The average number of mRNA molecules per active gene in yeast can be remarkably low. Consequently, the relative number of copies of each transcript per cell can vary greatly from moment to moment. When these transcripts are encoding metabolic enzymes, how do the resulting variations in enzyme concentrations affect the regulation of metabolic intermediates? Using a kinetic model of galactose utilization in yeast, we analysed the transmission of noise from transcription and translation on metabolic intermediate regulation. In particular, the effect of the kinetic properties of the galactose-1-phosphate uridylyltransferase reaction on the transmission of noise was analysed.

INTRODUCTION

The cellular translation machinery allows the synthesis of large quantities of proteins [1] from a small number of transcripts [2]. The obvious consequence of such a small number of transcripts is that stochastic variations in this number can potentially translate into large oscillations in protein concentrations. Is this compatible with cellular homoeostasis? It has been suggested that noise in gene expression is a biologically important variable, is generally detrimental to organismal fitness, and is selected against [3]. Galactose uptake in the yeast Saccharomyces cerevisiae has been described extensively at the metabolic and genetic levels [48], and provides an attractive opportunity to analyse how these two levels integrate. It also provides an example of a metabolic intermediate that needs to be maintained at low levels (see below). Evolutionary forces may have optimized network structures [911]. We recently analysed how the structure of the genetic network for galactose uptake is affecting the transmission of intrinsic noise in yeast [12]. In the present paper, we describe how the kinetic properties of enzymes can also provide robustness against intrinsic noise in yeast.

Genetic regulatory interactions control switches that allow yeast cells to adapt to changing environments. Galactose uptake begins with the entry of galactose into the cell through a galactose-inducible transport process that is dependent on the protein Gal2p encoded by the gene GAL2, followed by the conversion of galactose into glucose 1-phosphate through the Leloir pathway (reviewed in [4,5]). This requires the action of three galactose-inducible enzymes, GK (galactokinase; EC 2.7.1.6), GT (galactose-1-phosphate uridylyltransferase; EC 2.7.7.12) and EP (UDP-galactose 4-epimerase; EC 5.1.3.2). These correspond to the proteins Gal1p, Gal7p and Gal10p encoded by the genes GAL1, GAL7 and GAL10 respectively. The system can be seen in 1.

Galactose uptake

Scheme 1
Galactose uptake

(a) Complete picture; (b) supply and demand blocks for GA1P. Full arrows represent mass transformation; broken lines represent regulatory interactions, with arrowheads for activation and blunt ends for inhibition. TR, transport of galactose inside the cell.

Scheme 1
Galactose uptake

(a) Complete picture; (b) supply and demand blocks for GA1P. Full arrows represent mass transformation; broken lines represent regulatory interactions, with arrowheads for activation and blunt ends for inhibition. TR, transport of galactose inside the cell.

The galactose transporter and the three enzymes that metabolize galactose are encoded as part of the GAL regulon. The GAL regulon contains the genes that encode the transporter and the three enzymes, together with other genes that include MEL1, for external α-galactosidase activity, and GAL3 and GAL80, which encode two proteins, Gal3p and Gal80p, that are involved in the regulation of the complete regulon.

The induction of the GAL genes is subject to a dual regulation, where synthesis is induced by galactose, but is repressed by glucose (reviewed in [68]). Induction is co-ordinately mediated by the action of one common activator, Gal4p, a common repressor, Gal80p, and two common releasers of the repressor, Gal3p and Gal1p, which are activated in the presence of galactose. Glucose repression includes specific proteolysis of Gal2p [13] and transcriptional repression of the GAL genes.

The co-ordinated expression of GAL genes allows the entry of carbon, but it can have a secondary consequence in the form of accumulation of metabolic intermediates. These levels are a consequence of the combined effect of the enzyme concentrations (dependent on the gene induction) and also of the kinetic properties of the enzymes, their affinities for substrates and catalytic constants. GA1P (galactose 1-phosphate) is the product of GK and the substrate of GT, and an example of an intermediate that needs to be maintained at low levels. An accumulation of GA1P (e.g. deficiency in GT) causes an inability to grow in galactose conditions [14]. In humans, this deficiency produces the most common form of galactosaemia [15]. The mechanism of GA1P toxicity remains unclear; however, through experiments, in part in yeast, it has been suggested that accumulated GA1P inhibits the normal function of key enzymes, such as UDP-glucose pyrophosphorylase, inositol monophosphatase or phosphoglucomutase (discussed in [14,1619]). Lai et al. [19] have found in human fibroblasts that intracellular concentrations of GA1P, as found in classic galactosaemia, inhibit competitively the UDP-glucose pyrophosphorylase, and reduce the intracellular concentrations of UGL (UDP-glucose) and UGA (UDP-galactose), potentially affecting membrane-bound glycoproteins and glycolipids.

The level of mRNA per cell for GK has been measured as 33 mRNA per cell by Iyer and Struhl [20] at saturation levels of GAE (external galactose). For lower levels of galactose, lower mRNA levels can be expected, as Biggar and Crabtree [21] have shown graded changes in GAL1 promoter activity.

Are the low levels of transcripts compatible with GA1P homoeostasis? Do stochastic variations in the number of mRNA molecules cause significant fluctuations in GA1P concentration? To answer these questions, we simulated the steady state of a kinetic model that includes the metabolic pathway for galactose uptake, and the stochastic transcription and translation of the transporter and enzymes. We focused on partial (10%) induction in the galactose-utilization pathway, when the number of transcripts per cell is low, resulting in the greatest variability in all concentrations. We showed that there is some transmission of transcriptional intrinsic noise. However, the system appears to be robust at 10% induction, and this transmission does not imply very high variations in the level of GA1P. This appears to depend primarily on the kinetic properties of the GT encoded by GAL7.

EXPERIMENTAL

Modelling strategy

An ODE (ordinary differential equation) model was used to describe the system behaviour. The equations are described in Table 1, the parameter values are provided in Table 2, and the steady-state values are provided in Table 3. This model includes two parts, one describing the transporter and enzymic part, and the other describing transcription and translation. ODE simulations are restricted to the transporter and enzymic part of the model, using the Mathematica program. The input pattern was obtained from stochastic simulations for transcription and translation. We modelled only intrinsic noise [22] in transcription and translation (chemical noise due to low copy numbers) because we were interested only in the effect of noise on relative protein concentrations. These latter simulations involved a small number of molecules per cell, and were performed using the Gillespie algorithm (implemented in Dizzy, http://labs.systemsbiology.net/bolouri/software/Dizzy/), which differs from the ODE approach in that it handles integer (rather than continuous) numbers of molecules per cell, and uses a Monte Carlo approach to account for the inherently stochastic nature of chemical reactions [23].

Table 1
Kinetic models

Using i and j as indexes, vi is the reaction rate catalysed by i, Ri is the mRNA concentration of gene GALi, Gi is the protein concentration of protein Galip encoded by GALi, Km(i,j) is the Km of enzyme i for substrate j, kcat(i) is the kcat of enzyme i, kdil is the cell volume dilution rate constant, kS, Ri is the initiation rate constant for mRNA transcription of gene GALi, kD, Ri is the intrinsic degradation rate constant of mRNA of gene GALi, kS, Gi is the initiation rate constant for synthesis of protein Galip and kD, Gi is the intrinsic degradation rate constant of protein Galip. α is the interactive constant [46]. F is the fractional saturation of the regulatory complex on DNA. See Table 2 for parameter values and calculation.

graphic
 
graphic
 
Table 2
Parameter values

Using i as an index, τ50,Ri is the half life of mRNA of gene GALi. For conversion from molecules per cell (m/c) into mM, we assume 15×10−12 g of cell dry weight per yeast haploid cell [50] and 2.38 ml of cell volume per g of cell dry weight [18].

Parameter Value Units Note 
vTRKm(TR, GA) mM Value for the inducible high-affinity transport from [29
 α – [47
k 4350 min−1 Adjusted value (see text) 
vGKkcat(GK) 3350 min−1 [31]. Other measured values are 720 min−1 and 1.2 mM from [32
Km(GK, GAI) 0.6 mM [31
KIU 19.1 mM [32
KIC 160 mM [32
vGTkcat(GT) 59200 min−1 [34]. kcat is per enzyme molecule (dimer) 
Km(GT, GA1P) 4.0 mM [34
Km(GT, UGL) 0.26 mM [34
vEPkcat(EP) 40×3890 min−1 [30]. kcat is per enzyme molecule (dimer) (adjusted by multiplying by 40; see text) 
Km(EP, UGA) 0.22 mM [30
Keq 3.5 – [30]. Keq refers to the ratio UGL/UGA [30
Km(EP, UGL) 0.25 mM In Vicia faba [48
F Variable – Fractional saturation 
kS, R2 0.52 m/c·min−1 kS, R2=(kD, R2+kdil)·R2/F 
kS, R1,7,10 1.09 m/c·min−1 kS, R1,7,10=(kD, R1,7,10+kdil)·R1,7,10/F 
kD, R2 0.014 min−1 kD, R2=(ln2)/τ50,R2 
kD, R1,7,10 0.032 min−1 kD, R1,7,10=(ln2)/τ50,R1,7,10 
kS, G2 6.94 min−1 kS, G2=(kD, G2+kdil)·G2/R2 
kS, G1,7,10 9.92 min−1 kS, G1,7,10=(kD, G1,7,10+kdil)·G1,7,10/R1,7,10 
kD, G2,1,7,10 0.022 h−1 Average for 50 proteins [39
kdil 0.097 h−1 [18
Values used only for the deduction of values above:  
τ50,R2 49 min Poly(A) half-life estimates are from supplementary material of [49] (the value for R1, R7 and R10 is the average of values reported) 
τ50,R1,7,10 22 min  
R1 33 (F=1) m/c [20
R2,7,10 R1 m/c [25
G2/R2 3500 – Approximate average protein/mRNA molecule for “Transport facilitation” and 
G1,7,10/R1,7,10 5000 – “Metabolism” functional categories (see supplementary material of [1]) 
Parameter Value Units Note 
vTRKm(TR, GA) mM Value for the inducible high-affinity transport from [29
 α – [47
k 4350 min−1 Adjusted value (see text) 
vGKkcat(GK) 3350 min−1 [31]. Other measured values are 720 min−1 and 1.2 mM from [32
Km(GK, GAI) 0.6 mM [31
KIU 19.1 mM [32
KIC 160 mM [32
vGTkcat(GT) 59200 min−1 [34]. kcat is per enzyme molecule (dimer) 
Km(GT, GA1P) 4.0 mM [34
Km(GT, UGL) 0.26 mM [34
vEPkcat(EP) 40×3890 min−1 [30]. kcat is per enzyme molecule (dimer) (adjusted by multiplying by 40; see text) 
Km(EP, UGA) 0.22 mM [30
Keq 3.5 – [30]. Keq refers to the ratio UGL/UGA [30
Km(EP, UGL) 0.25 mM In Vicia faba [48
F Variable – Fractional saturation 
kS, R2 0.52 m/c·min−1 kS, R2=(kD, R2+kdil)·R2/F 
kS, R1,7,10 1.09 m/c·min−1 kS, R1,7,10=(kD, R1,7,10+kdil)·R1,7,10/F 
kD, R2 0.014 min−1 kD, R2=(ln2)/τ50,R2 
kD, R1,7,10 0.032 min−1 kD, R1,7,10=(ln2)/τ50,R1,7,10 
kS, G2 6.94 min−1 kS, G2=(kD, G2+kdil)·G2/R2 
kS, G1,7,10 9.92 min−1 kS, G1,7,10=(kD, G1,7,10+kdil)·G1,7,10/R1,7,10 
kD, G2,1,7,10 0.022 h−1 Average for 50 proteins [39
kdil 0.097 h−1 [18
Values used only for the deduction of values above:  
τ50,R2 49 min Poly(A) half-life estimates are from supplementary material of [49] (the value for R1, R7 and R10 is the average of values reported) 
τ50,R1,7,10 22 min  
R1 33 (F=1) m/c [20
R2,7,10 R1 m/c [25
G2/R2 3500 – Approximate average protein/mRNA molecule for “Transport facilitation” and 
G1,7,10/R1,7,10 5000 – “Metabolism” functional categories (see supplementary material of [1]) 
Table 3
Steady-state values at 10% and 100% fractional saturation levels at 0.5 mM GAE

For conversion from μmol or mmol/g of dry weight into mM, we assume 2.38 ml of cell volume per g of dry wright as used by Ostergaard et al. [18] (*), and 2.7 ml of cell volume per g of dry weight as suggested by Daran et al. [28] (†). It should be noted that our model gives a value of 0.26 mM for GA1P at F=1 and 2% GAE, in agreement with the 0.3 mM from Daran et al. [28] (‡), and the flux of galactose uptake (J) reported by Ostergaard et al. [18] probably reflects the fact that cells were previously pre-incubated in conditions of full induction, and they were then allowed to deplete the GAE concentration to 0.47 mM, resulting in a higher flux (this situation would correspond to F=1) (§).

 F Reference 
 0.1 Converted Original 
GA1P (mM) 0.09 0.09 0.09* 0.21 μmol/g of dry weight [18
   0.3†‡ 0.81 μmol/g of dry weight (2% GAE) [28
UGL (mM) 0.82 0.82 0.82† 2.21 μmol/g of dry weight (2% GAE) [28
UGA (mM) 0.24 0.24 0.24† 0.65 μmol/g of dry weight (2% GAE) [28
J (mM/min) 0.5 7.4*§ 1.06 mmol/(g of dry weight·h) [18
UGL/UGA 3.4 3.4 3.4 (2% GAE) [28
   3.9 (0.1% GAE in human fibroblast) [19
 F Reference 
 0.1 Converted Original 
GA1P (mM) 0.09 0.09 0.09* 0.21 μmol/g of dry weight [18
   0.3†‡ 0.81 μmol/g of dry weight (2% GAE) [28
UGL (mM) 0.82 0.82 0.82† 2.21 μmol/g of dry weight (2% GAE) [28
UGA (mM) 0.24 0.24 0.24† 0.65 μmol/g of dry weight (2% GAE) [28
J (mM/min) 0.5 7.4*§ 1.06 mmol/(g of dry weight·h) [18
UGL/UGA 3.4 3.4 3.4 (2% GAE) [28
   3.9 (0.1% GAE in human fibroblast) [19

Model simulations were performed at low (10%) induction of gene expression. Biggar and Crabtree [21] have shown that control of the GAL1 promoter activity [measured as fluorescence intensity from cells with a reporter GFP (green fluorescent protein)–GAL1 promoter] can be either graded or binary. Interestingly, they have shown that stimulation with variable levels of galactose produces graded changes in GAL1 promoter activity. Li et al. [24] have measured induction from an initial GAE concentration of 0 g/l to 20 g/l (111 mM), and have shown that this graded induction has a maximum attained at an initial GAE concentration of between 1 and 3 g/l. The value of 33 transcripts of GAL1 per cell reported by Iyer and Struhl [20] was under full induction, and this level of induction should be similar for GAL2, 7 and 10, as observed when comparing results from microarray experiments of steady-state induction (galactose) with those of repression (glucose) [25]. Taking all these results together, in the present paper, full induction is assumed to result in 33 GAL2, 1, 7 or 10 mRNA molecules per cell, and low (10%) induction is assumed to produce 3.3 molecules per cell. Our simulations compute the mRNA and protein concentrations within a single cell over several generations of asexual cell division at low induction.

MCA (metabolic control analysis) was applied to the analysis of steady-state sensitivities (see [26,27] for a review). These are divided into system and local sensitivities. The latter denote the sensitivity of an isolated reaction step (an enzyme or a block of reactions) and are called elasticities:

 
formula

where vi refers to the rate of the reaction i, x refers to a variable that affects it, and ss refers to the steady state. System sensitivities are divided into control coefficients and response coefficients. Control coefficients can be defined as the sensitivity of a dependent variable Y (concentration, flux) with respect to the concentration of one enzyme, or more generally, with respect to the activity of a reaction step i:

 
formula

A response coefficient refers to a system sensitivity with respect to a perturbation of any parameter. When the parameter p acts only on the reaction step i, the response coefficient can be expressed as:

 
formula

System characteristics that the model reproduces

(1) The kinetic parameters of this model were adjusted to result in a steady state closely matching the values provided by Ostergaard et al. [18] for GA1P, and by Daran et al. [28] for UGL and UGA and the ratio UGL/UGA.

(2) Ostergaard et al. [18] analysed the steady state in a chemostat continuous culture at 0.47 mM GAE. The value for GA1P was adjusted by changing the value of the parameter k for the transporter. The resulting value for the parameter k provides a Vmax close to that published by Reifenberger et al. [29] for inducible high-affinity transport.

(3) UGL+UGA is a conserved quantity in the model, and can be fixed to that published by Daran et al. [28]. The equilibrium constant for the EP reaction was measured from purified samples by Fukasawa et al. [30], Keq=UGL/UGA=3.5, which agrees closely with the ratio UGL/UGA=3.4, as measured by Daran et al. [28] from cell samples. We adjusted the value of kcat(EP) to reproduce the reported UGL/UGA ratio.

(4) Schell and Wilson [31], and Timson and Reece [32] have reported product inhibition for GK at high concentrations of GA1P. This product inhibition has been included in the model from the data provided by Timson and Reece [32], who analysed the GK reaction mechanism, and showed that GA1P is a mixed inhibitor in the GK reaction (see Tables 1 and 2 for details).

(5) To simulate low induction (10%), we set the fractional saturation regulating transcription (parameter F in Tables 1 and 2) to 0.1, and GAE to 0.5 mM (F=1 for full induction). The value of 0.5 mM GAE for low induction was selected because Li et al. [24] have reported gene expression rates of 10% at around 0.1 g/l GAE.

(6) The model was validated by analysing the dependence of GA1P on all parameters of the transporter and enzymic parts of the model. Each parameter was analysed in a range from 0.1- to 10-fold change, and the most significant are plotted in Figure 1. The activities can be changed by adjusting the concentration of transporter or enzymes, or through changes of catalytic constants. The simple nature of the metabolic pathway results in a significant dependence on the transporter activity and the enzymic activities for GK and GT (discussed below) and on the Km of GT for GA1P [Km(GT, GA1P)]. This last point is not surprising considering that GA1P is the substrate of the enzyme encoded by GAL7.

Model sensitivity with respect to GA1P

Figure 1
Model sensitivity with respect to GA1P

The curves show the steady-state value of GA1P for ±10-fold changes of all transporter and enzyme activities, and the only parameters of the metabolic pathway model [Km(GT, GA1P) and GAE] that can produce a change in GA1P greater than 0.2 mM. The steady state corresponds to GAE=0.5 mM.

Figure 1
Model sensitivity with respect to GA1P

The curves show the steady-state value of GA1P for ±10-fold changes of all transporter and enzyme activities, and the only parameters of the metabolic pathway model [Km(GT, GA1P) and GAE] that can produce a change in GA1P greater than 0.2 mM. The steady state corresponds to GAE=0.5 mM.

System characteristics not included in the model

(1) Galactose transport can be divided into three components [29,33]: two galactose-inducible transport systems, a high-affinity process and a low-affinity process, both dependent on Gal2p; and one residual constitutive low-affinity transport system, independent of Gal2p. The mechanism of this inducible two-component transport machanism is unknown, and we are assuming only the high-affinity component.

(2) GK is described by an irreversible reaction dependent only on GAI (internal galactose). The only available information refers to the forward reaction. However, a scenario of reversibility is implausible because it would imply that the Leloir pathway could reverse-metabolize GL1P (glucose 1-phosphate) into GAI, which would induce the GAL regulon in the absence of GAE.

(3) GK and GT catalyse bi-substrate reactions, but the Km used in the model corresponds to an apparent Km obtained by fixing the concentration of the other substrate [31,34]. The effect of variations in these Kms is negligible with respect to GA1P accumulation, with the exception of the Km of GT for GA1P (Figure 1).

(4) EP is a bifunctional enzyme with aldolase 1-epimerase (mutarotase) activity [35], not modelled here.

(5) UGL has a strategic position in yeast metabolism, and, as for GA1P, its regulation appears to be important. It is fundamental for the synthesis of the cell wall, the synthesis of glycogen and trehalose, N-glycosidation of proteins, and other roles (see [28]). Among the enzyme activities that affect UGL is that of UDP-glucose pyrophosphorylase, where GL1P with UTP is transformed into UGL and PPi by the action of UDP-glucose pyrophosphorylase. Also, UDP-glucose pyrophosphorylase has been shown to be a bifunctional enzyme [17,19], which catalyses the conversion of GA1P into UGA, although its activity is not relevant under physiological conditions [19]. All these transformations affecting UGL are not included in the model. The reaction catalysed by EP appears to be enough to reproduce the ratio UGL/UGA measured by Daran et al. [28] (discussed above).

(6) We assumed that induction in continuous chemostat culture results in a steady-state response, as assumed by Ostergaard et al. [18]. However, Braun and Brenner [36] have reported recently that, over several generations, the system converges to a single robust steady state, independent of the carbon source. The steady state we refer to should therefore be considered meta-stable on long time scales.

(7) Yeast growth involves relatively fast asexual cell division by budding [37]. The way that the induced state is transmitted to the daughter cells could impose fluctuations in the steady state in a continuous culture, and is not addressed by our model.

(8) Based on the experimental evidence of Gal7p subcellular localization, which requires co-expression of Gal1p and Gal10p, Christacos et al. [38] have suggested that substrate/product channelling or other interactions could alter the kinetics of the enzymes.

(9) mRNA and protein turnover are modelled with first-order degradation. Both decays depend on two rate constants, the cell volume dilution rate constant (kdil) and the intrinsic decay constant (kD, R or kD, G). The growing machinery of yeast, a mixture of cell growing and cell division by budding [37], is responsible for cell volume dilution. We adopted, as an estimate of the cell volume dilution rate, the dilution rate of 0.097 h−1 reported for the chemostat steady-state culture of Ostergaard et al. [18].

(10) Protein degradation and the ratio of protein/mRNA are based on general measures in S. cerevisiae, not specific for GAL products [1,39].

(11) We did not consider differences between mRNA and protein expression [40] owing to a lack of pertinent data.

RESULTS AND DISCUSSION

Transmission of transcriptional noise

Figure 2 illustrates transmission of noise from mRNA and protein concentrations to the level of GA1P under low induction. However, this transmission does not produce significant variations in the level of GA1P, and we can conclude that the system is robust. The transmission of noise depends on the relative time scales affecting the components of the network. The turnover of mRNA, proteins and metabolic intermediaries represent three different time scales. Inspection of rate constants in Table 2 provides an idea of the difference between mRNA and proteins. The time scale for proteins to return to steady state after perturbations is clearly larger than that of the other species. A longer time scale for turnover implies a longer time to steady state after a perturbation. The consequence of this is that the effect of noise in mRNA concentrations, with apparent oscillations with short periods, is smoothed into small oscillations of longer period. In Figure 2(b), the steady-state concentration of GA1P can be estimated at each time point using the corresponding transporter and enzyme concentrations at this point. Interestingly, this steady-state prediction closely matches the dynamic time course plotted for GA1P. This shows that the large periods of the apparent oscillations that affect the levels of proteins are larger than the time that GA1P needs to change its steady state.

Transmission of noise to GA1P

Figure 2
Transmission of noise to GA1P

(a) Protein time courses. (b) GA1P time course. The 2000 min time course shown corresponds to the period with the highest transmission in noise to GA1P in a total time course simulation of 30000 min. The values in parentheses correspond to the coefficients of variation (S.D./mean) estimated from the total time course. These are in broad agreement with recent results reported in [22].

Figure 2
Transmission of noise to GA1P

(a) Protein time courses. (b) GA1P time course. The 2000 min time course shown corresponds to the period with the highest transmission in noise to GA1P in a total time course simulation of 30000 min. The values in parentheses correspond to the coefficients of variation (S.D./mean) estimated from the total time course. These are in broad agreement with recent results reported in [22].

GA1P regulation

The fact that the apparent oscillations in the protein levels are much slower than the time that GA1P needs to change its steady state simplifies the analysis of noise transmission to the analysis of the steady-state regulation of GA1P. In order to analyse the steady-state regulation of a particular intermediate, a multi-step system can be simplified to a simple system with a supply block, the intermediate, and a demand block [41]. In 1(b) (galactose uptake), we show the supply and demand system around the GA1P intermediate as a simplification of the entire pathway in 1(a). In Figure 3, we apply a visual and quantitative analysis [41] to describe how the properties of supply and demand blocks control the steady-state flux, and contribute to the regulation of the intermediate, here GA1P. The six possible changes that we can introduce by changing the activities of supply and/or demand blocks are presented, where the numbered point 1 corresponds to the original steady state, and the others refer to the possible changes. The points in Figure 3 refer to the following situations: (1) reference steady state at 0.5 mM GAE and 10% induction (F=0.1), (2) proportional increase in the activities of both supply and demand blocks, (3) proportional decrease in the activities of both supply and demand, (4) increase in supply activity alone, (5) decrease in supply activity alone, (6) increase in demand activity alone, and (7) decrease in demand activity alone. Scenarios (2) and (3) correspond to co-ordinate regulation (constant ratio of supply and demand), while scenarios (4)–(7) correspond to changing the supply/demand ratio. In scenario (2), the co-ordinate expression of structural genes during galactose induction is in keeping with the observation that a co-ordinate increase in the activity of all the enzymes of the pathway allows an increase of flux without affecting the level of intermediates [42,43]. In scenario (5), the stronger repression of supply by glucose, through the proteolysis of Gal2p helps to ensure against transient increases in GA1P. The simultaneous change in absolute values in scenario (2) affects the flux, but not GA1P levels, while the relative change in scenario (5) affects both flux and GA1P. These observations are a consequence of the summation theorems of MCA (see [26,27] for a review):

 
formula

Supply and demand analysis

Figure 3
Supply and demand analysis

Reaction rates of the supply block (vsupply) against GA1P, and the demand block (vdemand) against GA1P. Broken lines correspond to increases or decreases in activities for supply or demand. The intersections of the two sets of curves correspond to steady-state flux and GA1P levels, and are represented with numbered points (see text for details). (a) Default model. (b) Modified model with a 20-fold increase of the ratio GA1P/Km(GT, GA1P).

Figure 3
Supply and demand analysis

Reaction rates of the supply block (vsupply) against GA1P, and the demand block (vdemand) against GA1P. Broken lines correspond to increases or decreases in activities for supply or demand. The intersections of the two sets of curves correspond to steady-state flux and GA1P levels, and are represented with numbered points (see text for details). (a) Default model. (b) Modified model with a 20-fold increase of the ratio GA1P/Km(GT, GA1P).

Thus the degree of transmission of noise depends on the relative activity of supply and demand blocks, and not on their absolute value.

How do the kinetic parameters of GT affect noise transmission?

A formalization of the relationships in Figure 3 for infinitesimal changes can be easily derived in terms of elasticities and control coefficients [41]:

 
formula

The slope of the curve at each (numbered) steady state in Figure 3 corresponds to the elasticity at that point. The slope of the demand curve is positive, i.e. εvdemandGA1P>0. For the supply block, the horizontal line for supply in Figure 3 corresponds to an elasticity εvsupplyGA1P=0. This is a consequence of both the assumption for GK of irreversibility and the fact that the effect of the inclusion of product inhibition is negligible.

Figure 1 shows that the concentration of GA1P depends mainly on the kinetic properties of GT. We can explore this fact algebraically in terms of MCA. If we analyse the demand block as a separate system, the block elasticity εvdemandGA1P of the whole system corresponds to the response coefficient of the flux J through the demand block with respect to GA1P (RJdemandGA1P). This response coefficient can be described by either of two expressions (see the Appendix for derivation):

 
formula

Numerical analysis of the values of the components of eqns 6 and 7 (Table 4) reveals that the only effective contributor to the value of RJdemandGA1P is the elasticity εvGTGA1P. Thus all flux control of the demand block is in GT (eqn 6), and the contribution of UGL variability is negligible (eqn 7). The elasticity εvGTGA1P is described by the following expression:

 
formula
Table 4
Numerical evaluation of the components of eqns 6 and 7
 Whole system Demand block alone Elasticities 
 CGTJ CGTJdemand RGA1PUGL εGA1PvGT εUGLvGT 
Default model 0.00 1.00 −0.01 0.97 0.01 
Model with 20-fold increased GA1P/Km(GT, GA1P) 0.00 1.00 −0.01 0.63 0.09 
 Whole system Demand block alone Elasticities 
 CGTJ CGTJdemand RGA1PUGL εGA1PvGT εUGLvGT 
Default model 0.00 1.00 −0.01 0.97 0.01 
Model with 20-fold increased GA1P/Km(GT, GA1P) 0.00 1.00 −0.01 0.63 0.09 

From this expression, we can see that the elasticity increases as the ratios GA1P/Km(GT, GA1P) or Km(GT, UGL)/UGL decrease, and vice versa, with a theoretical maximum value of one. Interestingly, the situation predicted by the model is close to the maximum value of 1 (Table 4). The resulting control coefficients (CGA1Psupply and CdemandGA1P) are both close to the minimum absolute value of 1, according to eqn 5.

We can analyse how the transmission of noise to GA1P concentration is affected by the kinetic parameters of GT. However, it is not possible to change the value for a parameter without changing the steady state. At the steady state, an equality involving the flux (J) on one side, and the equation describing vGT (see Table 1) on the other, constrains the values of VmaxGT, Km(GT, GA1P) and Km(GT, UGL). This constraint can be written by grouping the ratios involving substrates and Km as:

 
formula

If we want to maintain the reference steady state deduced from the bibliography, a decrease or increase in one parameter needs to be compensated for by a change in other parameters.

If we want to change the value for the ratio GA1P/Km(GT, GA1P) without changing the steady state for GA1P, it is necessary to impose a change on one of the other parameters in eqn 9. For example, a 20-fold increase of the ratio GA1P/Km(GT, GA1P) can be achieved by decreasing the Km(GT, GA1P) to a value of 0.2 mM and the kcat(GT) to 4600 min−1. All steady-state values for flux and intermediates are maintained, but at the cost of changes in the elasticities and control coefficients (see eqns 58). For the preceding example, εv demandGA1P≈0.6 and CGA1Psupply=−CdemandGA1P≈1.6. (Table 4). The consequence is higher transmission of noise to GA1P, as shown graphically in Figure 2. Figure 3(b) shows why the sensitivity to the transmission of noise to GA1P is increased.

More complex scenarios

The scenario analysed is the most simple, with the first block being described by an irreversible process with negligible product inhibition, and the second block being described by an equation showing a hyperbolic curve with respect to GA1P. In this scenario, εGA1Pvsupply=0 and εvdemandGA1P≈1 (Table 4). Optimal homoeostasis is provided by maximizing the value of the denominator of eqn 5 [44]. Thus more complex scenarios can be modelled with more optimal reductions of noise transmission as follows.

Supply block

The consequence of the assumption of irreversibility and negligible product inhibition for GK is that εGA1Pvsupply=0 and that all control in the flux is in the supply block, as is evident from Figure 3. As noted by Hofmeyr and Cornish-Bowden [41], when flux is controlled by one block, the other block determines the degree to which concentration of the linking metabolite is homoeostatically maintained. In other words, all the sensitivity of GA1P depends on εvdemandGA1P only. εGA1Pvsupply would be lower than 0 if either the reverse reaction for GK or product inhibition for GK were not negligible. It is interesting to note that, under such conditions, noise transmission would be reduced further, as the absolute value for CGA1Psupply and CGA1Pdemand would be lower.

Demand block

Eqns 6 and 7 show the contribution of the elasticity εGA1PvGT to the block elasticity εvdemandGA1P. For εGA1PvGT, more optimal situations could be reached: for example, a demand block reaction rate following co-operative kinetics with respect to its substrate. In this case εvdemandGA1P could be higher than 1. The other elements contributing to the value of εvdemandGA1P in eqns 6 and 7 can change according to UGL regulation. Our model does not include the UDP-glucose pyrophosphorylase and other enzyme activities that affect UGL. The activities of these reactions could produce a change in the value of eqns 6 and 7, potentially with worse noise transmission.

Is a very low level of induction possible?

Our simulations are based on a model that only tries to reproduce qualitatively the regulation of GA1P, and we cannot establish the exact level of GA1P that is toxic. However, there are measurements that allow estimation of the approximate GA1P concentration values that could be toxic. Lai et al. [19] analysed galactose toxicity in human fibroblasts, and reported that a GA1P concentration of 2.5 mM increases the apparent Km of purified UDP-glucose pyrophosphorylase for GL1P from 19.7 μM to 169 μM. Also, they reported a concentration of GA1P of 5.2 mM for GT-deficient cells. From our analysis, we can speculate that incrementally reducing the level of induction below 10%, one reaches a threshold beyond which intrinsic mRNA noise would translate to large GA1P oscillations. However, this situation is probably never reached. Multiple binding sites for the common activator Gal4p in all the genes for transporter and the enzymes could result in strong repression by Gal80p under non-induction that could also be effective at very low induction.

The repression of GAL genes is in part based on the common repressor Gal80p that interacts with the activator Gal4p. Under non-induced and non-repressed carbon conditions (for example in glycerol) a basal expression level is found for GAL3 and GAL80 (each with a single Gal4p-binding site), which contrasts with the non-expression of genes that encode the enzymes for galactose uptake [45]. Melcher and Xu [45] suggested a mechanism to explain the basal levels of the GAL genes with single Gal4p-binding site and the complete repression of GAL genes with multiple Gal4p-binding sites. They suggested that a Gal80p–Gal80p interaction on adjacent Gal4p-binding sites could stabilize the repression of Gal80p in genes with multiple Gal4p-binding sites. In addition, the subcellular localization and formation of a complex involving Gal7p, Gal1p and Gal10p [38] is another potential way to reduce noise.

Conclusions

We have shown that for biologically plausible concentrations of GA1P and mRNA in the yeast galactose-utilization pathway, the level of the toxic intermediate GA1P is well regulated at 10% induction. We traced this robustness to be primarily due to the kinetic parameters of one of the Leloir pathway enzymes, GT.

We thank Earl Solis, Vesteinn Thorsson, Daehee Hwang and Alistair Rust for their contributions to the model development. We thank the anonymous referees for their insightful comments during the review process.

Abbreviations

     
  • EP

    UDP-galactose 4-epimerase

  •  
  • GAE

    external galactose

  •  
  • GAI

    internal galactose

  •  
  • GA1P

    galactose 1-phosphate

  •  
  • GK

    galactokinase

  •  
  • GL1P

    glucose 1-phosphate

  •  
  • GT

    galactose-1-phosphate uridylyltransferase

  •  
  • MCA

    metabolic control analysis

  •  
  • ODE

    ordinary differential equation

  •  
  • UGA

    UDP-galactose

  •  
  • UGL

    UDP-glucose

APPENDIX

Derivation of eqn 6

GA1P acts only through GT and no other enzymes in the system, thus eqn 6 is a particular case of eqn 3. Note that the flux control coefficient for the demand block alone (CGTJdemand) and the flux control coefficient for the whole system (CGTJ) are different (see Table 4).

Derivation of eqn 7

Provided that the reaction rate through the demand block (vdemand) corresponds to the reaction rate through GT (vGT, which depends on GA1P and UGL), and that UGL depends on GA1P through the action of EP, then an expression for vdemand can be written as:

 
formula

In the context of a separate block demand, GA1P is an independent variable, and UGL is a dependent variable. By application of the chain rule, RGA1PJdemand can be derived:

 
formula

Now, considering the definitions for elasticity (eqn 1) and response coefficient (eqn 3), RGA1PJdemand can be described by the following expression:

 
formula

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