Gene Dosage Balance in Cellular Pathways

THE gene dosage balance hypothesis (GDBH) proposes, in a narrow sense, that stoichiometric imbalances in macromolecular complexes can be a source of dominant phenotypes. Gene dosage balance in such complexes is required as a “here and now” condition, as a partial aneuploid carrying a deletion or duplication of a dosage-sensitive gene will have a fitness defect (Birchleret al. 2001; Veitia 2002, 2003). Global evidence supporting the GDBH has been found in yeast. Focusing on essential genes (i.e., their homozygous deletion is lethal), Pappet al. (2003) have shown that dosage-sensitive genes (low heterozygote fitness) are at least two times more likely to encode proteins involved in complexes than are genes with low dosage sensitivity. Furthermore, a statistically significant proportion of genes whose overexpression is lethal encodes proteins involved in complexes. The concept of dosage balance is old. Consider, for instance, dosage compensation of the X chromosome. In Drosophila the transmission of the X from the female to the male is operationally equivalent to an entire chromosomal “deletion.” Balance is achieved by making the single X in the male about twice as active, transcriptionally, as either of the two X's in the female. In mammals, compensation is achieved by inactivating one X chromosome in the female (Marinet al. 2000). This clearly implies that at least some X-linked genes must respect a certain balance with autosomal products. The need of dosage balance also implies that single-gene duplications of certain types of subunits can be harmful. Consider a complex A-B-C. Under irreversible conditions, increasing the concentration of the bridge B can be detrimental, as inactive subcomplexes AB and BC may form (lowering the yield of ABC). Subunit B exerts a titrating power on A and C when overexpressed. On the contrary, increasing A and C is neutral, apart from the selective cost of their overproduction. Thus, some genes encoding interacting pairs should remain as single copies (case of B) or otherwise undergo coduplication with genes encoding their partners. Indeed, it can be shown that coduplication of B with A or C can overcome titration by excess of B (Teichmann and Veitia 2004; Veitia 2004). Accordingly, pairs of genes encoding interacting subunits tend to have the same number of paralogs and genes belonging to huge families seldom encode components of complexes (Pappet al. 2003). Recently, the validity of the GDBH was corroborated in yeast (Yanget al. 2003). Moreover, using human data, these authors found that the gene duplication level is higher for monomers than for components of protein complexes, which is consistent with the GDBH. Besides, the proportion of unduplicated genes was found to increase with the number of subunits in a complex. Here I show, with some examples, that the dosage balance notion is applicable to other cellular dynamic systems. I focus on gene dosage increase (duplication) but a similar reasoning holds for dosage reduction. The short-term outcome of a dosage balance alteration is relevant to understanding aspects of genetic dominance, as it may induce an immediate decrease of fitness, as mentioned above in the case of protein complexes. Classical genetics regards the phenomenon of dominance as a result of intralocus interactions. Stemming from the notion of balance itself, the models sketched below show how genetic dominance can arise also from interloci interactions, in line with the arguments of Omholtet al. (2000). Let us first consider a system displaying adaptation to a signal, which is the basis of a chemotactic mechanism (Macnab and Koshland 1972). A signal S controls (i) the translation of an RNA, constitutively present at a stable concentration, to produce a protein R and (ii) the synthesis of the protease X that degrades R (see the sniffer of Tyson et al. 2003). In response to changes in S, R undergoes transient changes but will go back to a steady state where its concentration is constant and independent of S (i.e., Rss in Figure 1) A simple network displaying adaptation to a stimulus (a) and the corresponding differential equations (b). All reactions have been assumed to be first order (linear) with respect to all reactants. The k's are specific rates. Positive/negative terms correspond to synthesis/degradation. Rss is [R] at steady state (i.e., when dR/dt = dX/dt = 0). Obviously, co-increase of R and X (represented, to simplify the notation, by 1.5 × k1 and 1.5 × k3, respectively) leads to the same Rss. . A transient increase of R above a threshold RThr may trigger an action that ceases when R comes back to Rss. The dynamics of the system can be represented by two simple differential equations (Figure 1). A 1.5-fold increase of the dosage of R, as in partial triploidy, implies increasing the steady-state concentration of its RNA and will be represented in the differential equations by 1.5 × k1. In such a case the steady-state R′ss would also increase by 1.5-fold, which can be above the threshold RThr. This is obviously a problem. However, a parallel increase of the rate of synthesis of X's mRNA (i.e., k3 becomes 1.5 × k3) will restore the normal amount of Rss. Thus, coduplication of X and R is harmless and the only visible effect is a faster adaptation. To explain dominance of the normal phenotype, Kacser and Burns (1981) showed that changing the amounts of most enzymes does not affect the visible phenotype. However, the concentration of an intermediate can vary a lot if the activity of an appropriate enzyme is raised or lowered substantially. So, even in a “Kacserian” context, the GDBH may apply to reactions of the form → A → Y → B → when the absolute level of Y (for instance) influences or determines a phenotype. Enzymes involved in signal transduction are expected to be particularly dosage sensitive. To illustrate this point, consider a system containing two converting enzymes E1 and E2 (for instance, a protein kinase and a phosphatase) that interconvert substrates W and W* (Goldbeter and Koshland 1981; Figure 2) A schematic Goldbeter-Koshland switch. The curves of the mole fraction of W* 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 parameters Km1/WTotal = Km2/WTotal = 0.01. The solid black sigmoid corresponds to gene dosage of E1 = E2. The gray curve corresponds to doubling E2, while the dotted curve represents doubling E1. For simplicity, activation of E1 is assumed to depend linearly on enzyme concentration [E] and stimulus strength (duration) for constant E2. . This topology [Goldbeter-Koshland (GK)] is common in signaling. In the GK system, the molar fraction of W* (i.e., W*/WTotal) is a function of the ratio of active E1/E2 (or more exactly of k1E1/k2E2; the k's are the catalytic constants of each enzyme; Goldbeter and Koshland 1981, 1984). E1 and E2 can be inducible or constitutive, but activated/deactivated differentially. One can be regulated and the other constitutive and even promiscuous (i.e., interacting with several partners). A curve representing the ratio of active k1E1/k2E2vs. the molar fraction of W* ranges from a hyperbola to a sigmoid whose steepness is governed 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 saturation of the converters, there is a threshold E1/E2 for which a jump from W → W* appears. Under these conditions, a gradual input is transformed into a switch-like response (Goldbeter and Koshland 1981). This amplification of the response to a stimulus that alters the ratio E1/E2 is called “zero-order ultrasensitivity.” The modular nature of the GK switch is explained by the fact that as E1 and E2 are saturated by their substrates (W and W*, respectively), the corresponding reaction rates do not depend on substrate concentration but only on the relative amounts of active converting enzymes. In the context of gene dosage balance, if we assume for simplicity that activation/deactivation of E1 and E2 depend linearly on the strength or duration of the stimulus, a parallel change (increase or decrease) of total E1 and E2 will change neither the position of the threshold nor the general shape of the sigmoid (Figure 2). This co-increase cannot lead E1 (E2) to become unsaturated by W (W*) because ultrasensitivity might vanish. Increasing the amount of one convertase alone (i.e., 1.5× in a triploid or 2× in a partial tetraploid) will be problematic as it shifts the position of the threshold. As shown in Figure 2, increasing E1 shifts the threshold to the left and the contrary for E2. Consider now that E2 participates in several reactions (promiscuous) and that the amount allocated to counteract the effect of E1 is fairly constant. To give rise to a GK switch, this fraction of E2 must be saturated by W*. Increasing only E1 is problematic; co-increase of E1 and E2 restores the normal activity of the switch E1/E2 but will perturb other switches in which E2 might be involved. Thus, even promiscuity imposes limits for duplication of dosage-sensitive genes. Goldbeter (1991) proposed an elementary mitotic clock with a chain of two GK switches responsive to cyclin. The output is the activation of a cyclin-degrading protease. To generate periodic changes of cyclin levels, the circuit requires delays introduced by the accumulation of cyclin itself and of a protease-activating enzyme, both of which must trespass their corresponding thresholds. Changing the dosage of one element arbitrarily may prevent cycling due to a shift in the threshold positions. A similar phenomenon is expected to arise according to more complex models of the cell cycle. In the model of Chenet al. (2000), including also two GK switches, the activity of cyclin B-dependent kinases (that defines the start and finish points of the cycle) depends explicitly on the ratio Cln2/Cdc20. The mitogen-activated protein kinase (MAPK) pathways are well-known intracellular signaling modules in eukaryotes. MAPKs are serine-threonine protein kinases that are activated by diverse stimuli ranging from cytokines, growth factors, neurotransmitters, hormones, cellular stress, and cell adherence. These cascades contain three levels: MAPKKK → MAPKK → MAPK with the corresponding deactivating enzymes (for review see Widmannet al. 1999). Each layer has the GK topology but they do not seem to be GK switches (Bluthgen and Herzel 2003). Modularity, to avoid cross-talk among the pathways, is ensured by tethering the kinases to scaffold proteins as well as by direct interaction between the former. Simulations show that the MAPK pathway can convert a gradual input into a switch-like output. This property makes the cascade suitable for mediating processes like mitogenesis, cell fate induction, and oocyte maturation, where a cell switches from one discrete state to another. However, sigmoidicity depends on the assumptions of the current models [Huang and Ferrell 1996 (HF); Bhalla and Iyengar 1999 (BI)]. Sigmoidicity can be studied by fitting the curves to the Hill sigmoid (y = xn/(K + xn)), where n is the Hill coefficient (the higher it is, the steeper the sigmoid, the sharper the threshold). According to the HF model, dosage alterations of either MAPKK or MAPK-phosphatase induce important effects on sigmoidicity but according to the BI model, changes in almost all components individually lead to striking changes in sigmoidicity (Figures 3 and 4 A genetic toggle. Gene u encodes a repressor of v, which is in turn a repressor of u. I1 and I2 are the inducers. Presence of I1 will trigger synthesis of u and repression of v, which persists even after removal of the inducer. (a) The null clines (containing the loci of du/dt = 0 and dv/dt = 0) intersect at three points when the repressive activities of u and v are balanced and in the presence of cooperative repression. This translates into the existence of two stable and one unstable steady states (stable state 1/high v, state 2/high u). (b) When there is an imbalance in the amounts of the repressors (i.e., relative excess of u) the system is not bistable anymore. (c) Phase diagram of the system. The lines mark the transition between bistability (the systems can flip between state 1 and state 2) and monostability (the system is unable to flip). The bistable region lies inside of each pair of curves. The small horizontal arrow represents an increase of dosage of v for the same dosage of u. The system crosses the bifurcation line and crashes (becomes monostable, permanently in state 1). The small diagonal arrow (OK) represents a co-increase of u and v: the system remains in the bistability region. Increase of cooperativity (β and γ) leads to a broader region of bistability increasing the robustness of the system. This figure is courtesy of Gardneret al. (2000) and Nature (Macmillan Magazines; modified and reproduced with permission). Doubling the genomic content associated with an increase in nuclear and cellular volume can warrant successful duplication of certain dosage-sensitive genes. Consider that M is an inactive monomer and that Mn is the active multimer composed of n monomers. Synthesis (k1), degradation (k2), and interaction of the monomers are represented with chemical (a) and differential equations (b). For simplicity, let the association/dissociation reactions between monomers and oligomers be faster than synthesis/degradation. We can define a pseudo-equilibrium constant K in the steady state (i.e., when dM/dt = 0). As usual, increasing dosage of M (i.e., by 1.5×) will be represented by 1.5 × k1. If the initial volume does not change, this will imply a (1.5)n-fold increase of active Mn! Thus, maintaining a balance with another oligomer, say Nx, after strict coduplication in the same volume is possible only if the number of monomers involved in Mn and Nx are identical (n = x), for similar KM and KN. A polyploidization event increasing cell volume proportionally is more likely to restore balance whatever the K's, n, or x are. Similar arguments explain imbalance after a heterozygous deletion of M or N. of Bluthgen and Herzel 2003). Remarkably, co-increase or co-decrease of all components at once translates into minor changes in the position of the stimulus threshold and sigmoidicity. After doubling or halving all the components, HF curves have similar sigmoidicity and the threshold position (loosely defined as the x corresponding to y = 0.5 for a steep sigmoid) changes by only ∼25% with respect to the reference system described by Bluthgen and Herzel (2003). For the BI curves, the threshold changes by only ∼30% upon halving or doubling all components. On the basis of this evidence, it is safe to consider the whole pathway as the functional unit. Therefore, if the selectable property is a switch-like behavior, the whole MAPK module is likely to be duplicated or retained after a global duplication. Simulations also show that if MAPK accumulates in the nucleus, so that as much as 50% of cytoplasmic MAPK is sequestered after phosphorylation by active nuclear MAPKK (for constant concentrations of MAPK-pase), these converters operate at saturation and a true GK-switch behavior will appear (Ferrell 1998). To keep the proper balance, coduplication of MAPKK/MAPK-pase is required, as explained above. Note that in case of whole-genome duplication, the nucleus will enlarge, leading likely to similar concentrations of the relevant proteins as duplication (see Here u and v are the concentrations of the and and the rates of synthesis of u and v, and the cooperativity of repression of and (β and 1 imply a circuit can as it can be between two stable steady states u or using transient Each state is associated with the of of genes responsive to the Moreover, after the removal of the signal, the system remains in the state where it This is an mechanism to of a that can be a of cellular The of the system can be by the curves corresponding to du/dt = 0 and dv/dt = 0 or null define points corresponding to steady when there is cooperativity (β and 1) and the repressive activities of u and v are balanced the null intersect three This translates into the existence of two stable and one unstable steady states (Figure Thus, bistability depends on the cooperative repression of If the rates of synthesis of the two repressors are not the null clines will intersect only a single stable steady state (Figure One can a diagram to the system lies in a region of bistability (the system or in a region of monostability (the system does not Changing gene dosage of u v is represented by a change of (i.e., The outcome of this will depend on in the the system is (Figure When and are the system in a region of For instance, an increase of 1.5×) may be as this means the by 1.5 However, and imply a selective as synthesis of lot of repressor is systems are more likely to in a region of bistability with amounts of u and In such a a small shift to the can imply the line from a region of bistability to a region of However, of u and v the system to remain in the bistable region. how coduplication of dosage-sensitive genes can For instance, the to X Figure 1) or to E1 a GK Let B be titrating and genes and C be A and C can to yield and stoichiometric and should be present of B the concentration of and its say R in Figure 1). However, the increase in of and alone and their in the same is not by and they will tend to Thus, a small duplications to as as genes and This points an of global duplications to the existence of paralogs of dosage-sensitive genes. In an imbalance by changing dosage of one gene be by a parallel change of of the interacting partners. However, in the context of cellular pathways, this is clearly so only if both as monomers (case of X and R in the of Figure 1) at if they are the same number of monomers and similar pseudo-equilibrium which a co-increase of both would into a change in the concentration of active This is so because reactions in the same volume with a higher input of monomers. is to a general of this but an applicable to the models is in Figure This points to the need for a whole-genome duplication that implies an increase of cell volume which to restore the concentration of monomers and as duplication. yeast its this the cellular volume is about two times the volume are of whole-genome duplication in including and 2001; and 2004; et al. 2004). This is expected to be by deletion or leading in to of genes to avoid After duplication, the retained paralogs may in and of network modules unable to Indeed, and have found of paralogs in this In line with the they that the of on the of may be more important than the effect of the duplication of genes. Consider also that yeast alone MAPK the and cell pathways and two pathways involved in al. 1999). a common al. but according to the all of cannot result from single-gene of duplication associated with of dosage-sensitive interacting might explain the existence of and also the gene in the The is and does not the Indeed, and Veitia have shown the existence of an excess of gene pairs encoding subunits of stable protein complexes in yeast. We that these pairs may be modules by the of dosage-sensitive that may the of complexes upon duplication. The above show that the of dominant may have simple in terms of dosage imbalances and that the of a system to such alterations can be by illustrate also points of dosage-sensitive genes. The fact that there are changes of gene dosage does not the of compensation by or of in the same pathway as as fitness is not by the initial dosage Indeed, of is al. which may explain gene modules in and and as a to A duplications is to produce pathways more complex than studied by and and for in where single-gene duplication is or I Bluthgen for of the MAPK pathway and for on the I for both the and the and and for their