a theoretical interpretation of the transient sialic acid

doi:10.1016/j.jmb.2007.10.073 J. Mol. Biol. (2008) 375, 875–889

Available online at www.sciencedirect.com

A Theoretical Interpretation of the Transient Sialic AcidToxicity of a nanR Mutant of Escherichia coli

Dominique Chu1⁎, Jo Roobol2 and Ian C. Blomfield2

1Computing Laboratory,University of Kent,Canterbury CT2 7NF, UK2Biomedical Research Group,Department of Biosciences,University of Kent,Canterbury CT2 7NJ, UK

Received 11 June 2007;received in revised form23 October 2007;accepted 25 October 2007Available online7 November 2007

*Corresponding author. E-mail [email protected] used: Fru-6P, fructo

0022-2836/$ - see front matter © 2007 E

This article reports on experimental evidence that an Escherichia coli nanRmutant shows inhibited growth in N-acetylneuraminic acid. This effect isprevented when inocula are grown in an excess of glucose, but not in anexcess of glycerol. The nanATEK operon is controlled by cataboliterepression, suggesting that diminished expression of the nanATEK operonin the presence of glucose explains the inocula effects. Neither double nanR–nagC nor nanR dam mutants show growth inhibition in the presence ofN-acetylneuraminic acid. A theoretical model of N-acetylneuraminic acidmetabolism (i.e., in particular of the nanATEK and nagBACD operons) ispresented; the model suggests an interpretation of this effect as being due totransient high accumulations of GlcNAc-6P in the cell. This accumulationwould lead to suppression of central metabolic functions of the cell, thuscausing inhibited growth. Based on the theoretical model and experimentaldata, it is hypothesised that the nanATEK operon is induced in a two-stepmechanism. The first step is likely to be repressor displacement by N-acetylneuraminic acid. The second stage is hypothesised to involve Dammethylation to achieve full induction.

© 2007 Elsevier Ltd. All rights reserved.

Edited by J. Kam
Keywords: Escherichia coli; N-Acetylneuraminic acid; GlcNAc-6P toxicity
Introduction

Sialic acids comprise a family of nine-carbon ketoamino sugars of which N-acetylneuraminic acid isthe best studied. Largely restricted to animals andtheir pathogens, sialic acids commonly occupy theterminal position in glycoconjugates on eukaryoticcell surfaces. They are an important nutrient formany bacteria that colonize animal hosts, eventhough some, such as Escherichia coli, lack a sialidaseand hence cannot liberate free N-acetylneuraminicacid from host glycoconjugates directly. In addition,microbial pathogens frequently decorate their sur-faces with sialic acid as a means of circumventinghost defenses; in E. coli, N-acetylneuraminic acidcontrols the expression of type 1 fimbrial adhesinand hence is a regulatory signal (for a recent review,see Severi et al.1).Uptake of N-acetylneuraminic acid into the

periplasm of E. coli occurs either via the general

ess:

se-6P; WT, wild type.

lsevier Ltd. All rights reserve

porins OmpF and OmpC, or via a selective channel(NanC).2 NanT is required for the transport ofN-acetylneuraminic acid across the cytoplasmicmembrane, and metabolism of N-acetylneuraminicacid to GlcNAc-6P requires nanATEK.3–6 Expressionof the nanATEK operon is subject to cataboliterepression, and hence is inhibited by glucose andis activated by Crp.7 Furthermore, it is induced byN-acetylneuraminic acid, which inactivates theNanR repressor.7 Two further steps, catalysis byNagAB and repression by the GlcNAc-6P-respon-sive regulator NagC, generate the central metabolicintermediate fructose-6P (Fru-6P) from GlcNAc-6P8

(see Fig. 1).The operator site for NanR at the nanATEK

promoter forms part of a conserved 27-bp sequencethat is also found upstream of the nanC gene.9 Theconserved element includes a Dam methylation site(GATCNanR), and NanR binding to its operatorprevents methylation of GATCNanR in vivo at bothnanC10 and nanA (unpublished data). Expression ofboth the nanATEK and the nanC operons is de-creased in a dam mutant.11 Furthermore, GATCNanR

is found within a nucleotide sequence that isexpected to be a relatively poor substrate for Dam

d.

mailto:[email protected]

Fig. 1. Schematic representation of N-acetylneuraminic acid transport and metabolism; details of the pathway aredescribed in the main text. The bacterial outer membrane (OM) and inner membrane (IM) are represented by verticallines. Proteins are boxed, whereas N-acetylneuraminic acid (Neu5Ac) and metabolic intermediates are not. Straightarrows indicate the flow of metabolic intermediates. The conversion of NanR and NagC from active repressors (NanRAand NagCA) to inactivated forms (NanRI and NagCI) by Neu5Ac and GlcNAc-6P, respectively, is indicated by bentarrows. GlcN-1P enters glycolysis by conversion to Fru-6P.

876 Transient Sialic Acid Toxicity of a nanR Mutant

methylase.12 Thus, if methylation of GATCNanR con-trols the nanATEK and nanC promoters, full expres-sion of these operons should be delayed untilGATCNanR is methylated following dissociation ofNanR from its operator sites when N-acetylneur-aminic acid is first encountered.In this contribution, we will provide experimental

evidence that a mutation of nanR, depending onthe inoculum, may lead to strong growth inhibitionin the presence of N-acetylneuraminic acid. Simul-taneous mutations of nagC or dam suppress thiseffect and lead to growth rates that are more com-parable to those of the wild type (WT). We inter-pret the inhibited growth of the nanR mutant ina N-acetylneuraminic acid assay as a result ofGlcNAc-6P accumulation in the cell; this conditionhas previously been shown to have toxic effects,although the mechanism of toxicity is unknown. 13

We use a differential equation model of thecontrol mechanisms of the nanATEK and nagBACDoperons and a set of simulations in order to quali-tatively explain the experimentally observed beha-viour of the system. The model suggests that thetoxicity is a transient effect (i.e., does not affect thelong-term behaviour of the system). Furthermore, inconjunction with available experimental data, themodel implies that full induction of nanATEK isachieved by a two-step mechanism that we proposecould involve methylation of the operator site: Inthe first step, the turnover of N-acetylneuraminicacid in the cell is slowly increased during shortperiods of NanR displacement. The second phaseis reached when methylation of the operator siteallows full induction of the nanATEK operon. Inthis contribution, we suggest that this two-stepinduction mechanism is necessary in order for thecell to avoid high transient GlcNAc-6P accumula-tions (with toxic effect), while at the same timebeing able to benefit from high steady-state turn-over rates.

Results

N-acetylneuraminic acid delays growth in ananR mutant background

Expression of the nanATEK operon is suppressedby NanR, a FadR-like transcriptional regulator thatis inactivated by N-acetylneuraminic acid. In thecourse of growth experiments, we found that rapidexponential growth in a nanR mutant was delayedconsiderably following inoculation into Mops [3-(N-morpholino) propanesulphonic acid] medium con-taining N-acetylneuraminic acid as the sole carbonsource (Fig. 2a) and, even after 7 h, was not as rapidas the WT.In the experiments described above, inocula were

grown to saturation overnight in a glucose minimalmedium containing a limiting (1.11 mM) amount ofthe carbon source.We reasoned that the toxic effect ofN-acetylneuraminic acid was due to overexpressionof the nanATEK operon in the mutant background,leading to an accumulation of excessive levels of N-acetylneuraminic acid, or one or more of its metabolicproducts. The nanATEK operon is suppressed byglucose (catabolite expression) independently ofNanR.7 In support of the hypothesis that overexpres-sion of the nanA operonproducesN-acetylneuraminicacid toxicity, N-acetylneuraminic acid toxicity wassuppressed completely when the inoculum wasgrown overnight in a medium containing an excessof glucose (22.2 mM) (Fig. 2b). This effect was notobserved when glucose was replaced by glycerol, anon-catabolite-repressing substrate (data not shown).It has been shown previously that the accumula-

tion of N-acetylneuraminic acid in a nanA mutantis growth inhibitory.14 However, it seemed improb-able to us that an accumulation of N-acetylneurami-nic acid could explain N-acetylneuraminic acidtoxicity in the nanR mutant, since any excess

Fig. 2. Growth inhibition of a nanRmutant in Mops minimal medium containing N-acetylneuraminic acid. The WTormutant strain indicated was inoculated following overnight growth in Mops minimal medium containing (a) 1.11 mM or(b) 22.2 mM glucose as the sole carbon source. The cultures were incubated at 37 °Cwith rapid aeration. The growth of theWT and the mutant strains indicated was monitored spectrophotometrically at 600 nM.

877Transient Sialic Acid Toxicity of a nanR Mutant

N-acetylneuraminic acid taken up by NanT in thenanRmutant should be converted rapidly toGlcNAc-6P by the concerted actions of the NanA, NanE, andNanK. On the other hand, GlcNAc-6P is also knownto be toxic,13 and we reasoned that low initialinduction of the nagBACD operon, and hence anaccumulation of GlcNAc-6P, would more likely beresponsible for N-acetylneuraminic acid toxicity. Insupport of this hypothesis, a secondary mutation inthe nagBACD operon repressor nagC suppressedcompletely the toxic effects of N-acetylneuraminicacid in the nanR mutant (Fig. 2b). The mechanismof GlcNAc-6P toxicity is unknown, but is thoughtto involve a lack of intermediary metabolites.13

Consistent with this hypothesis, no growth defectwas observed in a nanRmutant grown in aMops-richdefined medium that contains exogenous bases andamino acids (data not shown).Expression of nanA is activated by Dam meth-

ylation.15 To probe further the potential role of Dammethylation in the regulation of the nanA operon,the ability of a dammutation to suppress the effect of

N-acetylneuraminic acid on the nanR mutant wasalso examined (Fig. 3). It was found that the dammutant restored the growth rate of the nanR mutantto near-WT levels. This result supports the hypoth-esis that overexpression of the nanA operon pro-duces N-acetylneuraminic acid toxicity. Further-more, it indicates that if methylation of GATCNanR

activates the expression of the nanA operon, thenthis effect occurs even in the absence of NanR.

Theory

Formulation of the Base, Extended, and FullModels will introduce three variants of a model ofN-acetylneuraminic acid metabolism in E. coli. Thefirst (“base”) model makes a number of simplifyingassumptions, but provides some intuition about thebehaviour of the system. The second (“extended”)model extends the base model by taking intoaccount the transcription/translation step in geneexpression. Finally, the third (“full”) model refinesthe extended model by introducing more realistic

Fig. 3. Suppression of N-acetyl-neuraminic acid toxicity in a nanRmutant by simultaneous mutationof dam. The WT or the mutantstrains indicated were inoculatedinto fresh Mops minimal mediumcontaining N-acetylneuraminic acidas the sole carbon source followingovernight growth in Mops minimalmedium containing 1.11 mM glu-cose as the sole carbon source. Thecultures where incubated at 37 °Cwith rapid aeration. The growth ofthe WT and the mutant strainsindicated was monitored spectro-photometrically at 600 nM.

Table 2. Summary of the main symbols used

s1 N-acetylneuraminic acid (external)e1 Products of nanATEKs2 GlcNAc-6P


kinetics. This latter model is more complex than thebase and extended models, but shows the samequalitative behaviour (at least for the purpose of thecurrent contribution).

nR NanR (kept fixed)x1 N-acetylneuraminic acid in the cytoplasme2 Products of nagBACDϱex mRNA for enzyme exgx Transcription efficiency of

nanATEK and nagBACD

Formulation of the base, extended, and full models

Our base model is defined by the following set ofdifferential equations (Tables 1 and 2):

ds1 ¼ 0 (1)

dx1 ¼ rs1 � rs2 (2)

de1 ¼ re1 � ce1e1 (3)

ds2 ¼ rs2 � rx2 (4)

de2 ¼ re2 � ce2e2 (5)

where the parameters of the system are defined inTable 3. This model describes the flow of external(periplasmic) N-acetylneuraminic acid (s1) into thecytoplasm (Eqs. (1) and (2)) via NanT (here rep-resented by e1). Cytoplasmic N-acetylneuraminicacid is denoted by x1; its breakdown into GlcNAc-6P(represented by s2) is catalysed by e1. Differing con-centrations of NanA/NanB and NanT are taken careof by the relevant constants describing uptake andmetabolic efficiencies. Regulation of the uptake andmetabolism of N-acetylneuraminic acid is repre-sented in the model by the transcription efficiencyg (see Table 3) describing the activity of nanATEK.If g=0, then no e1 is produced, and hence noN-acetylneuraminic acid is taken up or metabolized,whereas g=1 describes the case of full nanATEKderepression and e1 is produced at its maximumrate (defined by the value of Ve1). Generally, a highernR (representing the amount of NanR) leads to alower g, and a higher x1 leads to a higher g.Equations (4) and (5) are the GlcNAc-6P counter-

parts of Eqs. (2) and (3). The symbol e2 represents theproducts of the nagBACD operon. The autorepres-sive activity of NagC is reflected in the transcriptionefficiency g2 (see Table 3). Like its counterpart g, g2varies between 0 and 1. The higher is e2, the lower isg2. Yet, the higher is g2, the higher are the expressionlevels of e2. Similarly, the more s2 (GlcNAc-6P) thereis in the system, the higher is g2. The details of theGlcNAc-6P breakdown (and its products) are notrepresented in this model.This model contains a number of simplifying

assumptions. In particular, the influx of N-acetyl-

Table 1. Summary of the main symbols used

s1 N-acetylneuraminic acid (external)e1 Products of nanATEKs2 GlcNAc-6PnR NanR (kept fixed)x1 N-acetylneuraminic acid in the cytoplasme2 Products of nagBACDϱex mRNA for enzyme exgx Transcription efficiency of

nanATEK and nagBACD

neuraminic acid into the periplasm is not explicitlymodelled because N-acetylneuraminic acid diffusesthrough the outer cell wall through the nonspecificOmpF/OmpC porin;2 uptake into the periplasm istherefore only weakly regulated. Other simplifyingassumptions include the following:

• Themodel reduces themetabolism ofN-acetylneur-aminic acid (see Fig. 1) to a two-step process.

• Themetabolic enzymes encoded by nanATEK andnagBACD are summarised by e1 and e2, respec-tively. Essentially, this amounts to assuming thatthe concentrations of the proteins expressed fromthe same operon are in a fixed ratio.

These simplifications can be justified by consider-ing that (i) they facilitate the mathematical inter-pretation and treatment of the model. Furthermore,(ii) this article only considers qualitative models;simplifications resulting in quantitative inaccuraciesare inconsequential. Finally, (iii) more complexmodels would also introduce a number of additionalunknown parameters; these would need to beguessed and, by themselves, introduce additionaluncertainties.The base model (i.e., Eqs. (1)–(5)) does not rep-

resent mRNA expression. An extended model thatrepresents the mRNA expression is identical withthe base model, except for the following replace-ment of and addition to Eqs. (1)–(5) (Tables 4–7):

d|e1 ¼ d|e1ðre1 � |e1Þ (6)

de1 ¼ |e1 � ce1e1 (7)

d|e2 ¼ d|e2ðre2 � |e2Þ (8)

de2 ¼ |e2 � ce2e2 (9)

Note that the steady-state solutions of the extendedmodel are identical with those of the base model.From a dynamical point of view, the main differencebetween the models is that production of mRNA—here denoted as ϱex (where x=1, 2)—introduces adelay into the system. The size of the delay willcrucially depend on the translation efficiency of theenzymes dex

ϱ.

The extended model assumes that (i) mRNAbreakdown is negligible, and (ii) proteins are de-graded at a fixed rate. These assumptions are notnecessarily correct (e.g., see Narang16 and Narang

Table 3. Definition of the dynamical parameters used inthe model of N-acetylneuraminic acid metabolism

gW k2 þ k3x1k2 þ k3x1 þ k1nR

g2Wl2 þ l3s2

l2 þ l3s2 þ l1e2re1WgVe1 re2Wg2Ve2

rs1WVs1 e1s1

Ks1 þ s1rs2WVs2 e1

x1Kx1 þ x1

rx2WVce2s2

Ks2 þ s2cex—protein breakdown coefficient

The parameter k1 describes the affinity of nR for the operator site,k2 is the dissociation constant, and k3 is a constant describing theefficiency of nR displacement by x1. The parameters lx have ananalogous interpretation for e2 and s2. Vs1, Vs2, and Vsc are theenzymatic rates for N-acetylneuraminic acid transport into thecytoplasm (i.e., the s1→x1 conversion), the rate of N-acetylneur-aminic acid breakdown, and the rate of GlcNAc-6P breakdown,respectively. The parameters Kx are saturation parameters. Thetranscription efficiencies g,g2 are derived under the assumptionthat displacement of the repressor is a one-step process (i.e., nointermediate repressor–derepressor–nucleotide compounds areformed; see Appendix A for the derivation). All parameters areconstant and positive; in this contribution, we restricted the valueof the parameters to the interval (0, 1) (see Materials and Methodsfor an explanation).


and Pilyugin17). A more realistic model would takeinto account the limited lifetime of mRNA andreplace fixed-rate protein breakdown with a modelwhere protein is diluted by cell growth (i.e.,proportional to the efflux of GlcNAc-6P given byrx2). The resulting model (which we will henceforthrefer to as the full model) is then obtained byreplacing Eqs. (6)–(9) with the following:

d|e1 ¼ d|e1ðre1 � |e1Þ � c||e1 (60)

de1 ¼ |e1 � rx2ce1e1 (70)

d|e2 ¼ d|e2ðre2 � |e2Þ � c||e2 (80)

de2 ¼ |e2 � rx2ce2e2 (90)

The breakdown of mRNA with constant cϱ isrepresented by the last terms in Eqs. (6′) and (7′).

Derivation of steady-state values

In what follows, we will calculate the steady stateof the base model (which is identical with the steadystate of the extended model).From Eq. (3), we can calculate steady-state values

for e1:

gVe1 � ce1e1 ¼ Ve1

︸Wβ

k2 þ k3x1k2 þ k3x1 þ k1nR

� ce1e1 ¼ 0

YhVe1 � ce1he1 � ce1k1nRe1 ¼ 0

Ye1 ¼ hVe1

hce1 þ ce1k1nR(10)

Similarly, using Eq. (2), we can calculate the steady-state concentration of x1:

rs1� rs2 ¼ 0 ¼ Vs1e1︸Wσ

s1Ks1 þ s1

0@

1A�Vs2e1

x1Kx1 þ x1

(11)

Yx1 ¼ Vs1jKx1

Vs2 � Vs1j(12)

Due to the assumption that the metabolic proteinsNanA and NanB and the transporter NanT are ina fixed ratio, in this model, the concentration ofN-acetylneuraminic acid (x1) in the cytoplasm isindependent of the induction level of the nanATEKoperon (i.e., e1); this is clear from Eq. (12). Theamount of e1 regulates the turnover rate of N-acet-ylneuraminic acid (i.e., how fast it is taken up andconverted into GlcNAc-6P). Hence, the level of x1 isonly determined by the parameters of the systemand by the amount ofN-acetylneuraminic acid in theperiplasm.Using Eq. (4), we obtain a steady-state value for

s2—the GlcNAc-6P concentration in the cytoplasm.Setting aW x1

Kx1þx1

� �, we get:

rs2 � rx2 ¼ 0 ¼ Vs2e1a� Vce2s2

Kc þ s2YVs2e1aKc þ Vs2e1as2 � Vce2s2 ¼ 0

Ys2 ¼ Vs2e1aKc

Vce2 � Vs2e1a(13)

Here, s2 is a function of the inducers e1, e2, andx1. Keeping everything else fixed, an increase in e1(nanATEK induction) will lead to an increase ins2 (GlcNAc-6P concentration); increasing e1, how-ever, can only continue as long as the denominatorin Eq. (13) remains greater than zero. An increasebeyond this point will lead to an undefined steadystate (i.e., an unstable model). Apart from e1, thesteady-state concentration of GlcNAc-6P also de-pends on the external/periplasmic, but not the cyto-plasmic, N-acetylneuraminic acid concentration.

Necessary conditions for the stability of WT

For reasons of biological plausibility, we assumethat the WT will normally operate in a stable dyna-mical regime (see Computational Simulations for adiscussion of what we mean by stability). The abovesteady-state analysis allows one to establish somenecessary conditions for the existence of stablesolutions:

• From Eq. (12): Vs2NσVs1. This essentially meansthat the speed with which N-acetylneuraminicacid is converted must be higher than the speedwith which it is brought into the system.


• From Eq. (13): Vce2NVs2e1α. This means that theinflux of GlcNAc-6P must be lower than its efflux(i.e., metabolic breakdown).

Independent of the parameters, the model has, atmost, one steady-state solution (see Appendix A).

Results for the nanR single mutant

The nanR mutant corresponds to the limiting caseof nR≪k2+k3x1. In this case, e1 will approach thesteady-state value:

enR1 ¼ Ve1=ce1 (14)

Clearly, e1nRN e1 for any positive nR, confirming the

intuitive expectation that the turnover of N-acet-ylneuraminic acid is strictly higher in the nanRmutant than in the WT. As a consequence of thehigher levels of e1 in the mutant, it can be seen fromEq. (13) that the steady-state values of s2 will behigher than in the WT. Note, however, that thismodel predicts that the N-acetylneuraminic acidconcentration in the cytoplasm will not be affectedby the mutation.

Double mutant

The nanR–nagC double mutant can be calculatedexactly. It is defined by g2=1 and nR=0. In this case,all variables take very simple solutions in terms oftheir parameters. The concentrations of e1 and e2 canbe obtained from Eqs. (3) and (5), respectively:

e1 ¼ Ve1

ce1, e2 ¼ Ve2

ce2(15)

The steady-state amount of s2 can be obtained fromEq. (4):

s2 ¼Vs2

Ve1ce1

aKc

VcVe2ce2

� Vs2Ve1ce1

a(16)

Note that, in the limit of very large s2, the singlemutant will show the exact same behaviour as thedouble mutant because g2→1 for large s2.

Simulations

Approach of steady state in the extended model

As long as one is only interested in the steady-state values, the basic model of Eqs. (1)–(5) and theextended model of Eqs. (1)–(9) are identical. Thesemodels differ, however, in how they approach thesteady state. The transcription step introduces anadditional delay into the system that potentiallymodifies the transient behaviour of the model.In what follows, we will (unless stated otherwise)

assume initial conditions corresponding to very low(or vanishing) enzyme (i.e., e1 and e2) and mRNA

concentrations. In the case of the basic model, thereare only small delays in the system (dependingon the specific parameters), and the cytoplasmicsteady-state concentrations of x1 and s2 (N-acetyl-neuraminic acid and GlcNAc-6P, respectively) areapproached from below (i.e., the steady-state con-centration is the maximum, or is close to themaximum, concentration reached). If the delay inthe system is significant, then the transient concen-trations of s2 and/or x1 will initially overshoot theirsteady-state concentrations by possibly quite largeamounts. In the extended model, the parameterthat determines the delay resulting from mRNAexpression is dex

ϱ.

Figures 4 and 5 show how the steady state isapproached depending on the size of the delay inthe extended model. For a value of dex

ϱ=0.1, thesystem in Fig. 4 shows no significant overshoot ofthe steady-state values. Both the WT and the singlenanRmutant approach their respective steady statesfrom below. With increased delay (i.e., decreasingthe value of dex

ϱ), the qualitative behaviour changes,and both the WT and the mutant show transientaccumulations of s2 before the steady-state valueis reached. For long delays, these accumulationscan be very significant compared to the steady state.Accumulations occur in both the WT and themutant, but in the particular case of Fig. 4, theaccumulation in the mutant is higher than in the WTby a factor of 4. Note that, in the extendedmodel, thesteady-state value is independent of the delay dex

ϱ.

Similarly, for the parameters used in Fig. 5, thenanR mutant shows a transient accumulation of s2by a factor of N60 compared to the steady-state valuefor a delay of dex

ϱ= 0.001. Yet again, no accumulation

occurs in the simulation of the nanR–nagC doublemutant.

Stability of qualitative behaviour: The full model

While the full model is somewhat more compli-cated to analyse when compared to the base orextended model, most of the intuition developedfor the base model can be carried over to the fullmodel as well. What is more important, however,is that the full model approaches steady statequalitatively in the same way as the extendedmodel does. The model shows an overshoot of s2that depends on the delay in gene expression.Figure 6 is an example; here, the overshoot of s2 isrelatively modest, and the difference between WTand mutant is small. For different parameters, suchas in Fig. 7, the model demonstrates a very largedifference in transient GlcNAc-6P concentrations inthe WT and the mutant.

Discussion

N-Acetylneuraminic acid is a carbon source forE. coli that can be metabolized into GlcNAc-6P andFru-6P, eventually fueling the growth of the organism.However, our results show that too rapid utilization

Fig. 4. Comparison of how GlcNAc-6P approaches steady state for different delays and a different set of parameters.The double mutant (bottom right) approaches the steady state from below even for significant delays. For the parametersof this simulation, see Table 4.


of N-acetylneuraminic acid can lead to growthinhibition, depending upon the growth conditionsprior to N-acetylneuraminic acid exposure. We willhenceforth assume that this toxic effect is caused byGlcNAc-6P.In this section, we will interpret (and connect)

the experimental, theoretical, and simulationresults reported in Results; we will also discusshow they can be understood in the light ofprevious results. The main conclusion is a predic-tion that nanATEK will be regulated in a two-stepprocess, in that full induction of the gene is onlyreached after an initial period with intermittentinduction. The rationale is that, by using this two-step process, the cell can efficiently metabolizeN-acetylneuraminic acid and GlcNAc-6P in steadystate while avoiding the toxic effects of transient(high) accumulations of GlcNAc-6P.

Steady-state behaviour does not explainGlcNAc-6P toxicity

The lack of growth of the nanR mutant cannot beexplained by the steady-state properties of this (orindeed any other possible) model. To see this, con-sider the two possible scenarios that could explainthe observed experimental evidence:

(1) The mutant has a steady-state GlcNAc-6Pconcentration that is high enough to cause

toxicity, whereas the concentration in theWT istoo low to show any symptoms.

(2) The WT has a nontoxic steady state, but inthe mutant, GlcNAc-6P grows without bounds(i.e., it does not have a steady state). Translatedinto the language of dynamical systems, thiswould mean that the WT shows stable beha-viour, but the mutant is unstable.

A priori, both scenarios are plausible vis-à-vis themathematical model: There are sets of parameterswhere theWT has a steady state but the mutant doesnot; even if the mutant has a steady-state GlcNAc-6Pconcentration, for many sets of parameters, it will bemuch higher than the WT concentration, thuspotentially causing the toxic effect.We think that a steady-state explanation is not

correct for the following reasons: Firstly, the para-meters leading to a stable WT and an unstable mu-tant are rare in the parameter space of the extendedmodel (we estimated them to be about 3%; seeAppendix A). However, this estimate is most cer-tainly dependent on one of our modelling assump-tions, namely, that nR=0.7 (which we left fixed).Increasing this value would lead to an increase inthe proportion of models with stable WT but un-stable mutant. Moreover, one cannot a priori excludethat evolution chose a rare set of parameters.A stronger argument is provided by the second

reason: While a small proportion of parameter space

Fig. 5. Comparison of how s2 approaches steady state for different values of the (delay) parameter dexϱ. This is an

example simulation. The quantitative details depend on the parameters chosen for the model (listed in Table 5); however,the qualitative trend of increased transient accumulations for longer delays is independent of the parameters.


supports stable behaviour for the WT and unstablebehaviour for the nanR mutant, we did not find asingle set of parameters where the WT and singlemutant are stable and unstable, respectively, butthe double nanR–nagC mutant is stable. Thissuggests that the stable–unstable–stable combina-tion is at least extremely rare or even nonexistent.Theoretically, one would indeed not expect to finda stable–unstable–stable configuration because themodel predicts the nanR mutant and the doublenanR–nagC mutant to be equivalent for large s2(because g2→1 for s2→∞; i.e., the nagBACD operonis completely derepressed at high concentrations ofGlcNAc-6P).Thirdly, experimental evidence directly suggests

non-steady-state explanations. The toxic effectexperienced by the nanR mutant is transient in thesense that the population recovers and returns tonear-normal growth after a certain period (see Fig. 2)(i.e., there is no observable toxic effect of GlcNAc-6Pin the long run). Steady-state explanations wouldonly describe the behaviour of the system in the longterm.Fourthly, in systems where there is only one stable

steady state, the long-term behaviour of the systemis independent of the initial conditions ofthe system. Yet in the experiments, the toxic effectis only seen in carbon-starved inocula, but not ininocula grown in carbon-rich conditions. In themathematical model, different inoculations manifest

themselves in different initial conditions. If a systemhas a steady state that is unique and stable, then itslong-term behaviour does not depend on its initialconditions. Hence, the fact that the appearance oftoxicity depends on the inoculum indicates that theeffect is not due to the steady-state behaviour of thesystem.In summary, mathematical and experimental

evidence indicates strongly that GlcNAc-6P toxicityin the nanR mutant is a transient effect. Steady-stateGlcNAc-6P levels are nontoxic in the WT, in thenanR mutant, and in the double nanR–nagC mutant.

An ansatz based on transient behaviour

Rather than being due to the steady-state proper-ties of the system, we suggest a model based ontransient accumulations of GlcNAc-6P and subse-quent toxicity effects. The explanatory ansatz can besummarised as follows:

• Upon exposure to external N-acetylneuraminicacid, transient GlcNAc-6P concentrations transi-ently rise to high levels both in the WT and in thenanR. These transient accumulations tend toincrease when a delay in the nagBACD geneexpression is introduced.

• The mathematical models can explain the experi-mentally observed transient toxicity of the nanRmutant (and its absence in the WT) when

Fig. 6. Comparison of how s2 approaches steady state for different values of the (delay) parameter dexϱ for the full

model. Here, nanR repression is efficient; hence, there is a smaller transient effect (see parameters in Table 6).


transient NanATEK levels are low in the WT buthigh in the mutant.

• A large difference in transient NanATEK concen-trations between the nanRmutant and theWTcanbe explained when nanATEK derepression by N-acetylneuraminic acid is inefficient.

• This suggests a two-step model of nan induction,whereby full induction levels of the operon areonly reached after a period of gradual buildup ofenzyme levels in the cell. This two-step mechan-ism can avoid toxicity in the WT while stillallowing full utilization of the carbon source.

• This explains available experimental evidence(i.e., the experiments with the dam mutant in Fig.3 and data from the literature, especially Kalivodaet al.7).

In the remainder of this section, we will justify thismodel in detail.

Delays cause transient accumulations, but notin the double mutant

The two most relevant qualitative features of boththe extended and the full models are as follows: (i)The transient GlcNAc-6P accumulations in theextended/full model last longer and are greaterthe longer the delays (see Figs. 4–7). (ii) Independentof the delay, the nanR–nagC mutant approachesthe steady-state GlcNAc-6P concentration from

below (i.e., shows no transient accumulations of s2;GlcNAc-6P).An interpretation of experimental data based on

the assumption that GlcNAc-6P toxicity is a tran-sient phenomenon immediately explains aspects ofthe available data: The toxicity effect in the nanRmutant, one can hypothesize, is caused by transientaccumulations of GlcNAc-6P, which in turn arecaused by a (normal) delay in induction of thesystem. Once the toxic limit is reached, this normaldelay is exacerbated by even higher GlcNAc-6Paccumulations until eventually the cell manages toreturn to normal function. In the double mutant,the nagBACD operon will always be fully expressed(due to the absence of the repressing NagC), hencesteady-state concentrations of the relevant enzymeswill be reached; breakdown of the GlcNAc-6Pcommences without delay upon the cell's exposureto N-acetylneuraminic acid.This ansatz has a potential problem apparent from

Figs. 4 and 6: Unlike the solutions in Figs. 5 and 7,GlcNAc-6P accumulates in these simulations tosimilar levels in both the WT and the nanR mutant.This suggests that there are regions in parameterspace where both the WT and the nanRmutant tran-siently accumulate toxic concentrations of GlcNAc-6P. This leaves a crucial aspect of the model under-explained, namely, the experimentally observeddifference in qualitative behaviour between the WTand the single mutant.

Fig. 7. Comparison of how s2 approaches steady state for different values of the (delay) parameter dexϱ for the full

model. Here, nanR repression is inefficient (see parameters in Table 7); hence, there is a large transient effect. The last panelshows s2 on a log scale because of the great differences between the three curves.


One possible explanatory strategy is to simplyacknowledge that the parameters of the real systemshappen to be such that the WT does not display ahigh transient GlcNAc-6P accumulation while thenanRmutant does (clearly, there are parameters thatsupport this). The force of this argument is, in fact,much stronger than it seems at a first glance: Theabsence of toxic GlcNAc-6P accumulations in theWT per se does not require any explanation becausetoxicity impairs viability. What remains to beexplored, at this point, is under which conditionsthe system would accumulate high transientGlcNAc-6P concentrations in the mutant, while theWT only shows small accumulations.In Figs. 4 and 5 (for the extended model) and in

Figs. 6 and 7 (full model), the GlcNAc-6P accumula-tion in the mutant is always higher than in the WT.Theoretically, this is expected to be the case for allpossible parameter sets because the nanR mutanthas a higher influx of GlcNAc-6P than the WT, whilethe regulation of the GlcNAc-6P metabolism isassumed to be the same as in the WT. Yet, thedifference between the accumulation in the WT andthe accumulation in the mutant will cruciallydepend on the parameters of the system; it couldrange from very small to very large (as illustratedby the examples in Figs. 4–7). This leads to twoquestions: (i) Theoretically, which parameters deter-mine the difference in transient behaviour? (ii) Are

there any plausible biological reasons to expect abig/small difference in transient GlcNAc-6Pbetween the WT and the mutant in the cell?Addressing (ii) first, biologically, it is plausible

to expect that the maximum GlcNAc-6P concen-tration in the WT will be well below toxic levels.If steady-state GlcNAc-6P values in the WT wereclose to the toxic limit, random fluctuationswould take the cell over this limit. Combinedwith the experimental evidence on growth inhibi-tion in the mutant, this suggests that the dif-ference between maximum transient GlcNAc-6Paccumulation in the WT and maximum transientGlcNAc-6P accumulation in the nanR mutantshould be large.

What determines transient GlcNAc-6Paccumulations?

In order to get a better theoretical understandingof the conditions that lead to a large difference intransient GlcNAc-6P levels, consider that the max-imum transient GlcNAc-6P accumulation willdepend on the rate of GlcNAc-6P synthesis (i.e.,GlcNAc-6P influx) and its breakdown rate (i.e.,efflux). Assuming (for the moment) a fixed effluxmechanism, then accumulation levels will mainlydepend on the N-acetylneuraminic acid turnoverrate. This rate itself depends on the induction levels

Fig. 8. The transient values ofGlcNAc-6P depend on the concen-tration of the enzyme regulatingN-acetylneuraminic acid through-put. In these simulations, x1 waskept constant at 1, and e1 was set tovarious levels ranging from 0.3 to1.5. Enzyme levels will be high inthe nanR mutant compared to theWT if NanR is an effective repressorover physiological levels of N-acet-ylneuraminic acid (see the main textfor details).


of nanATEK (e1 in the model). Figure 8 shows s2 (i.e.,GlcNAc-6P) accumulations for various levels of e1;in this graph, the parameters regulating the break-down of s2 have been kept constant. The figureclearly illustrates that (at least for this particular setof parameters) increasing the amount of e1 leads to ahigher transient GlcNAc-6P accumulation. Fig. 8only describes the behaviour of a specific set ofparameters for GlcNAc-6P breakdown. However,any set of parameters will show a qualitativelysimilar trend, with only the quantitative details (i.e.,the absolute values of s2 accumulation) being dif-ferent. In terms of the regulation of the GlcNAc-6Pmetabolism, a high difference in e1 in the modeltranslates into a high difference in nanATEK repres-sion in the cell between the WT and the nanRmutant. Since the nanR mutant is fully derepressed,this means that the WT is, at most, partially inducedeven in the presence of external N-acetylneuraminicacid. In short, the difference in transient GlcNAc-6Plevels between the WT and the mutant will be big ifN-acetylneuraminic acid does not efficiently dere-press nanATEK.In the full model, the same qualitative relation

between the overshoot in the WT and the mutantholds true. Figures 6 and 7 are two examples ofthe possible behaviour of the full model underdifferent delays. The qualitative differencebetween the two figures is reflected in theparameters and in the gene transcription efficiencyg. Its value (for the WT) in the simulation in Fig. 6is around 0.9, indicating nearly full induction ofthe genes; the corresponding value in Fig. 7, onthe other hand, never rises above 0.24, indicatingpoor derepression for this set of parameters (datanot shown).

A delay explains the experimental data if NanRdisplacement is inefficient

In terms of the understanding of the qualitativeproperties of the N-acetylneuraminic acid meta-bolic network, this points to a scenario whereby

nanATEK is strongly repressed by NanR even inthe presence of N-acetylneuraminic acid. This is sofor the following reason: If steady-state amounts ofN-acetylneuraminic acid could efficiently derepressnanATEK, then one would expect only small dif-ferences in the transient phases of the WT and themutant because, even in the former, nanATEKwould be derepressed; the situation is then morelike the example simulations shown in Fig. 4or Fig. 6, rather than the ones shown in Fig. 5 orFig. 7.This interpretation is consistent with experimental

evidence reported by Kalivoda et al., who used invitro gel mobility experiments to demonstrate thatNanR displacement by N-acetylneuraminic acid isinefficient.7 However, in the same article, theauthors also reported very high in vivo inductionlevels of nanATEK in the presence of sialic acid.Hence, overall, the evidence was somewhat contra-dictory. Kalivoda et al. speculated (and providedsome experimental evidence) that the differencebetween the in vivo and the in vitro results might bedue to the specificity of N-acetylneuraminic aciduptake for a specific isomer of N-acetylneuraminicacid (the α-isomer).While the isomer specificity of NanT could

provide a partial resolution of this contradiction,our mathematical model indicates that inefficientnanATEK induction by N-acetylneuraminic acidcould be an adaptive feature with biological realityin vivo: At the level of the individual cell, inefficientNanR displacement means intermittent periodsof nanATEK activation; the resulting short pulseswould gradually build up enzyme levels (andN-acetylneuraminic acid turnover rates) and, assuch, significantly reduce the risk for GlcNAc-6Paccumulations.This ansatz explains why the WT can avoid the

toxic effect; however, it would also imply a lowersteady-state N-acetylneuraminic acid turnover ratefor the WT, and hence suboptimal nutrient utiliza-tion (suboptimal because, as the nanR mutantindicates, the fully derepressed phenotype is, in


steady state, fully viable while providing a highernutrient uptake rate). This is not compatible withavailable experimental evidence showing similargrowth rates for nanR mutants (taken from theinocula that are not toxic) and the WT. A possibleway to circumvent this problem is a two-stepinductionmodel, whereby the initial phase of pulsednanATEK activity is followed by full induction of theoperon; this would require a second mechanism toeffect the full induction. Such a two-step model ofnanATEK induction could combine avoidance oftoxic transient GlcNAc-6P accumulations with highsteady-state N-acetylneuraminic acid turnover.We propose that methylation of the NanR-binding

site at the nanA promoter could participate in thetwo-step induction of the nanATEK operon. Thereare several possible scenarios as to how methylationcould work. One possibility is that the methylatedoperator activates nanATEK independently of NanRbinding; another is that the affinity of NanR for themethylated operator is sufficiently lowered toenable more efficient displacement by N-acetylneur-aminic acid. Both possibilities require that (i) NanRbound to the operator acts as a methylation-blocking factor, which we know it does, and (ii)methylation of the free operator is (relatively)inefficient. As the nucleotide sequence surroundingGATCNanR (CCAGATCAAT) is expected to makethe GATC site a relatively poor substrate for Dammethylase,12 this second requirement also seemslikely to be fulfilled.Since we have shown that mutation of dam can

suppress N-acetylneuraminic acid toxicity, ourexperimental evidence suggests that methylation ofGATCNanR per se, rather than an effect of GATCNanR

methylation on NanR binding, is likely to partici-pate in the control of the nanA operon. However,further work is needed to determine if and howmethylation of GATCNanR affects the expressionof the nanATEK operon directly and to rule outthe possibility that methylation does affect theaffinity of NanR for its operator in the presence ofN-acetylneuraminic acid.Theoretically, the two-step model could work as

follows: Assuming that the concentrations of N-acet-ylneuraminic acid and NanR are constant, and thatthe association and displacement rates for NanRare λ and γ, respectively, then the probability ofnanATEK being repressed (at timeTr followingNanRbinding) or induced (at time Ti following NanRdisplacement) will be distributed exponentially:

PrðTr ¼ tÞ~exp �Etð Þ, PrðTi ¼ tÞ~exp �gtð Þ (17)

Formally, the system can be modelled as a two-statecontinuous-time Markov chain. Then the probabilityof finding the system in either of its state at a giventime tk is determined by the quotient of the transitionrates, that is,

time spent in repressed statetime spent in induced state

¼ g

E(18)

At the level of the individual cell, inefficient dis-placement of NanR by N-acetylneuraminic acidmeans precisely that this quotient γ/λ is very small(i.e., that nanATEK is repressedmost of the time). Thisagain means that there will only be short pulses ofnanATEK expression and hence low initial ratesof N-acetylneuraminic acid turnover, preventingGlcNAc-6P toxicity.These short pulses of gene activity could not

explain the experimentally observed full nanATEKinduction as reported by Kalivoda et al. Yet,methylation could come in as a plausible second-step mechanism to effect full operon induction:Assuming a constant N-acetylneuraminic acid con-centration, then the time to methylation Tm willagain be exponentially distributed, but with themethylation rate of the unprotected GATC sitemodified by the probability of NanR not beingbound to the operator:

Pr Tm ¼ tð Þ~exp � g

Eyt

� �(19)

This suggests the plausibility of a two-step model ofnanATEK induction. During the first phase, theoperon is expressed intermittently in order to slowlyincrease N-acetylneuraminic acid turnover and toavoid toxic GlcNAc-6P levels.

Conclusion

The overall picture emerging from this is as fol-lows: Exposure of the nanR mutant to N-acetylneur-aminic acid leads to a transient accumulation ofGlcNAc-6P in the cell, with a toxic effect. The exis-tence of GlcNAc-6P toxicity in E. coli is welldocumented in the literature (and corroborativeevidence is reported in this contribution), althoughwe do not currently have a mechanistic understand-ing of this effect. Delays in cell functions such as geneexpression are a result of the GlcNAc-6P toxic effect.These delays lead to even higher transient accumula-tions of GlcNAc-6P, triggering a positive feedback oftoxicity. Eventually, however, the GlcNAc-6P concen-tration will fall back to the (nontoxic) steady-stateconcentration. At this point, the toxic effect disap-pears, and cell function returns to normal. The doublenanR–nagC mutant does not show any toxic effectbecause even in the absence of external N-acetylneur-aminic acid, nan is fully expressed, and hence thereexists no delay in gene expression and GlcNAc-6Plevels are approached from below.The WTwould be susceptible to the same toxicity

effect as the nanR mutant, unless N-acetylneurami-nic acid turnover in the cell is initially increasedsignificantly more slowly in theWT than in the nanRmutant. This could be achieved if, in E. coli, nanRdisplacement by N-acetylneuraminic acid was in-efficient, only allowing intermittent activity ofnanATEK. Such a model is consistent with measure-ments of the in vitro displacement efficiency ofN-acetylneuraminic acid as reported by Kalivoda

Table 5. The parameters used to simulate the extendedmodel in Fig. 5

Ks1 0.059473Vs1 0.277749Ve1 0.186220Ks2 0.159184Ve2 0.455408k1 0.706329k 0.354706


et al.; however, it does not account for empiricalevidence for full nanATEK induction in vivo (assuggested by the experiments reported in thiscontribution and by Kalivoda et al.).7 In order toaccount for this, we predict a second mechanismthat can lead to full induction of the operon after atransient period. It is likely that this secondmechanism involves methylation.

2k3 0.134793l1 0.974713l2 0.175839l3 0.948983Kc 0.067786ce2 0.097309ce1 0.716721Vc 0.040482Vs2 0.863023Vs0 0.826344Ks0 0.286851

Materials and Methods

Bacterial strains, plasmids, media, and growthconditions

Bacterial strains, which were all derivatives of the E. coliK-12 strain BGEC905, have been described in detailpreviously.18

Minimal Mops medium was prepared as described byNeidhardt et al. and supplemented with 10 mM thiamine,1.32 mM phosphate, and glycerol or glucose, and/orN-acetylneuraminic acid was added to a final concentra-tion of 1.11 mM.19 All reagents were obtained from Sigma,except for N-acetylneuraminic acid, which was purchasedfrom Merck Biosciences.Cells were grown aerobically at 37 °C in minimal Mops

medium, with the carbon source as indicated andsubcultured to a starting OD600 of 0.01. Culture densitieswere then monitored hourly spectrophotometrically at600 nm.

Computational simulations

The quantitative details of the system modelled byEqs. (1)–(9) strongly depend on the particular choice ofthe unknown parameters. Determining these parameterseither experimentally or through literature search isresource demanding, and the results are likely to be subjectto largeuncertainties.Hence, insteadof guessing the “right”parameters of the system, the properties of the model wereexamined for a wide range of possible parameters.Throughout the text, we refer to “stable” systems or

models; by this, we mean a set of parameters such that, forall variables of the system, there exists at least one stablesteady state. In particular, a stable system will not show

Table 4. The parameters used to simulate the extendedmodel in Fig. 4

Ks1 0.342733Vs1 0.310270Ve1 0.606901Ks2 0.452776Ve2 0.599438k1 0.249651k2 0.528698k3 0.474912l1 0.703337l2 0.217218l3 0.712827Kc 0.395445ce2 0.541125ce1 0.686598Vc 0.487362Vs2 0.372910Vs0 0.564737Ks0 0.713628

unbounded growth of the concentrations of one or moremolecular species.We estimate the proportion of parameters that lead to

stable models by creating a large number of randommodels [i.e., instantiations of the above (basic) modelwith random parameter values]. We check each of thesemodels for stability and are thus able to estimate theproportion of stable models in the space of all models.For practical reasons, we limit the value of parameters tovalues between 0 and 1 (drawn from a uniformdistribution); restricting the range of possible parametersto this interval does not limit the generality of ourconclusions because we are not interested in the absolutevalue of the parameters, but in their relative magnitudes.All searches of the parameter space reported in thiscontribution were unrestricted (i.e., also included thoseregions of parameter space that are known to beunstable). The simulated WT was always assumed tohave an arbitrary but fixed value of nR=0.7.The simulations of the transient behaviour were

performed using the standard numerical integrator of theMaple9 software package. The value of s1 was keptconstant at a high (saturation level) of 4000 (arbitraryunits); independent of the specific parameter values, thisamount saturates the uptake mechanism in the model.

Table 6. The parameters used to simulate the full modelin Fig. 6

Ks1 0.848758Vs1 0.886621Ve1 0.001469Ks2 0.631901Ve2 0.665286k1 0.430014k2 0.789845k3 0.470588l1 0.799339l2 0.555222l3 0.748967Kc 0.838957ce2 0.137088ce1 0.516116Vc 0.523218Vs2 0.894033Vs0 0.517968Ks0 0.254791cρ 0.075890

Table 7. The parameters used to simulate the full modelin Fig. 7

Ks1 0.391264Vs1 0.004042Ve1 0.137534Ks2 0.866446Ve2 0.328774k1 0.937549k2 0.195214k3 0.290259l1 0.710602l2 0.412784l3 0.179312Kc 0.385129ce2 0.026382ce1 0.489789Vc 0.721505Vs2 0.700199Vs0 0.364209Ks0 0.999872cρ 0.175391


The single nanR mutant was simulated by setting thevalue of nR to 0. The nagC mutant was simulated bysetting the value of g2 to 1. Furthermore, in the simulationsof the transient behaviour of the nagC mutant, the initialconditions of ϱe2 were determined as follows:

• Set the value of s1 (external N-acetylneuraminic acid)to 0.

• Determine the steady-state concentrations e2ss and ϱe2

ss

of e2 and ϱe2, respectively, through simulation.• Simulate the system again using the standard value of

s1=4000, with initial conditions of e2 and ϱe2 set to e2ss

and ϱe2ss, respectively.

Acknowledgements

I.C.B. thanks Jackie Plumbridge for discussionson N-acetylneuraminic acid toxicity. I.C.B. and J.R.were supported by grant 076360/Z/05/Z from theWellcome Trust.

Appendix A. The Base Model Has, atMost, One Solution

Solving Eq. (5) gives the following expression fore2:

e2 ¼ 12ce2 l1

�T1FffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT21 þ T2

q� �(A1)

where T1=ce2l2+ce2s2l3 and T2=4ce2l1Ve2l2+4ce2l1Ve2s2l3.It is immediately clear that only the “+” solutiongives physically meaningful (i.e., positive real)steady-state values for e2. In order for the “+”solution to be positive, the expression under thesquare root needs to be greater than T2. This istrivially always the case. Hence, e2 has exactly thesame number of steady-state solution as s2.

Using computational algebra systems, it is feasibleto generate explicit solutions of s2 in terms ofparameters of the system. These solutions, however,are extremely complicated and, therefore, notinsightful. In the general case, s2 will have severalsolutions. The relevant question in the currentcontext is how many physically realistic (i.e.,positive real) solutions s2 has. Addressing thisquestion does not require a full solution of s2.From Eq. (13), we can see that s2 only has a steady-state solution if e2N e1αVs2/Vc; here, e1 and α can besubstituted by combinations of parameters only.This means that, if the system has at least one steadystate, then e2 must be greater than some positiveconstant C0 (without worrying, for the moment,about its exact value):

e2 � C0N0 (A2)

Given the known expression for the steady state of e2from Eq. (A1) and given that we require e2 to bepositive real, we obtain:

�T1 þ ðT21 þ T2Þ1=2NC1 (A3)

Again, here, C1=2ce2l1C0 summarises a combinationof some constant parameters. This equation meansthat there exists a positive constant K such that:

T22� 2T1C1 � C2

1 � K ¼ 0Ys2

¼ �ð4ce2 l1Ve2 l2 � 2ce2 l2C� 2ce2 l2K�C2�2CK�K2Þ2ce2 l3ð2l1Ve2�C�KÞ

(A4)

This tells us that, depending on the parameters, s2has either no or a unique steady-state solution; e2has always the same number of steady-statesolutions as s2.The conclusion that there exists, at most, one

steady state might have to be modified if oneassumed additional nonlinearities in the model.These could, for example, arise from cooperativitybetween derepressors. We have investigated thisquestion numerically (data not shown) and foundthat, over a wide range of possible cooperativityassumptions, themodel still has, atmost, one steady-state solution. Furthermore, whenever we foundmore than one steady-state solution, then there wasstill only one attractor solution. While our numericalexperiments do certainly not amount to a mathe-matical proof, there is a general argument to supportthe conclusion: Multistability in biochemical net-works is mostly associated with activators ratherthan with repressors.20 Given that the interactionsin this system are mostly repressive, the existence ofmultiple stable steady states would be unexpected.We can therefore exclude the possibility of multiplestable steady states.We also solved a number of random instantiations

of the extend model (see Materials andMethods). Astheoretically predicted, we found not a single set ofparameters for which there were multiple steady-state solutions. Of a sample of size of 81,000, 32.5%


leads to stable behaviour in the WT with nR=0.7.Based on a sample of 33,000, an estimated 29.5%shows stable behaviour for both the WT and theNanR mutant. Finally, we found not a single ins-tance where the WT and the double mutant werestable but the nanR mutant was unstable. Note thatthese experiments were performed over the entireparameter space, ignoring the restriction to theidentified feasible values reported above.The density of steady-state solutions for the full

model crucially depends on the mRNA breakdownrate. For high mRNA breakdown rates, steady-statesolutions become very sparse, particularly for longerdelays. This is so because mRNAwill increasingly bebroken down before it could be translated; this, inturn, could lead to metabolic bottlenecks, wherethe efflux rate falls below the influx rate, leading tothe accumulation of metabolites. Our numericalsimulations have shown that, similarly to theextended model, there exists, at most, one positivereal solution for the full model.

Appendix B. Derivation of TranscriptionEfficiencies

We assume that NanR is displaced by GlcNAc-6Pin one step; that is, using the variable names fromthe model, we have the following reaction schemefor displacement:

gþ nR⇌k1

k2ð1� gÞ (B1)

ð1� gÞ þ x1Yk3

gþ x1 þ nR (B2)

Here, g is the probability with which the relevantgene is turned on; this quantity is described by thefollowing differential equation:

�g ¼ �gnRk2 þ ð1� gÞk2 þ ð1� gÞx1k3 (B3)

By setting g ̇=0, one obtains an equation that can besolved for g:

g ¼ k2 þ x1k3nRk1 þ k2 þ x1k3

(B4)

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http://dx.doi.org/10.1016/j.jtbi.2006.08.007

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