an extremely large sample would be needed to detect the difference in variance between the F2 and F1 populations at a sufficient level of significance in situations with many genes and relatively small gene effects. As sample sizes are limited by considerations of time, money, and space, while environmental influences on behavioral characters are usually very pronounced, one should normally expect F2 variances not to differ significantly from F1 variances. Due to sampling error, they may then even be smaller, although usually not significantly so. Homeostatic processes may be responsible if the latter situation occurs [27].

The problem therefore boils down to one of statistical power. It should be recalled here that larger sample sizes are needed to obtain accurate estimates of the variance of a population than for estimating its mean. By analogy, this also goes for covariances. In sum, unless rather huge sample sizes are used, classical crosses will generally lack the statistical power needed to accurately estimate genetical correlations.

Another reason that the classical cross is less suited for bi- and multivariate studies is the fact that results are not generalizable, but based on a restricted sample of two inbred strains only: even if genetic correlations would be estimated correctly with this design (see [24] for appropriate statistical methods), there exists a not negligible probability that they would be due to a linkage disequilibrium instead of pleiotropy. Of course, it is exactly the latter property that researchers employ when localizing genes. Note, however, that this is a special case in which one character, the molecular marker, is completely determined by the genotype (see also the following section).

A more suitable method is the diallel cross, for which a bivariate extension is available [8], whereas an interesting shortcut is offered by using a panel of inbred strains. Correlations between strain means either permit the estimation of additive-genetic correlations (using the methods described in [25]), or provide a direct lower-bound estimate of additive-genetic correlations (if the traits to be correlated have been measured in different individuals from these strains).

From the foregoing, it may easily be seen that the variance of the means of a set of inbred strains equals

D + E/n (19)
where n is the harmonic mean of the number of subjects per strain. The covariance between the means obtained for two characters x and y can then be expressed as
Dxy + Exy/n                                                                                                         (20a)
in case both characters are being measured on the same individuals or as
Dxy                                                                                                                 (20b)
in case characters x and y are being measured on different individuals from the same strains. The correlation between the strain means in these two situations will now equal
(Dxy + Exy/n)/                                                                     (21a)
Dxy/                                                                                     (21b)
respectively. Especially if environmental effects are small and large numbers of subjects are being used, the correlation between inbred strain means will approach the genetical correlation. In addition, it can easily be seen that equation 21b will always render a lower-bound estimate of the genetical correlation, even if n is small or if environmental effects are large. It should be realized that equation 21a may render a significant correlation even in the complete absence of any genetical effects. In the latter case, equation 21a reduces to the environmental correlation. By using the within-strain variation and covariation as estimates of the environmental variances and covariances, respectively, equations 19 and 20 render unbiased estimates of environmental and genetical correlations for both cases (for more details, see [25]).

Recombinant inbred strains (RIS) have sometimes also been used to estimate genetic correlations between phenotypes, using the above-described methods for correlations using ordinary inbred strains. It should be realized, however, that there is a considerable risk that any genetic correlations thus found will be due to a linkage disequilibrium because RIS have been derived from an F2 between two inbred strains. As was the case with the classical cross itself, this property is of course of interest for researchers hoping to localize QTL (but see ch. 3.3.3).

Except in the case of correlations between inbred strain means, testing the significance of genetic correlations is often problematic and the power of available tests is not yet well known at this moment. Fortunately, when the environmental and genetic correlations are used as input for further, multivariate, analyses, the possible significance or lack thereof of an individual correlation is no longer very important.


3.4.1. Rearing behavior in an open-field and hippocampal mossy fibers in mice

Crusio et al. [12] carried out a diallel cross study in which 5 different inbred strains were crossed in all possible ways. In 150 male mice from the 25 resulting crosses they measured the rearing-up frequency during a 20 min session in an open field and the extent of the hippocampal intra- and infrapyramidal mossy fiber projection (IIPMF). They obtained a marginally significant phenotypical correlation of 0.138 (df = 148, 0.05 < P < 0.10). Ordinarily, one would take such a result as evidence that variations in the size of the IIPMF is not related to behavioral variation. However, a quantitative-genetic partitioning of the covariation showed that the genetic correlation was quite sizable: 0.479. The low phenotypical correlation was explained by the modest heritability for rearing (0.25 Vs 0.53 for the IIPMF [10]) and by the fact that the (low) environmental correlation had a sign opposite to that of the genetic correlation. The genetic relationship between rearing, on the one hand, and the IIPMF, on the other hand, was confirmed by the finding that a line selected for high rearing frequency had larger IIPMF projections than a line selected for low rearing frequencies [13], exactly as would be predicted from a positive genetic correlation between these characters. From these data, it was concluded that the IIPMF play an important role in the regulation of open-field rearing [9].

3.4.2. Nerve conduction velocity and IQ in man

Lately, human behavior geneticists have also started to use genetic correlations to uncover brain-behavior relationships, especially the very active group around Dorret Boomsma at the Free University of Amsterdam (The Netherlands). In a recent experiment they used the twin method (see chapter 12) to examine the possible existence of a genetic correlation between speed-of-information processing (SIP) and IQ [33]. It was postulated that SIP, as derived from reaction times on experimental tasks, measures the efficiency with which subjects can perform basic cognitive operations underlying a wide range of intellectual abilities. Phenotypic correlations generally range from 0.2 to 0.4. Rijsdijk et al. [33] showed that genetic correlations also fell in this range at ages 16 and 18 years, whereas environmental correlations were essentially zero. A common, heritable biological basis underlying the SIP-IQ relationship is thus very probable.

3.4.3. Localization of QTL

Currently, Recombinant Inbred Strains are widely used as a tool to localize QTL. Unfortunately, problems of statistical power (often also due to multiple testing) lead to many false positives and negatives, as illustrated by the studies of Mathis et al. [31] and Gershenfeld et al. [22]. In the first study, a number of QTL associated with open-field behavior were "identified" using a large set of RIS between the inbred mouse strains C56BL/6J and A/J. In the second one, an F2 generation between these same inbred strains was studied, again leading to the "identification" of a number of QTL for the same behavioral phenotypes. However, none of the QTL found was common to both studies, aptly demonstrating the lack of reliability of current QTL methods.

4.  Some common misapplications

 A final point of caution is at its place here. Quantitative-genetic methods are not only used, but also, unfortunately, regularly abused. One problem already addressed above concerns whether information obtained at one level (on the components of means, say) can render information about another level (on the components of variance or covariance, for instance). This is sometimes the case, sometimes not. A few frequently made mistakes are mentioned here.

I. If some genetic effects are found for one character but not for another, this implies that these characters are influenced by different genes. Wrong! The equations given above should already make it abundantly clear that this is not true. An example may illustrate this.

Albinism in mice, for instance, is a character that is completely recessive as far as coat color is concerned. A quantitative-genetic analysis would indicate the significant presence of both d and h of equal size, in such a way that the heterozygote would completely resemble the non-albino homozygote. However, if we now would perform a quantitative-genetic analysis of the phenotype "activity of the enzyme tyrosinase", we would find that dominance is completely absent for this character, the heterozygote being completely intermediate between the two homozygotes. Still, as we know very well, only one and the same gene is involved here, which acts as a recessive on the level of coat color but shows intermediate inheritance on the level of the activity of the responsible enzyme.

In fact, only the presence of a genetic correlation between two phenotypes provides evidence that (a) gene(s) is (are) simultaneously influencing both. As was pointed out above, the reverse need not be true.

II. If two characters are correlated between two parental strains but not in their F2, they segregate independently and, hence, are influenced by different genes (in other words, there is no genetic correlation between them). Wrong! In fact, this observation may be true, but in only one exceptional situation: if at least one of the characters we are dealing with (for instance, a molecular-genetic marker) is completely determined by the genotype. This latter condition is called complete penetrance or, in quantitative-genetic terms, the heritability in the broad sense is said to equal 1. For behavioral and neural phenotypes, this condition almost never occurs. A few further observations should be made.

First, in all situations (including the above one), the phenotypical correlation within an F2 generation will be a function of the heritabilities of both characters and the sizes and signs of the genetic and environmental correlations between them. If, to simplify the equation, we suppose dominance effects to be absent for both characters x and y, then

rP = hxhyrD + exeyrE                                                                                                 (22)
where rP is the phenotypical correlation, hx and hy are the square roots of the narrow-sense heritabilities of characters x and y, ex and ey are the square roots of the "environmentalities" (the proportion of the phenotypical variance due to the environment; ex2 = 1 - hx2), and rE is the environmental correlation.

Equation 22 has a number of important implications. For instance, the size and sign of rP evidently do not render any information at all about the size and sign of rD. It can easily be seen that a significant phenotypical correlation may even be completely absent (rP not significantly different from zero) in case rD and rE have opposite signs and the absolute value of hxhyrD comes close to that of exeyrE. In recent years, experiments attempting to localize polygenes (called Quantitative Trait Loci, QTL) have become ever more popular. In such experiments one character, the molecular-genetic marker whose possible linkage to a putative QTL is being tested, will have a heritability of 1 and zero environmentality. In such a case, equation 22 reduces to

rP = hxrD                                                                                                                                                   (23)
This latter equation implies that there is an upper bound to the correlation between a behavioral or neural phenotype, on the one hand, and a marker locus, on the other hand, even if this marker locus would not just be linked to the hypothetical QTL but actually be identical with it. Equation 23 explains why, especially when heritabilities are low (and also because of the lack of statistical power of estimates of variance and covariance in F2 populations referred to in section 3.3), the power to reliably detect QTL is often very low, leading to many false positives and negatives [32]).

Second, note that in the above erroneous statement the words "between two parental strains" were used. The correlation within such strains has obviously no bearing at all on the eventual presence or absence of genetic correlations. As all individuals within inbred strains have the same genotype, any correlations occurring within such a strain are of environmental origin, of course. This is not to say, of course, that at any given point in time, two individuals belonging to the same inbred strain may not have different profiles of gene expression. If such is the case, however, then such differences in gene expression itself must be due to environmental influences (assuming, of course, that both individuals are at similar stages of development).

III. To determine the heritability of a character one should carry out a selection study.

This proposition is perhaps formally not incorrect (heritabilities can be derived from selection studies), but it contains in fact two conceptual errors. The first one is that there is some information to be gained from a heritability coefficient. Actually, except as an intermediate step in estimating genetic correlations, heritabilities do not have any intrinsic value. The only interesting facts about heritabilities are whether they differ significantly from zero (meaning that there is significant genetic variation) or from unity (meaning that there is significant environmental variation). There is one further use of heritability estimates, which is that their size predicts the eventual effects of selection pressures (whether artificial or natural [16]). This second conceptual error derives from this fact: once a selection study has been carried out and selection has led to the successful establishment of divergent lines, knowledge about heritability become more or less useless: it is not interesting any more to determine whether selection might be successful! Thus, there is only a single situation in which one would perform a selection experiment to estimate heritabilities: when one wishes to estimate genetic correlations and for some reason inbred strains are not available.

5.  Conclusion

In conclusion, if the above pitfalls are avoided, quantitative-genetic experiments can render valuable information on the genetic architecture of a trait. In addition, they can provide information about the multivariate genetic structure of complexes of traits. Because of this last property, quantitative genetics may serve as a valuable additional tool in the neuroscientist's arsenal and may greatly enhance our understanding of the genetic and neural mechanisms underlying individual differences in behavior.

6.  Acknowledgments

The preparation of this manuscript was supported by the Centre National de la Recherche Scientifique (UPR 9074), Ministry for Research and Technology, Région Centre, and Préfecture de la Région Centre. UPR 9074 is affiliated with INSERM and the University of Orléans. This chapter is dedicated to the memory of my teacher, mentor, and dear friend, Hans van Abeelen (1936-1998).

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