DOI: 10.1534/genetics.104.029785
Letter to the Editor
Gene Dosage Balance in Cellular Pathways: Implications for
Dominance and Gene Duplicability
Reiner A. Veitia
1Universite´ Denis Diderot/Paris VII and INSERM U361/E0021, Hoˆpital Cochin, 75014 Paris, France
Manuscript received April 7, 2004 Accepted for publication May 27, 2004
T
HE gene dosage balance hypothesis (GDBH) pro- from the selective cost of their overproduction. Thus, poses, in a narrow sense, that stoichiometric imbal- some genes encoding interacting pairs should remain ances in macromolecular complexes can be a source of as single copies (case ofB) or otherwise undergo co-dominant phenotypes. Gene dosage balance in such duplication with genes encoding their partners. Indeed, complexes is required as a “here and now” condition, it can be shown that coduplication of B with A or C as a partial aneuploid carrying a deletion or duplication can overcome titration by excess ofB(Teichmannand of a dosage-sensitive gene will have a fitness defect Veitia2004;Veitia2004). Accordingly, pairs of genes (Birchler et al.2001;Veitia2002, 2003). Global evi- encoding interacting subunits tend to have the same dence supporting the GDBH has been found in yeast. number of paralogs and genes belonging to huge fami-Focusing on essential genes (i.e., their homozygous dele- lies seldom encode components of complexes (Papption is lethal),Pappet al.(2003) have shown that dosage- et al. 2003). Recently, the validity of the GDBH was sensitive genes (low heterozygote fitness) are at least corroborated in yeast (Yanget al.2003). Moreover, us-two times more likely to encode proteins involved in ing human data, these authors found that the gene complexes than are genes with low dosage sensitivity. duplication level is higher for monomers than for com-Furthermore, a statistically significant proportion of ponents of protein complexes, which is consistent with genes whose overexpression is lethal encodes proteins the GDBH. Besides, the proportion of unduplicated involved in complexes. The concept of dosage balance genes was found to increase with the number of subunits is old. Consider, for instance, dosage compensation of in a complex. Here I show, with some examples, that the X chromosome. In Drosophila the transmission of the dosage balance notion is applicable to other cellular the X from the female to the male is operationally equiv- dynamic systems. I focus on gene dosage increase (dupli-alent to an entire chromosomal “deletion.” Balance is cation) but a similar reasoning holds for dosage reduc-achieved by making the single X in the male about twice tion. The short-term outcome of a dosage balance alter-as active, transcriptionally, alter-as either of the two X’s in ation is relevant to understanding aspects of genetic the female. In mammals, compensation is achieved by dominance, as it may induce an immediate decrease inactivating one X chromosome in the female (Marinet of fitness, as mentioned above in the case of protein al.2000). This clearly implies that at least some X-linked complexes. Classical genetics regards the phenomenon genes must respect a certain balance with autosomal of dominance as a result of intralocus interactions.
Stem-products. ming from the notion of balance itself, the models
The need of dosage balance also implies that single- sketched below show how genetic dominance can arise gene duplications of certain types of subunits can be also from interloci interactions, in line with the argu-harmful. Consider a complexA-B-C. Under irreversible ments of Omholtet al.(2000).
conditions, increasing the concentration of the bridge Let us first consider a system displaying adaptation to Bcan be detrimental, as inactive subcomplexesABand a signal, which is the basis of a chemotactic mechanism BC may form (lowering the yield of ABC). Subunit B (MacnabandKoshland1972). A signalScontrols (i) exerts a titrating power onAandCwhen overexpressed. the translation of an RNA, constitutively present at a On the contrary, increasingA and C is neutral, apart
stable concentration, to produce a protein R and (ii) the synthesis of the proteaseXthat degradesR(see the sniffer ofTysonet al.2003). In response to changes in
1Address for correspondence:Universite´ Denis Diderot/Paris VII and
S, Rundergoes transient changes but will go back to
INSERM U361/E0021, Hoˆpital Cochin, Bat. Baudelocque, 123 bd
Port Royal, 75014 Paris, France. E-mail: [email protected] a steady state where its concentration is constant and
by the ratios Km1/WTotal and Km2/WTotal (Km’s are the
Michaelian constants of the enzymes). High values of these parameters (i.e., 1) yield a hyperbola. For low values of the parameters (i.e., 0.01), which means satura-tion of the converters, there is a threshold E1/E2 for
which a jump fromW→W* appears. Under these condi-tions, a gradual input is transformed into a switch-like response (GoldbeterandKoshland1981). This am-Figure 1.—A simple network displaying adaptation to a
plification of the response to a stimulus that alters the stimulus (a) and the corresponding differential equations (b).
ratio E1/E2 is called “zero-order ultrasensitivity.” The
All reactions have been assumed to be first order (linear) with
respect to all reactants. The k’s are specific rates. Positive/ modular nature of the GK switch is explained by the
negative terms correspond to synthesis/degradation.Rssis [R] fact that asE
1 andE2are saturated by their substrates
at steady state (i.e., whendR/dt⫽dX/dt⫽0). Obviously,
co-(WandW*, respectively), the corresponding reaction increase ofRandX(represented, to simplify the notation, by
rates do not depend on substrate concentration but only 1.5⫻k1and 1.5⫻k3, respectively) leads to the sameRss.
on the relative amounts of active converting enzymes. In the context of gene dosage balance, if we assume for simplicity that activation/deactivation ofE1 andE2
independent ofS(i.e.,Rssin Figure 1). A transient
in-crease ofRabove a thresholdRThrmay trigger an action depend linearly on the strength or duration of the
stim-ulus, a parallel change (increase or decrease) of total that ceases whenRcomes back toRss. The dynamics of
the system can be represented by two simple differential E1andE2will change neither the position of the
thresh-old nor the general shape of the sigmoid (Figure 2). equations (Figure 1). A 1.5-fold increase of the dosage
ofR, as in partial triploidy, implies increasing the steady- This co-increase cannot leadE1(E2) to become
unsatu-rated byW(W*) because ultrasensitivity might vanish. state concentration of its RNA and will be represented
in the differential equations by 1.5⫻k1. In such a case Increasing the amount of one convertase alone (i.e.,
1.5⫻ in a triploid or 2⫻in a partial tetraploid) will be the steady-stateR⬘sswould also increase by 1.5-fold, which
can be above the threshold RThr. This is obviously a problematic as it shifts the position of the threshold. As
shown in Figure 2, increasingE1shifts the threshold to
problem. However, a parallel increase of the rate of
synthesis of X’s mRNA (i.e.,k3 becomes 1.5 ⫻ k3) will the left and the contrary forE2. Consider now that E2
participates in several reactions (promiscuous) and that restore the normal amount ofRss. Thus, coduplication
of XandR is harmless and the only visible effect is a the amount allocated to counteract the effect of E1 is
fairly constant. To give rise to a GK switch, this fraction faster adaptation.
To explain dominance of the normal phenotype, of E2 must be saturated by W*. Increasing only E1 is
problematic; co-increase ofE1 andE2restores the
nor-Kacser and Burns (1981) showed that changing the
amounts of most enzymes does not affect the visible mal activity of the switch E1/E2 but will perturb other
switches in whichE2might be involved. Thus, even
pro-phenotype. However, the concentration of an
interme-diate can vary a lot if the activity of an appropriate miscuity imposes limits for duplication of dosage-sensi-tive genes.
enzyme is raised or lowered substantially. So, even in a
“Kacserian” context, the GDBH may apply to reactions Goldbeter(1991) proposed an elementary mitotic clock with a chain of two GK switches responsive to of the form→A→Y→B→when the absolute level of
Y(for instance) influences or determines a phenotype. cyclin. The output is the activation of a cyclin-degrading protease. To generate periodic changes of cyclin levels, Enzymes involved in signal transduction are expected to
be particularly dosage sensitive. To illustrate this point, the circuit requires delays introduced by the accumula-tion of cyclin itself and of a protease-activating enzyme, consider a system containing two converting enzymes
E1andE2(for instance, a protein kinase and a phospha- both of which must trespass their corresponding
thresh-olds. Changing the dosage of one element arbitrarily tase) that interconvert substrates W and W* (
Gold-beter and Koshland 1981; Figure 2). This topology may prevent cycling due to a shift in the threshold positions. A similar phenomenon is expected to arise [Goldbeter-Koshland (GK)] is common in signaling. In
the GK system, the molar fraction ofW* (i.e.,W*/WTotal) according to more complex models of the cell cycle. In
the model ofChenet al.(2000), including also two GK is a function of the ratio of activeE1/E2(or more exactly
of k1E1/k2E2; thek’s are the catalytic constants of each switches, the activity of cyclin B-dependent kinases (that
defines thestartandfinishpoints of the cycle) depends enzyme;GoldbeterandKoshland1981, 1984).E1and
E2can be inducible or constitutive, but activated/deacti- explicitly on the ratio Cln2/Cdc20.
The mitogen-activated protein kinase (MAPK) path-vated differentially. One can be regulated and the other
constitutive and even promiscuous (i.e., interacting with ways are well-known intracellular signaling modules in eukaryotes. MAPKs are serine-threonine protein kinases several partners). A curve representing the ratio of
ac-tivek1E1/k2E2vs.the molar fraction ofW* ranges from that are activated by diverse stimuli ranging from
Figure 2.—A schematic Goldbeter-Koshland switch. The curves of the mole fraction ofW* at the steady state as a function of stimulus strength (duration) were established from Equation 7 of
Goldbeter and Koshland (1981, corrected
form in Goldbeter and Koshland 1984), for
the parametersKm1/WTotal⫽Km2/WTotal⫽0.01. The
solid black sigmoid corresponds to gene dosage ofE1⫽ E2. The gray curve corresponds to
dou-blingE2, while the dotted curve represents
dou-blingE1. For simplicity, activation ofE1is assumed
to depend linearly on enzyme concentration [E] and stimulus strength (duration) for constantE2.
lular stress, and cell adherence. These cascades contain Simulations also show that if MAPK accumulates in
three levels: MAPKKK → MAPKK → MAPK with the the nucleus, so that as much as 50% of cytoplasmic
corresponding deactivating enzymes (for review see MAPK is sequestered after phosphorylation by active
Widmannet al.1999). Each layer has the GK topology nuclear MAPKK (for constant concentrations of MAPK-but they do not seem to be GK switches (Bluthgen pase), these converters operate at saturation and a true andHerzel2003). Modularity, to avoid cross-talk among GK-switch behavior will appear (Ferrell1998). To keep the pathways, is ensured by tethering the kinases to the proper balance, coduplication of MAPKK/MAPK-scaffold proteins as well as by direct interaction between pase is required, as explained above. Note that in case
the former. of whole-genome duplication, the nucleus will enlarge,
Simulations show that the MAPK pathway can convert leading very likely to similar intranuclear concentrations a gradual input into a switch-like output. This property of the relevant proteins as before duplication (see below). makes the cascade suitable for mediating processes like Finally, let us examine a minimal genetic toggle based mitogenesis, cell fate induction, and oocyte maturation, on two repressible promoters arranged in a mutually where a cell switches from one discrete state to another. inhibitory network. This system can be represented with However, sigmoidicity depends on the assumptions of the following rate equations (Gardneret al.2000 and the current models [Huangand Ferrell 1996 (HF); references therein):
BhallaandIyengar 1999 (BI)]. Sigmoidicity can be
studied by fitting the curves to the Hill sigmoid (y⫽ du
dt ⫽ ␣1
1⫹v⫺u; dv dt ⫽
␣2
1⫹ u␥⫺ v. xn/(K⫹xn)), wherenis the Hill coefficient (the higher
it is, the steeper the sigmoid, the sharper the threshold).
Hereuandvare the concentrations of the repressors, According to the HF model, dosage alterations of either
and␣1and␣2represent the effective rates of synthesis
MAPKK or MAPK-phosphatase induce important effects
ofuandv, respectively. Parameters and␥ represent on sigmoidicity but according to the BI model, changes
the cooperativity of repression of promoters 2 and 1, in almost all components individually lead to striking
respectively (and␥ ⬎1 imply cooperativity). Such a changes in sigmoidicity (Figures 3 and 4 ofBluthgenand
circuit can display bistability, as it can be flipped
be-Herzel2003). Remarkably, co-increase or co-decrease of
tween two stable steady states (either highuor highv) allcomponents at once translates into minor changes in
using transient inductive signals. Each state is associated the position of the stimulus threshold and sigmoidicity.
with the expression of different sets of genes responsive After doubling or halving all the components, HF curves
to the repressors. Moreover, after the removal of the have similar sigmoidicity and the threshold position
inducing signal, the system remains in the state where (loosely defined as thexcorresponding toy⫽0.5 for
it was. This is an epigenetic mechanism to ensure stabil-a steep sigmoid) chstabil-anges by only ⵑ25% with respect
ity of a developmental decision that can be subsequently to the reference system described by Bluthgen and
reverted: a kind ofconditionalcellular memory.
Herzel(2003). For the BI curves, the threshold changes
The stability of the system can be explored by studying by only ⵑ30% upon halving or doubling all
compo-the curves corresponding todu/dt⫽0 anddv/dt⫽ 0 nents. On the basis of this evidence, it is safe to consider
or null clines. Their intersections define points corre-the whole pathway as corre-the functional unit. Therefore, if
sponding to steady states. Here, when there is coopera-the selectable property is a switch-like behavior, coopera-the
tivity ( and␥ ⬎ 1) and the repressive activities of u whole MAPK module is likely to be duplicated or
inter-and/orvis represented by a change of␣(i.e.,␣1and/or
␣2, respectively). The outcome of this operation will
critically depend on where, in the phase diagram, the system is (Figure 3c). When ␣1 and ␣2 are high, the
system operates in a wide region of bistability. For in-stance, an increase of␣2(by 1.5⫻) may be
inconsequen-tial as this means shifting the point (log ␣2, log␣1) by
ⵑlog 1.5 units rightward. However, high␣1and␣2imply
a selective disadvantage, as synthesis of lot of repressor is costly. Natural systems are more likely to work in a region of bistability with limited amounts of u and v. In such a case, a small shift to the right can imply trespassing the (bifurcation) line from a region of bista-bility to a region of monostabista-bility. However, co-increas-ing of u andvauthorizes the system to remain in the bistable region.
Before closing, note how difficult coduplication of unlinked dosage-sensitive genes can be. For instance, equate the trimer ABC to X (from Figure 1) or to E1
(from a GK switch). Let componentBbe titrating and genes A, B, and C be unlinked. Genes A and C can duplicate independently to yieldA⬘andC⬘without stoi-chiometric interference. BothA⬘andC⬘should be pres-ent before “coduplication” ofB(to get the right concen-tration of trimer) and its partner, sayR(as in Figure 1). Figure3.—A genetic toggle. Geneuencodes a repressor
However, the increase in frequency ofA⬘andC⬘alone ofv, which is in turn a repressor ofu.I1andI2are the inducers.
and their coexistence in the same individuals is not Presence ofI1will trigger synthesis ofuand repression ofv,
justified by any advantage and they will tend to disap-which persists even after removal of the inducer. (a) The null
clines (containing the loci ofdu/dt⫽0 anddv/dt⫽0) inter- pear. Thus, a scenario involving small duplications to
sect at three points when the repressive activities ofuandv get as few as five genes coduplicated, partly sequentially
are balanced and in the presence of cooperative repression.
and partly concertedly, seems virtually impossible. This This translates into the existence of two stable and one
unsta-points toward an underestimated role of global duplica-ble steady states (staduplica-ble state 1/highv, state 2/highu). (b)
tions to justify the existence of paralogs of dosage-sensi-When there is an imbalance in the amounts of the repressors
(i.e., relative excess ofu) the system is not bistable anymore. tive genes. In principle, an imbalance caused by
chang-(c) Phase diagram of the system. The lines mark the transition ing dosage of one gene could be rescued by a parallel
between bistability (the systems can flip between state 1 and
change of expression of the interacting partners. How-state 2) and monostability (the system is unable to flip). The
ever, in the context of cellular pathways, this is clearly bistable region lies inside of each pair of curves. The small
so only if both partners act as monomers (case ofXand horizontal arrow represents an increase of dosage ofvfor the
same dosage ofu. The system crosses the bifurcation line and Rin the example of Figure 1) or, at most, if they are
crashes (becomes monostable, permanently in state 1). The homo-oligomers having the same number of monomers
small diagonal arrow (OK) represents a co-increase ofuand
and similar apparent pseudo-equilibrium constants, which
v: the system remains in the bistability region. Increase of
seems unlikely. Otherwise, a co-increase of both partners cooperativity (and␥) leads to a broader region of bistability
would translate into a disproportionate change in the increasing the robustness of the system. This figure is courtesy
ofGardneret al.(2000) andNature(Macmillan Magazines; concentration of active oligomers. This is so because
modified and reproduced with permission). oligomerization reactions take place in the same volume
with a higher input of monomers. It is difficult to pro-vide a general proof of this conjecture but an illustrative sect three times. This translates into the existence of example, directly applicable to the models outlined here, two stable and one unstable steady states (Figure 3a). is discussed in Figure 4. This merely biochemical argu-Thus, bistability depends on the cooperative repression ment points to the need for a whole-genome duplication of transcription. If the rates of synthesis of the two re- that implies an increase of cell volume (Gregory2001), pressors are not balanced, the null clines will intersect which tends to restore the concentration of monomers only once, producing a single stable steady state (Figure and multimers as before duplication. Even yeast during 3b, monostable system). One can draw a phase diagram its life cycle respects this constraint: the diploid cellular to represent whether the system lies in a region of bista- volume is about two times the haploid volume (Walker
bility (the system works) or in a region of monostability 1998;Sherman2002).
from sequential single-gene duplications. Multiple rounds of genome duplication associated with preferen-tial retention of dosage-sensitive interacting partners might explain the existence of paralogous networks and also the tendency toward nonrandom gene association in the eukaryotic genome. The perspective outlined here is compatible with, and does not diminish the evolutionary importance of, segmental duplications. In-deed,Teichmann andVeitia(2004) have shown the existence of an excess of linked gene pairs encoding subunits of stable protein complexes in yeast. We specu-lated that these pairs may be modules (perhaps gener-ated by the nonrandom retention of dosage-sensitive genes) that may maintain the right stoichiometry of complexes upon segmental duplication.
The examples discussed above show that the appear-ance of dominant phenotypes may have simple physio-Figure4.—Doubling the genomic content associated with
logical explanations in terms of dosage imbalances and an increase in nuclear and cellular volume can warrant
success-that the resilience of a system to such alterations can ful duplication of certain dosage-sensitive genes. Consider that
be adjusted by selection. They illustrate also key points
Mis an inactive monomer and thatMnis the active multimer
composed ofnmonomers. Synthesis (k1), degradation (k2), concerning duplicability of dosage-sensitive genes. The
and interaction of the monomers are represented with chemi- fact that there are rules governing changes of gene
cal (a) and differential equations (b). For simplicity, let the
dosage does not exclude the possibility of compensation association/dissociation reactions between monomers and
by up- or downregulation of partners in the same path-oligomers be faster than synthesis/degradation. We can define
a pseudo-equilibrium constantKin the steady state (i.e., when way as long as fitness is not severely compromised by
dM/dt⫽0). As usual, increasing dosage ofM(i.e., by 1.5⫻) the initial dosage alteration. Indeed, coevolution ofcis
-will be represented by 1.5⫻k1. If the initial volume does not regulatory sequences is commonplace (Wray et al.
change, this will imply a (1.5)n-fold increase of activeM n! Thus,
2003), which may explain gene synexpression (coregula-maintaining a balance with another oligomer, sayNx, after
tion within modules in time and space; Nierhs and strict coduplication in the same volume is possible only if the
number of monomers involved in Mn and Nx are identical Pollet1999), as a way to ensure balance. A scenario
(n⫽x), for similarKMandKN. A polyploidization event increas- involving massive duplications is crucial to produce par-ing cell volume proportionally is more likely to restore balance
alogous pathways (probably more complex than those whatever theK’s,n, orxare. Similar arguments explain
imbal-studied byTeichmannandBabu2004) and raw mate-ance after a heterozygous deletion ofMorN.
rial for evolution in cases where single-gene duplication is difficult or impossible.
I thank Nils Bluthgen for his kind help concerning simulations of
tion events in many eukaryotes including yeast, plants,
the MAPK pathway and for helpful comments on the manuscript. I
and vertebrates (Makalowski2001;BlancandWolfe
thank Jim Collins for his comments concerning both the original and
2004;Kelliset al.2004). This process is expected to be
the revised manuscript and Sandrine Caburet, Indrani Bose, and Vidya
followed by deletion or rearrangements leading in part Nanjundiah for their interesting suggestions. to preferential retention of “interacting” genes to avoid
imbalances. After genome duplication, the retained par-alogs may diverge concertedly in sequence and pattern
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