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<h1 class="top">The Basis of Dominance</h1>
<p class="header">This page is the second of a series of four containing Chapter 1
<q>The basis of dominance</q> by
Athel Cornish-Bowden & Vidyanand Nanjundiah (2006), pp. 1–16 in
<cite>The Biology of Genetic Dominance</cite>
(ed. R. A. Veitia)
Landes Bioscience, Georgetown, Texas.
<h2 class="framed">Dominance as a product of evolution </h2>
<p><em>Fisher’s model.</em> All discussions under this head must begin
with Fisher’s theory of the evolution of dominance via modifier genes. He
considered that <q>it is rather a peculiarity of the wild type to be generally
dominant than a peculiarity of the mutant to be recessive to the type from
which it arose</q>. He believed that in the absence of any mechanism to adjust
the results, the intensity of any character would be proportional to the gene
dose, so that for genotypes <i>AA, Aa</i> and <i>aa</i> the phenotypes would show 100%, 50%
and 0% respectively of normal level of the character. He further considered
that this result would be harmful not only to the homozygote <i>aa</i> but also to the
heterozygote <i>Aa.</i> Over many generations, therefore, natural selection would have
provided <q>modifier genes</q> to bring the heterozygote phenotype to
equivalence with that of the normal homozygote. The modifier would act on the
heterozygote but would not affect fitness by itself. Selection would act in
favour of the modifier at a rate proportional to the product of two factors:
the chance that the modifier was present in the heterozygous individual and the
fitness advantage caused by its presence.
<div class="sidebar-medium">
<p class="ref-in-sidebar">13. Bagheri-Chaichian H, Wagner GP. Evolution of dominance through incidental
selection. SFI working paper at http://www.santafe.edu/sfi/publications/
wpabstract/200211064 (2002)
</div>
<p><em>Difficulties with the Fisher model.</em> Strictly speaking, if one
wishes to explain the evolution of presentday wild type alleles starting from a
condition in which they arose as recessives, modifier action should be invoked
only in the (original) recessive homozygote — that is, if one wishes to say
that the mutation was fully recessive at the beginning. But going by reasonable
population sizes and what we know of mutation rates (about 1 in 10<sup>6</sup> to 1 in 10<sup>5</sup>
per locus), the population is hardly likely to contain individuals that are
homozygous for the mutant allele right from the start. Even if we say that the
modifier acts on the heterozygote, which amounts to assuming that the mutation
had a weak dominant effect at the beginning, the pool of individuals on whom
selection can act remains miniscule because the mutation rate is extremely low.
The number of relevant individuals in reasonably sized interbreeding groups
could be so small that evolution for dominance, besides being slow, would run
the risk of being overwhelmed by selection acting on the modifier gene
itself.<sup>7,8</sup> Fisher discounted this possibility: he thought that populations
would always be large enough and that one could envisage modifiers that had no
direct effect on their own. It can be shown that if the modifier evolves
because of a direct effect on fitness, selection for dominance need not be
hampered by the fact that the mutation rate, or the pool of heterozygotes, is
small.<sup>13</sup>
<div class="sidebar-medium">
<p class="ref-in-sidebar">6. Haldane JBS. A note on Fisher’s theory of the origin of dominance and
a correlation between dominance and linkage. <acronym class="deja" title="American Naturalist">Amer Nat</acronym> 1930; 64:87–90.
<p class="ref-in-sidebar"> 14. Haldane JBS. The theory of selection for melanism in Lepidoptera. <acronym class="deja" title="Proceedings of the Royal Society of London, Series B">Proc Roy
Soc Lond B</acronym> 1956; 145:303–306.
<p class="ref-in-sidebar"> 15. Charlesworth D. Evidence against Fisher’s theory of dominance.
Nature 1979; 278:848–849
</div>
<p>Haldane<sup>6</sup> also criticized Fisher’s theory on grounds somewhat similar
to those of Wright. He put forward an alternative explanation in which
dominance had coevolved as a byproduct of selection in favour of buffering the
genotype against environmental or genetic perturbations. Selection for
buffering would gradually change the phenotypic value of the mutant
heterozygote to the same level as in the wild-type homozygote. Much later,
Haldane<sup>14</sup> drew on his investigations into the evolution of industrial melanism
(in which the melanic form is both advantageous and dominant) and suggested
that dominance might have evolved from a low level, conceivably by a Fisher-type
process, concomitantly with the spread of the allele. Note that in
Haldane’s scheme the pool of heterozygotes is large and keeps increasing.
However, in common with Fisher’s model both his proposals share the
assumption that selection acts on the heterozygote.
<p>The criticisms made by Wright and Haldane were on the grounds of
plausibility. Besides that, there are two problems with Fisher’s model,
both based on observational grounds. The first is that it does not explain why,
in the heterozygous state, the presence of a strongly deleterious allele is not
as apparent as the presence of a weak allele. In other words, the model cannot
account for the observed negative correlation between the degree of dominance
and the fitness of the mutant homozygote.<sup>15</sup> The second problem is more serious.
Fisher included in his explanation of dominance a feature that almost precluded
any possibility of experimental verification, namely that the selection of
modifier genes would require an enormous number of generations before the
heterozygous phenotype became indistinguishable from that of the normal
homozygote. This would mean that one could not study successive generations for
a sufficiently long time to observe the gradual evolution of dominance.
<p>Nonetheless, the model does lead to a specific prediction that is open to
testing in an appropriate organism. It is obvious that concepts such as
dominance and recessivity have no meaning in haploid organisms; in addition,
less obviously, Fisher’s model implies that modifier genes for
dominance, and hence the evolution of dominance, cannot occur in haploid
organisms, and no dominance should be observed in a haploid organism that
passes through occasional diploid or polyploid generations. <i>Chlamydomonas
reinhardtii</i> is such an organism: it exists normally as a haploid organism, with
one nucleus per cell, and with no more than a single copy of each chromosome,
but it also has diploid or polyploid generations only occasionally. These occur
too infrequently for the organism to experience the consequences of
heterozygosis to drive the selection of suitable modifier genes. If
Fisher’s hypothesis is correct, therefore, one should see no sign of
dominance in diploid generations of <i>C. reinhardtii.</i>
<div class="sidebar-medium">
<p class="ref-in-sidebar">6. Haldane JBS. A note on Fisher’s theory of the origin of dominance and
a correlation between dominance and linkage. <acronym class="deja" title="American Naturalist">Amer Nat</acronym> 1930; 64:87–90.
<p class="ref-in-sidebar">9. Kacser H, Burns JA. The molecular basis of dominance. Genetics 1981;
97:639–666.
<p class="ref-in-sidebar">16. Orr HA. A test of Fisher’s theory of dominance.
<acronym class="deja" title="Proceedings of the National Academy of Sciences of the USA">Proc Natl Acad Sci USA</acronym>
1991; 88:11413–11415.
</div>
<p>What is observed is quite different. Orr<sup>16</sup> examined numerous mutations in
diploid cells and found that the great majority were recessive, exactly as
would be expected from the theory of Kacser and Burns<sup>9</sup> developed from
Wright’s ideas, but inconsistent with that of Fisher. Obviously, if a
property that can only evolve if it is selected in diploid (or polyploid) cells
proves to be exactly the same in a species that is nearly always haploid, an
explanation for that property that requires the longterm persistence of
diploidy cannot be a universal explanation. Orr noted that his observations
disposed not only of Fisher’s model but also of the two proposed by
Haldane.<sup>6</sup> Haldane’s models also required selection in heterozygotes, and
therefore could not explain why mutant genes are as likely to be recessive in
the very rare heterozygotes of a principally haploid species like <i>C.
reinhardtii</i> as they are in species that are always diploid.
<h2 class="framed"><a name="wri"></a>Wright’s theory: dominance as a physiological
phenomenon </h2>
<div class="sidebar-medium">
<p class="ref-in-sidebar">7. Wright S. Fisher’s theory of dominance. <acronym class="deja" title="American Naturalist">Amer Nat</acronym> 1929; 63:274–279.
</div>
<p>We have seen how Wright<sup>7</sup> considered that the selection pressure would be too
weak for Fisher’s explanation to work, and later8 he pointed out that the
modifier genes were themselves subject to mutation, and thus in need of their
own modifier genes to protect them from the effects of such mutations, and so
on, with obvious possibilities of infinite regress. He argued that an
explanation for dominance should be sought in mechanistic terms, meaning in
terms of immediate physiology. Even with care to avoid reading more into
Wright’s words than he intended, it is clear that his statement that
<q>the curve expressing the relation of <i>the product</i> to <i>enzyme amount</i> is a
hyperbola, asymptotic at its upper limit. Doubling the quantity of <i>enzyme</i> will
less than double the amount of <i>product</i></q> (in which the italicized words and
phrases replace algebraic symbols in the original) contained two important
ideas. First, there is nothing here that can be interpreted as a reference to
genes, so we are dealing with an explanation at a physiological or biochemical
level; second, he recognizes at the outset that biochemical responses to enzyme
levels are nonlinear.
<div class="sidebar-medium">
<p class="ref-in-sidebar">17. Haldane JBS. In: Enzymes. London: Longmans Green, 1930.
<p class="ref-in-sidebar">9. Kacser H, Burns JA. The molecular basis of dominance. Genetics 1981;
97:639–666.
<p class="ref-in-sidebar">18. Sheppard PM, Ford EB. Natural selection and the evolution of dominance.
Heredity 1966; 21:139–147.
<p class="ref-in-sidebar">19. Dawkins R. In: The Extended Phenotype: the Long Reach of the Gene, Oxford:
Oxford University Press 1981.
</div>
<p>As we shall see, this makes his explanation of recessivity far closer to
current ideas than anything that Fisher or Haldane wrote, even though Haldane
was much better informed about biochemistry in general and enzymes in
particular than either of his rivals: his book,<sup>17</sup> written at about the time of
the controversy, is a classic account of enzymes that is still worth reading
for its essential insights more than 70 years afterwards.
<p>Nonetheless, it was Fisher’s ideas that were generally accepted, not
those of Haldane or Wright. As late as 1966, Wright’s view of recessivity
as a physiological phenomenon was rejected as impossible;<sup>18</sup> and in 1981 an
important book<sup>19</sup> by an author justifiably famous for the clarity and cogency of
his writing could still contain some obscure pages presenting a vague exposition
of Fisher’s ideas barely more intelligible or convincing than
Fisher’s own. In the same year, however, Kacser and Burns<sup>9</sup> provided a
fully modern version of the physiological theory, in which they showed that
dominance and recessivity follow automatically from the known kinetic behaviour
of enzymes in isolation and when embedded in metabolic pathways.
<h2 class="framed"><a name="eff"></a>Effect of enzyme activity on metabolic flux </h2>
<p>For almost any enzyme in isolation, for example in a spectrophotometric
assay, the rate of reaction at specified concentrations of substrates, products
and any inhibitors or activators present in the mixture is proportional (at
least approximately) to the concentration of the enzyme: if the activity of
enzyme is doubled, the rate is doubled. Exceptions occur if the enzyme exists
in several different states of association with different degrees of catalytic
activity, and the experiment is done in the concentration range where the
degree of association varies. Other exceptions arise if the enzyme is unstable
at high dilution, a problem common enough in kinetic studies to oblige
experimenters to take precautions to handle it properly. However, it is an
artificial problem in the physiological context, because the high enzyme
dilutions commonly used to make steady-state rates slow enough to study
conveniently in the spectrophotometer are themselves unphysiological.
Nonetheless, the usual proportional dependence of rate on enzyme concentration
observed in steady-state kinetic experiments is misleading as a guide to
behaviour in the cell, because in the cell the concentrations of substrates,
products and so on are not constants fixed by an external agent, the
experimenter, but variables that depend on the activities not only of the
enzyme of interest but of all the enzymes in the system. The product of almost
any enzyme is also the substrate of one or more other enzymes; the substrate of
almost any enzyme is also the product of one or more other enzymes.
<div class ="centred">
<h4><img src="images/vidyaf1.gif" width="673"
height="531" alt="Abrupt change in enzyme activity"></h4></div>
<p class="legend">Fig. 1. Effect of an abrupt change in enzyme activity in a system of two
enzymes in steady state. The system considered is shown in the inset at
bottomright. In the initial steady state the two rates <i>v</i><sub>1</sub> and <i>v</i><sub>2</sub> are equal to
one another at some arbitrary value <i>J</i><sub>0</sub>. If the activity of the second enzyme is
abruptly decreased by 50%, this must produce a corresponding abrupt decrease in
<i>v</i><sub>2</sub> to 0.5<i>J</i><sub>0</sub>, but the system is then no longer in steady state because <i>v</i><sub>1</sub> > <i>v</i><sub>2</sub>,
and so the intermediate S is being released faster than it is being consumed.
Its concentration [S] must therefore increase, with two effects: as S is the
product of the first enzyme its increased concentration increases the product
inhibition and so <i>v</i><sub>1</sub> decreases; at the same time it is the substrate of the
second enzyme, and so by the usual effects of substrates on enzymes the rate <i>v</i><sub>2</sub>
increases. Eventually a new steady state is reached in which the two rates are
again equal, at a value smaller than <i>J</i><sub>0</sub> but larger than 0.5<i>J</i><sub>0</sub>.
<p>Consider now what will happen in response to a twofold decrease in the
activity of the second of two enzymes in a pathway that constitutes the whole
of a two-enzyme system (Fig. 1). The immediate effect will be the loss of any
steady state that existed before the change, because the first enzyme will
initially continue to work at the original rate, with the second working at
only half that rate. The common intermediate, the product of the first enzyme
and substrate of the second, is thus no longer in steady state, because it is
being produced faster than it is being consumed: its concentration must
therefore increase, but that has its own effects, increased inhibition of the
first enzyme and increased saturation of the second. Eventually these will
produce a new steady state in which the rates are again balanced, but notice
that the increased inhibition of the first enzyme and increased saturation of
the second mean that the new steady-state rate must be less than the original
rate but greater than half the original rate. In other words the effect of
decreasing the activity of the second enzyme is less than proportional. This
result is general, applying to changes more or less than twofold, to changes in
the activity of the first enzyme as well as to that of the second, and to
systems with any number of enzymes greater than one. In all these cases the
result of changing any enzyme activity is a less-than-proportional change in the
steady-state flux through the system.
<div class="sidebar-medium">
<p class="ref-in-sidebar">20. Kacser H, Burns JA. The control of flux. <acronym class="deja" title="Symposia of the Society for Experimental Biology">Symp Soc Exp Biol</acronym> 1973; 27:65–104.
<p class="ref-in-sidebar">21. Fell D. In: Understanding the Control of Metabolism, London: Portland
Press 1997.
<p class="ref-in-sidebar">22. Cornish-Bowden A. In: Fundamentals of Enzyme Kinetics (3rd edn), London:
Portland Press 2003.
</div>
<p>The original analysis was made by Kacser and Burns<sup>20</sup> before they analysed
dominance and recessivity, and is now discussed in textbooks,<sup>21,22</sup> so here it
is sufficient to give the main result. The effect on the metabolic flux of any
enzyme activity can be expressed as a flux control coefficient, defined as the
logarithmic derivative of the flux <i>J</i> with respect to a perturbation <i>p</i> divided
by the logarithmic derivative of the enzyme activity <i>v<sub>i</sub> </i>with respect to the
same perturbation <i>p</i>:
<p>(1) <i>C<sup>J</sup><sub>i</sub></i> = (∂ln<i>J</i>/∂ln<i>p</i>)/(∂ln<i>v<sub>i</sub></i>/∂ln<i>p</i>)
= (∂ln<i>J</i>/∂ln <i>v<sub>i</sub></i>)
<p>The anonymous perturbation <i>p</i> is introduced to avoid the mathematical
looseness in the simpler form at the right derived from the fact that <i>v<sub>i</sub> </i> is not
a true independent variable of the system; the simpler form is acceptable as
long as it is remembered that an external perturbation is implied even if not
specified. Moreover, to the extent that enzyme activities are proportional to
enzyme concentrations one can regard the enzyme concentration as the
perturbation <i>p</i>: this was done in the original paper of Kacser and Burns<sup>20</sup> but
in more recent work most authors have preferred to avoid an assumption that is
not only unnecessary but also, more seriously, was responsible for a widespread
misconception that metabolic control analysis deals only with changes in
activity brought about by changes in enzyme concentration. By a more rigorous
analysis along the lines of what was done above for a two-enzyme system, one can
readily prove the fundamental property of flux control coefficients expressed by
the following equation:
<p>(2) Σ<sup><i>n</i></sup><sub><i>i</i>=1</sub> <i>C<sup>J</sup><sub>i</sub></i> = 1
<p>in which <i>n</i> is the number of enzymes in the system. This is called the
summation property for flux control coefficients, and expresses the fundamental
idea that flux control is shared among all the enzymes in the system. As even
the simplest real cell contains hundreds of different enzymes the average share
must be very small. More realistically, perhaps, noting that most enzymes have a
very small share of the control of flux through pathways other than those in
which they occur, one can say that most of the control of the flux through any
pathway is shared among the enzymes of that pathway. This still implies an
average share of no more than 20% for most pathways, and often less.
<p>There is a complication in this argument that needs to be made explicit. In
referring to shares we are tacitly assuming that all the control coefficients
in the sum shown in eqn. (1) are positive, but that is not strictly true. The
flux control coefficient of an enzyme for a flux through a part of the
metabolic network not in series with the reaction that it catalyses can
certainly be negative, and often is. However, in nearly all real cases such
negative flux control coefficients are small enough to have little effect on
the general argument. Instead of saying that control of flux through any
pathway is exactly shared among all the enzymes of the system we must say that
most of it is shared among the enzymes of the pathway. This statement is not
very different from what we said in the previous paragraph.
<p>Before leaving the question of negative control coefficients, we should note
that these can be defined for other system variables as well as fluxes; for
example, the effect of enzyme activities on any metabolite concentration [S]
can be expressed by concentration control coefficients with definitions very
similar to that in eqn. (1). The resulting quantities are often not merely
negative but large in magnitude as well, and the summation property
corresponding to eqn. (2) has a righthand side of zero, not unity. In such a
case the idea of sharing loses all meaning, so we should emphasize that the
idea of sharing flux control comes not merely from the existence of a summation
property but from two additional points, one theoretical and general, the other
from observation: first the sum is unity; second, the individual elements in
the sum are mostly positive and, when negative, are normally small. On a
preliminary view the summation property in eqn. (2) already provides the
explanation of dominance and recessivity, as it suggests that changing any
enzyme activity will produce a much less than proportional change in metabolic
flux, and hence a much less than proportional change in metabolic output, such
as the amount of a pigment produced by a pea plant. However, that is too
simple, as it applies only to small — strictly infinitesimal — changes in
activity, whereas we need to consider the effects of changes in activity of the
order of twofold. For increases in activity the naive view remains valid, but
for decreases, which are more relevant to heterozygotes in diploid organisms,
it does not, because any flux control coefficient typically increases when the
enzyme activity is decreased, eventually reaching a value of 1 when the
activity is close to zero.
<p>Both simple models and experimental observations in numerous cases indicate
that for finite changes in activity the curve relating metabolic flux to the
activity of any enzyme in the pathway resembles a rectangular hyperbola. In the
simplest case where all enzymes operate far enough below saturation for the
kinetics with respect to to their substrates to be of first-order the curve is
exactly a rectangular hyperbola; it remains quite similar to one in the more
realistic case where some or all of the enzymes operate in a range with
detectable saturation. In Michaelis–Menten terms this implies substrate
concentrations of the order of the relevant Michaelis constants. What eqn. (2)
now means is that most enzymes are located near or on the flat part of the
hyperbola in the state corresponding to the normal homozygote. If any
particular enzyme happens to have a flux control coefficient close to unity —
unusual, but not impossible — then all the others in the pathway need to move
further to the right along the curve.
<p>Consider now the expected metabolic fluxes <i>J</i><sub>AA</sub>, <i>J</i><sub> Aa</sub> and <i>J</i><sub> aa</sub> for the normal
homozygote, heterozygote and abnormal homozygote respectively of the gene for
an enzyme normally located in the typical position on the hyperbola, as shown
in Fig. 2. It is immediately evident not only that <i>J</i><sub>AA</sub> ≅ <i>J</i><sub> Aa</sub> >> <i>J</i><sub> aa</sub> = 0, but
also, more important, that this relation will survive even quite large
variations in the assumption about where the normal enzyme lies on the curve:
it can be displaced any distance to the right without invalidating the
relationship, and significantly to the left also.
<p>This, in essence, is the explanation of dominance and recessivity proposed
by Kacser and Burns (1981). It may be objected that it predicts only that the
heterozygote phenotype will be more similar to the normal homozygote phenotype
than to the abnormal homozygote phenotype, not that it will be identical to it.
This is true, but in practice the difference would not be likely to have been
noticed in Mendel’s observations, or indeed in most observations of gross
phenotypes since then. In summary, therefore, the theory of Kacser and Burns
(1981), which was an elaboration and extension of that suggested by Wright
(1934), is the only one currently available that provides a satisfactory
explanation of dominance and recessivity.
<p class="navigation"><a href="vidya3.htm">Continued...</a>
<div class="sidebar-tail">
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Page created on 30 January 2006<br>
Last update: 22 December 2008<br>
Last significant update: 28 October 2008<br>
<em>Comments to <a href="mailto:acornish@ifr88.cnrs-mrs.fr">Athel Cornish-Bowden</a></em><br>
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