Gene regulation in continuous cultures: A unified theory for bacteria and yeasts
aa r X i v : . [ q - b i o . CB ] J a n Gene regulation in ontinuous ultures: Auni(cid:28)ed theory for ba teria and yeastsJason T. NoelDepartment of Chemi al Engineering, University of Florida, Gainesville,FL 32611-6005.Atul NarangDepartment of Chemi al Engineering, University of Florida, Gainesville,FL 32611-6005.Abstra tDuring bat h growth on mixtures of two growth-limiting substrates, mi robes on-sume the substrates either sequentially or simultaneously. These growth patternsare manifested in all types of ba teria and yeasts. The ubiquity of these growthpatterns suggests that they are driven by a universal me hanism ommon to allmi robial spe ies. In previous work, we showed that a minimal model a ountingonly for enzyme indu tion and dilution explains the phenotypes observed in bat h ultures of various wild-type and mutant/re ombinant ells. Here, we examine theextension of the minimal model to ontinuous ultures. We show that: (1) Sev-eral enzymati trends, usually attributed to spe i(cid:28) regulatory me hanisms su h as atabolite repression, are ompletely a ounted for by dilution. (2) The bifur ationdiagram of the minimal model for ontinuous ultures, whi h lassi(cid:28)es the substrate onsumption pattern at any given dilution rate and feed on entrations, provides aa pre ise explanation for the empiri ally observed orrelation between the growthpatterns in bat h and ontinuous ultures. (3) Numeri al simulations of the modelare in ex ellent agreement with the data. The model aptures the variation of thesteady state substrate on entrations, ell densities, and enzyme levels during thesingle- and mixed-substrate growth of ba teria and yeasts at various dilution ratesand feed on entrations. (4) This variation is well approximated by simple analyt-i al expressions that furnish physi al insights into the steady states of ontinuous ultures. Sin e the minimal model des ribes the behavior of the ells in the absen eof any regulatory me hanisms, it provides a framework for rigorously quantitatingthe e(cid:27)e t of these me hanisms. We illustrate this by analyzing several data sets fromthe literature.Key words: Mathemati al model, mixed substrate growth, substitutablesubstrates, hemostat, gene expression.Preprint submitted to Elsevier O tober 24, 2018 Introdu tionThe me hanisms of gene regulation are of fundamental importan e in biology.They play a ru ial role in development by determining the fate of isogeni embroyoni ells, and there is growing belief that the diversity of biologi alorganisms re(cid:29)e ts the variation of their regulatory me hanisms (Carroll et al.,2005; Ptashne and Gann, 2002).Many of the key prin iples of gene regulation, su h as positive, negative, andallosteri ontrol, were dis overed by studying the growth of ba teria andyeasts in bat h ultures ontaining one or more growth-limiting substrates.A model system that played a parti ularly important role is the growth ofEs heri hia oli on la tose and a mixture of la tose and glu ose (Müller-Hill,1996).It turns out that the enzymes atalyzing the transport and peripheral atabolismof la tose, su h as la tose permease and β -gala tosidase, are synthesized orindu ed only if la tose is present in the environment. The mole ular me ha-nism of indu tion was dis overed by Monod and oworkers (Ja ob and Monod,1961). They showed that the genes en oding the peripheral enzymes for la -tose are ontiguous and trans ribed sequentially, an arrangement referred toas the la operon. In the absen e of la tose, the la operon is not trans ribedbe ause the la repressor is bound to a spe i(cid:28) site on the la operon alledthe operator. This prevents RNA polymerase from atta hing to the operon andinitiating trans ription. In the presen e of la tose, trans ription of la is trig-gered be ause allola tose, a produ t of β -gala tosidase, binds to the repressor,and renders it in apable of binding to the operator.When Es heri hia oli is grown on a mixture of glu ose and la tose, la tran-s ription, and hen e, the onsumption of la tose, is somehow suppressed untilglu ose is exhausted. During this period of preferential growth on glu ose, theperipheral enzymes for la tose are diluted to very small levels. The onsump-tion of la tose begins upon exhaustion of glu ose, but only after a ertainlag during whi h the la tose enzymes are built up to su(cid:30) iently high lev-els. Thus, the ells exhibit two exponential growth phases separated by anintermediate lag. Monod referred to this phenomenon as diauxi or (cid:16)double(cid:17)growth (Monod, 1942).Two major mole ular me hanisms have been proposed to explain the repres-sion of la trans ription in the presen e of glu ose:(1) Indu er ex lusion (Postma et al., 1993): In the presen e of glu ose, en-Email address: narang he.ufl.edu (Atul Narang). Corresponding author. Tel: + 1-352-392-0028; fax: + 1-352-392-95132yme IIA glc , a peripheral enzyme for glu ose, is dephosphorylated. Thedephosphorylated IIA glc inhibits la tose uptake by binding to la tose per-mease. This redu es the intra ellular on entration of allola tose, andhen e, the trans ription rate of the la operon.Geneti eviden e suggests that phosphorylated IIA glc a tivates adenylate y lase, whi h atalyzes the synthesis of y li AMP ( AMP). Sin e glu- ose uptake results in dephosphorylation of IIA glc , one expe ts the AMPlevel to de rease in the presen e of glu ose. This forms the basis of these ond me hanism of la repression.(2) AMP a tivation (Ptashne and Gann, 2002): It has been observed thatthe re ruitment of RNA polymerase to promoter is ine(cid:30) ient unless aprotein alled atabolite a tivator protein (CAP) is bound to a spe i(cid:28) CAP site on the promoter. Furthermore, CAP has a low a(cid:30)nity for theCAP site, but when bound to AMP, its a(cid:30)nity in reases dramati ally.The inhibition of la trans ription by glu ose is then explained as follows.In the presen e of la tose alone (i.e., no glu ose), the AMP level ishigh. Hen e, CAP be omes AMP-bound, atta hes to the CAP site, andpromotes trans ription by re ruiting RNA polymerase. When glu ose isadded to the ulture, the AMP level de reases by the me hanism de-s ribed above. Consequently, CAP, being AMP-free, fails to bind to theCAP site, and la trans ription is abolished.Me hanisti mathemati al models of the glu ose-la tose diauxie appeal to boththese me hanisms (Kremling et al., 2001; Santillán et al., 2007; Dedem and Moo-Young,1975; Wong et al., 1997)Over the last few de ades, mi robial physiologists have a umulated a vastamount of data showing that diauxi growth is ubiquitous (reviewed in Egli,1995; Harder and Dijkhuizen, 1982; Kovarova-Kovar and Egli, 1998). It o - urs in diverse mi robial spe ies and numerous pairs of substitutable sub-strates (satisfying identi al nutrient requirements). In many of these systems,the above me hanisms play no role. For instan e, AMP is not dete table ingram-positive ba teria (Ma h et al., 1984), and has little e(cid:27)e t on the tran-s ription of peripheral enzymes in pseudomonads (Collier et al., 1996) andyeasts (Eraso and Gan edo, 1984). It has been proposed that in these ases,the me hanisms of repression are analogous to those of the la operon, butinvolve di(cid:27)erent trans ription fa tors, su h as C pA in gram-positive ba te-ria (Chauvaux, 1996), Cr in pseudomonads (Morales et al., 2004), and MigAin yeasts (Johnston, 1999).Paradoxi ally, there is growing eviden e that AMP a tivation and indu erex lusion are not su(cid:30) ient for explaining la repression. Not long after the dis- overy of AMP a tivation (Perlman and Pastan, 1968; Ullmann and Monod,1968), Ullmann presented several lines of eviden e showing that this me h-anism played a relatively modest role in diauxi growth (Ullmann, 1974).3ut the most ompelling eviden e was obtained by Aiba and oworkers, whoshowed that the AMP levels are essentially the same during the (cid:28)rst andse ond growth phases; furthermore, glu ose-mediated la repression persistsin the presen e of high (5mM) exogenous AMP levels, and in mutants ofE. oli in whi h the ability of AMP to in(cid:29)uen e la trans ription is om-pletely abolished (Inada et al., 1996; Kimata et al., 1997). In Se tion 3.1, weshall show that the positive orrelation between la expression and intra el-lular AMP levels (Epstein et al., 1975), whi h forms the foundation of the AMP a tivation me hanism, does not re(cid:29)e t a ausal relationship: The ob-served variation of la expression is almost entirely due to dilution (ratherthan AMP a tivation).The persisten e of la repression in AMP-independent ells has led to the hy-pothesis that indu er ex lusion is the sole ause of repression (Kimata et al.,1997). However, this me hanism exerts a relatively mild e(cid:27)e t. The la tose up-take rate is inhibited only ∼
50% in the presen e of glu ose (M Ginnis and Paigen,1969).Thus, trans riptional repression, by itself, annot explain the several hundred-fold repression of la during the (cid:28)rst exponential growth phase of diauxi growth. In earlier work, we have shown that omplete repression is predi tedby a minimal model a ounting for only indu tion and growth (Narang, 1998a,2006).Yet another phenomenon that warrants an explanation is the empiri ally ob-served orrelation between the substrate onsumption pattern and the spe- i(cid:28) growth on the individual substrates. For instan e, in the ase of diauxi growth, it has been observed that:In most ases, although not invariably, the presen e of a substrate per-mitting a higher growth rate prevents the utilization of a se ond, `poorer'substrate in bat h ulture (Harder and Dijkhuizen, 1982).It turns out that sequential onsumption of substrates is not the sole growthpattern in bat h ultures. Monod observed that in several ases of mixed-substrate growth, there was no diauxi lag (Monod, 1942, 1947), and subse-quent studies have shown that both substrates are often onsumed simulta-neously (Egli, 1995). The o urren e of simultaneous substrate onsumptionalso appears to orrelate with the spe i(cid:28) growth rates on the individual sub-strates. Based on a omprehensive review of the literature, Egli notes that:Espe ially ombinations of substrates that support medium or low maxi-mum spe i(cid:28) growth rates are utilized simultaneously (Egli, 1995).Re ently, we have shown that the minimal model provides a natural explana-tion for the foregoing orrelations, whi h hinges upon the fa t that the enzyme4a) (b)Figure 1. If the indu tion rate of the peripheral enzyme is independent of theindu er level, its a tivity de reases with the dilution rate. (a) The a tivities of β -gala tosidase (Silver and Mateles, 1969, Fig. 1) and amidase (Clarke et al., 1968,Fig. 2) during growth of the onstitutive mutants, E. oli B6b2 and P. aerugi-nosa C11, on la tose and a etamide, respe tively. (b) The a tivity of formaldehydedehydrogenase (FDH) during glu ose-limited growth of H. polymorpha and C. boi-dinii (Egli et al., 1980, Fig. 1 ,d). The urves in (a) and (b) shows the (cid:28)ts to eqs. (17)and (18), respe tively.dilution rate is proportional to the spe i(cid:28) growth rate (Narang and Pilyugin,2007a). Roughly, substrates that support high growth rates lead to su h highenzyme dilution rates that the enzymes of the (cid:16)less preferred(cid:17) substrates arediluted to near-zero levels. On the other hand, substrates that support lowgrowth rates fail to e(cid:30) iently dilute the enzymes of the other substrates. Amore pre ise statement of this argument is given in Se tion 3.3.Now, all the experimental results des ribed above were obtained in bat h ul-tures. However, sin e the invention of the hemostat, mi robial physiologistshave a quired extensive data on mi robial growth in ontinuous ultures. De-tailed analyses of this data have revealed well-de(cid:28)ned patterns or motifs thato ur in diverse mi robial spe ies growing on various substrates (Egli, 1995;Harder and Dijkhuizen, 1982; Kovarova-Kovar and Egli, 1998). The goal ofthis work is to show that the minimal model also a ounts for the patternsobserved in ontinuous ultures. We begin by giving a brief preview of thesepatterns in single- and mixed-substrate ultures.In hemostats limited by a single substrate, steady growth an be maintainedat all dilution rates up to the riti al dilution rate at whi h the ells washout. At sub riti al dilution rates, the peripheral enzyme a tivities of various ell types and growth-limiting substrates invariably show one of the followingtwo trends (Harder and Dijkhuizen, 1982):(1) If the peripheral enzyme is fully onstitutive, its a tivity de reases mono-toni ally with the dilution rate (Fig. 1a). The very same trend is observedeven if the peripheral enzyme is indu ible, but the nutrient medium la kssubstrates that an indu e the synthesis of the enzyme (Fig. 1b and5a) (b)Figure 2. The a tivity of indu ible enzymes passes through a maximum at an in-termediate dilution rate. (a) A tivities of PtsG (Seeto et al., 2004, Fig. 1) and β -gala tosidase (Silver and Mateles, 1969, Fig. 4a) during glu ose-limited growthof E. oli K12 and la tose-limited growth of E. oli B6, respe tively. (b) A tivitiesof al ohol oxidase (Egli and Harder, 1983, Fig. 1) and amidase (Clarke et al., 1968,Fig. 1) during methanol-limited growth of H. polymorpha and a etamide-limitedgrowth of P. aeroginosa, respe tively.Fig. 6d).(2) If the peripheral enzyme is indu ible and the medium ontains the in-du ing substrate, the enzyme a tivity passes through a maximum at anintermediate dilution rate (Fig. 2).Clarke and oworkers were the (cid:28)rst to observe these trends during a etamide-limited growth of wild-type and onstitutive mutants of P. aeruginosa (Clarke et al.,1968). They hypothesized that in wild-type ells, the amidase a tivity exhibitsa maximum (Fig. 2b) due to the balan e between indu tion and atabolite re-pression. Spe i(cid:28) ally,at low dilution rates, atabolite repression is minimal and the rate of ami-dase synthesis is dependent mainly on the rate at whi h a etamide is pre-sented to the ba teria. Thus, with in rease in dilution rate the amidasespe i(cid:28) a tivity under steady state onditions in reases. However, above D = 0 . h − the growth rate has in reased to the point where metaboli intermediates are being formed at a su(cid:30) iently high rate to ause signi(cid:28) ant atabolite repression. At higher dilution rates atabolite repression be omesdominant and at D = 0 . h − the enzyme on entration has de reased on-siderably (Clarke et al., 1968, p. 232).In onstitutive mutants, the enzyme a tivity de reases monotoni ally (Fig. 1a)be ause (cid:16)they la k ompletely the part where, in the urves for the wild-type strain, . . . indu tion is dominant.(cid:17) These hypotheses have subsequentlybeen invoked to rationalize the similar trends found in the wild-type ellsand onstitutive mutants of many other mi robial systems (reviewed in Dean,1972; Matin, 1978; Toda, 1981). We shall show below that the de line of theenzyme a tivity (at all D in onstitutive mutants, and at high D in wild-type6igure 3. Simultaneous and preferential growth patterns in ontinuous ultures.(a) During growth of H. polymorpha on a mixture of xylose and gly erol, both sub-strates are onsumed at all dilution rates up to washout ( al ulated from Fig. 3 ofBrinkmann and Babel, 1992). Closed and open symbols show the substrate on en-trations during single- and mixed-substrate growth, respe tively. (b) During growthof E. oli B on a mixture of glu ose and la tose, onsumption of la tose eases ata dilution rate below the (washout) dilution rate at whi h onsumption of glu ose eases (Silver and Mateles, 1969, Fig. 3).(a) (b)Figure 4. Variation of the substrate on entrations and ell density during mixed(cid:21)substrate growth of H. polymorpha on a mixture of glu ose and methanol (Egli et al.,1986, Fig. 2). The total feed on entration of glu ose and methanol was 5 g L − inall the experiments. The per entages show the mass per ent of methanol in the feed.The riti al dilution rates on pure methanol ( urve labeled 100%) and pure glu ose( urve labeled 0%) are ∼ ∼ − , respe tively. ells) is entirely due to dilution, rather than atabolite repression.In hemostats limited by pairs of arbon sour es, two types of steady statepro(cid:28)les have been observed. Importantly, these pro(cid:28)les orrelate with, andin fa t, an be predi ted by, the substrate onsumption pattern observed insubstrate-ex ess bat h ultures. Indeed, pairs of substrates onsumed simul-taneously in substrate-ex ess bat h ultures are onsumed simultaneously in ontinuous ultures at all dilution rates up to washout (Fig. 3a). In ontrast,pairs of substrates that show diauxi growth in substrate-ex ess bat h ultures7re onsumed simultaneously in ontinuous ultures only if the dilution rateis su(cid:30) iently small. For instan e, during growth of Es heri hia oli B on amixture of glu ose and la tose, onsumption of the (cid:16)less preferred(cid:17) substrate,la tose, de lines sharply at an intermediate dilution rate, beyond whi h onlyglu ose is onsumed (Fig. 3b). This growth pattern has been observed in sev-eral other systems (reviewed in Harder and Dijkhuizen, 1976, 1982), but themost omprehensive data was obtained in studies of the methylotrophi yeasts,Hansenula polymorpha and Candida boidinii, with mixtures of methanol + glu- ose (Egli et al., 1980, 1982b,a, 1986). These studies, whi h were performedwith various feed on entrations of glu ose and methanol, revealed severalwell-de(cid:28)ned patterns:(1) The sharp de line in methanol onsumption at an intermediate dilutionrate is analogous to the phenomenon of diauxi growth in bat h ulturesinasmu h as it is triggered by a pre ipitous drop in the a tivities of theperipheral enzymes for methanol, the (cid:16)less preferred(cid:17) substrate. The tran-sition dilution rate was empiri ally de(cid:28)ned by Egli as the dilution rate atwhi h the on entration of the (cid:16)less preferred substrate(cid:17) a hieves a su(cid:30)- iently high value, e.g., half of the feed on entration (Egli et al., 1986,Fig. 1).(2) The transition dilution rate is always higher than the riti al dilutionrate on methanol (Fig. 4a). This was referred to as the enhan ed growthrate e(cid:27)e t, sin e methanol was onsumed at dilution rates signi(cid:28) antlyhigher than the riti al dilution rate in ultures fed with pure methanol.(3) The transition dilution rate varies with the feed omposition (cid:22) the largerthe fra tion of methanol in the feed, the smaller the transition dilutionrate (Fig. 4a).These and several other general trends, dis ussed later in this work, have beenobserved in a wide variety of ba teria and yeasts (reviewed in Egli, 1995;Kovarova-Kovar and Egli, 1998).The o urren e of the very same growth patterns in su h diverse organismssuggests that they are driven by a universal me hanism ommon to all spe ies.Thus, we are led to onsider the minimal model, whi h a ounts for only thosepro esses (cid:22) indu tion and growth (cid:22) that o ur in all mi robes. In earlierwork, omputational studies of a variant of the minimal model showed thatit aptures all the growth patterns dis ussed above (Narang, 1998b). Here,we perform a rigorous bifur ation analysis of the model to obtain a omplete lassi(cid:28) ation of the growth patterns at any given dilution rate and feed on- entrations. The analysis reveals new features, su h as threshold e(cid:27)e ts, that We shall onstantly appeal to this extensive data on the growth of methylotrophi yeasts. A detailed exposition of the metabolism and gene regulation in these organ-isms an be found in a re ent review (Hartner and Glieder, 2006).8igure 5. Kineti s heme of the minimal model (Narang, 1998a).were missed in our earlier work. It also provides simple explanations for thefollowing questions:(1) During single-substrate growth of wild-type ells, why does the a tivityof the peripheral enzymes pass through a maximum?(2) During growth on mixtures of substrates, what are onditions underwhi h (a) both substrates are onsumed at all dilution rates up to washout(Fig. 3a), and (b) onsumption of one of the substrates eases at an in-termediate dilution rate (Fig. 3b)?(3) Why does the transition dilution rate exist? Why is it always larger thanthe riti al dilution rate of the (cid:16)less preferred(cid:17) substrate, and why doesit vary with the feed on entrations (Fig. 4a)?Finally, we show that under the onditions typi ally used in experiments,the physiologi al (intra ellular) steady states are ompletely determined byonly two parameters, namely, the dilution rate and the mass fra tion of thesubstrates in the feed (rather than the feed on entrations per se). In fa t, wederive expli it expressions for the steady state substrate on entrations, elldensity, and enzyme levels, whi h provide physi al insight into the variationof these quantities with the dilution rate and feed on entrations.2 TheoryFig. 5 shows the kineti s heme of the minimal model. Here, S i denotes the i th exogenous substrate, E i denotes the (cid:16)lumped(cid:17) peripheral enzymes for S i , X i denotes internalized S i , and C − denotes all intra ellular omponents ex ept E i and X i (thus, it in ludes pre ursors, free amino a ids, and ma romole ules).We assume that: 91) The on entrations of the intra ellular omponents, denoted e i , x i , and c − , are based on the dry weight of the ells (g per g dry weight of ells,i.e., g gdw − ). The on entrations of the exogenous substrate and ells,denoted s i and c , are based on the volume of the rea tor (g L − andgdw L − , respe tively). The rates of all the pro esses are based on thedry weight of the ells (g gdw − h − ). We shall use the term spe i(cid:28) rateto emphasize this point.The hoi e of these units implies that if the on entration of any intra el-lular omponent, Z , is z g gdw − , then the evolution of z in a ontinuous ulture operating at dilution rate, D , is given by dzdt = r + z − r − z − D + 1 c dcdt ! z, where r + z and r − z denote the spe i(cid:28) rates of synthesis and degradationof Z , and Dz is the spe i(cid:28) rate of e(cid:31)ux of Z from the hemostat. It isshown below that the sum, Dz + (1 /c )( dc/dt ) z , is pre isely the rate ofdilution of Z due to growth.(2) The transport and peripheral atabolism of S i is atalyzed by a uniquesystem of peripheral enzymes, E i . The spe i(cid:28) uptake rate of S i , denoted r s,i , follows the modi(cid:28)ed Mi haelis-Menten kineti s r s,i ≡ V s,i e i s i K s,i + s i , where V s,i is a (cid:28)xed onstant, i.e., indu er ex lusion is negligible.(3) Part of the internalized substrate, denoted X i , is onverted to C − . Theremainder is oxidized to CO in order to generate energy.(a) The onversion of X i to C − and CO follows (cid:28)rst-order kineti s, i.e., r x,i ≡ k x,i x i . (b) The fra tion of X i onverted to C − , denoted Y i , is onstant. It isshown below that Y i is essentially identi al to the yield of biomasson S i . This assumption is therefore tantamount to assuming thatthe yield of biomass on a substrate is una(cid:27)e ted by the presen e ofanother substrate in the medium.(4) The internalized substrate also indu es the synthesis of E i , whi h is as-sumed to ontrolled entirely by the initiation of trans ription (i.e., me h-anisms su h as attenuation and proteolysis are negle ted).(a) The spe i(cid:28) synthesis rate of E i follows the hyperboli Yagil & Yagilkineti s (Yagil and Yagil, 1971), r + e,i ≡ V e,i x i α i + x i , (1)where V e,i is a (cid:28)xed onstant, i.e., atabolite repression is negligible.In the parti ular ase of negatively ontrolled operons (modulated10y repressors), V e,i and α i an be expressed in terms of parametersasso iated with kineti s of RNAP-promoter and repressor-operatorbinding, respe tively. In other ases, (1) must be viewed as a phe-nomenologi al des ription of the indu tion kineti s.(b) Enzyme degradation is a (cid:28)rst-order pro ess, i.e., the spe i(cid:28) rate ofenzyme degradation is r − e,i = k e,i e i . ( ) The enzymes are synthesized at the expense of C − , and their degra-dation produ es C − .Given these assumptions, the mass balan es yield the equations ds i dt = D ( s f,i − s i ) − r s,i c, (2) dx i dt = r s,i − r x,i − D + 1 c dcdt ! x i , (3) de i dt = r + e,i − r − e,i − D + 1 c dcdt ! e i , (4) dc − dt = ( Y r x, + Y r x, ) − (cid:16) r + e, − r − e, (cid:17) − (cid:16) r + e, − r − e, (cid:17) − D + 1 c dcdt ! c − , (5)where s f,i denotes the on entration of S i in the feed to the hemostat.Following Fredri kson, we observe that eqs. (3)(cid:21)(5) impli itly de(cid:28)ne the spe- i(cid:28) growth rate and the evolution of the ell density (Fredri kson, 1976).Indeed, sin e x + x + e + e + c − = 1 , addition of (3)(cid:21)(5) yields r g − D + 1 c dcdt ! ⇔ dcdt = ( r g − D ) c, (6)where r g ≡ X i =1 r s,i − X i =1 (1 − Y i ) r x,i (7)is the spe i(cid:28) growth rate. As expe ted, it is the net rate of substrate a u-mulation in the ells. It follows from (6) that the last term in eqs. (3)(cid:21)(5)is pre isely the dilution rate of the orresponding physiologi al (intra ellular)variables.The equations an be simpli(cid:28)ed further be ause k − x,i , the turnover time for X i , is small (on the order of se onds to minutes). Thus, x i rapidly attains11uasisteady state, resulting in the equations ds i dt = D ( s f,i − s i ) − r s,i c, (8) de i dt = r + e,i − ( r g + k e,i ) e i , (9) dcdt = ( r g − D ) c, (10) x i ≈ V s,i e i s i / ( K s,i + s i ) k x,i , (11) r g ≈ Y r s, + Y r s, , (12)where (11)(cid:21)(12) follow from the quasisteady state relation, ≈ r s,i − r x,i .It is worth noting that:(1) The parameter, Y i , whi h was de(cid:28)ned as the fra tion of X i onverted to C − , is essentially the yield of biomass on S i . Indeed, substituting therelation, r x,i ≈ r s,i in (7) shows that after quasisteady state has beenattained, Y i approximates the fra tion of S i that a umulates in the ell.(2) Although we have assumed that Y i is onstant, the validity of (12) doesnot rest upon this assumption. It is true even if the yields are variable(i.e., depend on the physiologi al state).(3) Enzyme indu tion is subje t to positive feedba k. This be omes evidentif (11) is substituted in (1): The quasisteady state indu tion rate, r + e,i ,is an in reasing fun tion of e i . The positive feedba k re(cid:29)e ts the y li stru ture of enzyme synthesis that is inherent in Fig. 5: E i promotes thesynthesis of X i , whi h, in turn, stimulates the indu tion of E i .We shall appeal to these fa ts later.The equations for bat h growth are obtained from the above equations by let-ting D = 0 . In what follows, we shall be parti ularly on erned with the initialevolution of the enzymes in experiments performed with su(cid:30) iently small in-o ula. Under this ondition, the substrate onsumption for the (cid:28)rst few hoursis so small that the substrate on entrations remain essentially onstant attheir initial levels, s ,i . In the fa e of this quasi onstant environment, the en-zyme a tivities move toward a quasisteady state orresponding to exponential(balan ed) growth. This motion is given by the equations de i dt = r + e,i − ( r g + k e,i ) e i , (13) x i ≈ V s,i e i s ,i / ( K s,i + s ,i ) k x,i , (14) r g ≈ Y V s, e s , K s, + s , + Y V s, e s , K s, + s , , (15)12btained by letting s i ≈ s ,i in eqs. (11)(cid:21)(12). The steady state(s) of theseequations represent the quasisteady state enzyme a tivities attained duringthe (cid:28)rst exponential growth phase.We note (cid:28)nally that the ells sense their environment through the ratio, σ i ≡ s i K s,i + s i , whi h hara terizes the extent to whi h the permease is saturated with S i .Not surprisingly, we shall (cid:28)nd later on that the physiologi al steady statesdepend on the ratio, σ i , rather than the substrate on entrations per se. It is,therefore, onvenient to de(cid:28)ne the ratios σ ,i ≡ s ,i K s,i + s ,i , σ f,i ≡ s f,i K s,i + s f,i . For a given ell type, σ i , σ ,i , σ f,i are in reasing fun tions of s i , s i, , s f,i , re-spe tively, and an be treated as surrogates for the orresponding substrate on entrations. In what follows, we shall frequently refer to σ i , σ ,i , and σ f,i as the substrate on entration, initial substrate on entration, and feed on- entration, respe tively.3 Results3.1 The role of regulationBefore analyzing the foregoing equations for bat h and ontinuous ultures,we pause to onsider the role of regulatory me hanisms. This is importantbe ause the literature pla es great emphasis on the regulatory me hanisms,whereas the model negle ts them ompletely ( V s,i and V e,i are assumed to be onstants). It is therefore relevant to ask: To what extent do the regulatoryme hanisms ontrol gene expression? To address this question, we shall fo uson the well- hara terized la operon in E. oli. The analysis shows that AMPa tivation and indu er ex lusion annot explain the strong repression observedin experiments.3.1.1 AMP a tivationInterest in AMP as a la regulator began when it was dis overed that the la indu tion rate in reased ∼ β -gala tosidase a tivity with intra ellular AMP level dur-ing exponential growth of the la - onstitutive strain E. oli X2950 on various ar-bon sour es (Epstein et al., 1975, Fig. 2). (b, ) During bat h growth of (b) E. oliMC4100 λ CPT100 (Kuo et al., 2003, Fig. 2) and ( ) E. oli NC3 (Wanner et al.,1978, Fig. 1) on various arbon sour es, the steady state β -gala tosidase a tivityis inversely proportional to the spe i(cid:28) growth rate on the arbon sour e. More-over, the β -gala tosidase a tivity in reases at most 2-fold if 5mM AMP is addedto the medium ( (cid:4) ). (d) The inverse relationship is satis(cid:28)ed at D & . h − in ontinuous ultures of E. oli K12 W3110 grown on glu ose and glu ose + 1mMIPTG (Kuo et al., 2003, Fig. 1). The urves in (b)(cid:21)( ) and (d) show the (cid:28)ts to (16)and (17), respe tively.rate in reased with the intra ellular AMP levels (Fig. 6a). They obtainedthis relationship by measuring the β -gala tosidase a tivities and intra ellular AMP levels during exponential growth of the la - onstitutive strain E. oliX2950 on a wide variety of arbon sour es. The use of a onstitutive strainserved to eliminate indu er ex lusion: In su h ells, the repressor-operatorbinding is impaired so severely that α i ≈ , and the indu tion rate is una(cid:27)e tedby the indu er level ( r + e,i ≈ V e,i ). The diverse arbon sour es provided a way ofvarying the intra ellular AMP levels: In general, the lower the spe i(cid:28) growthrate on a substrate, the higher the intra ellular AMP level (Buettner et al.,1973).Although Fig. 6a shows a orrelation between the AMP level and the la indu tion rate, it does not prove that there is a ausal relationship between14hem. Indeed, sin e protein degradation is negligible in substrate-ex ess bat h ultures (Mandelstam, 1958), (13) yields e i ≈ r + e,i r g ≈ V e,i r g . (16)Thus, the steady state enzyme level depends on the ratio of the spe i(cid:28) indu -tion and growth rates. Sin e the substrates that yield high intra ellular AMPlevels also support low spe i(cid:28) growth rates, it is on eivable that Fig. 6a re-(cid:29)e ts the variation of the spe i(cid:28) growth rate, r g , rather than the spe i(cid:28) indu tion rate, V e,i .Eq. (16) provides a simple riterion for he king if the spe i(cid:28) indu tion ratevaried in the experiments: Evidently, V e,i varies if and only if e i is not inverselyproportional to r g . Unfortunately, we annot he k the data in Fig. 6a withthis riterion, sin e the authors did not report the spe i(cid:28) growth rates of the arbon sour es. However, the very same experiments have been performed byothers (Kuo et al., 2003; Wanner et al., 1978), and this data shows that the β -gala tosidase a tivity is inversely proportional to r g (Figs. 6b, ). It followsthat in Figs. 6b and 6 , the β -gala tosidase a tivity varies almost entirely dueto dilution. The in(cid:29)uen e of AMP on la trans ription is very modest, a fa tthat is further on(cid:28)rmed by the very marginal (<2-fold) improvement of the β -gala tosidase a tivity in the presen e of 5mM AMP (Figs. 6 ). The sameis likely to be true of the data in Fig. 6a. Thus, the data from Epstein et al.,1975, whi h is extensively ited to support the existen e of AMP-mediatedla ontrol, for es upon us just the opposite on lusion: AMP has virtuallyno e(cid:27)e t on the la indu tion rate.The data from ontinuous ultures also implies the absen e of signi(cid:28) ant AMP a tivation. Indeed, (9) implies that when la - onstitutive mutants of E. oli are grown in ontinuous ultures, the steady state β -gala tosidase a tivityat su(cid:30) iently high dilution rates is given by the relation e i ≈ V e,i D . (17)This relationship also holds for wild-type ells if the medium ontains satu-rating on entrations of the gratuitous indu er, IPTG. In both ases, AMP ontrol is signi(cid:28) ant only if e i is not inversely proportional to D . But exper-iments performed with la - onstitutive mutants (Fig. 1a) and wild-type ellsexposed to saturating IPTG levels (Fig. 6d) show that β -gala tosidase a tivityis inversely proportional to D for su(cid:30) iently large D . At small D , the enzyme a tivity is signi(cid:28) antly lower than that predi ted by (17).Evidently, this dis repan y annot be due to AMP a tivation. It annot be resolvedby enzyme degradation either: Although the data an be (cid:28)tted to the equation, e i = V e,i / ( D + k e,i ) , the values of k e,i required for these (cid:28)ts are physiologi ally implausible15atabolite repression also appears to be insigni(cid:28) ant in ell types other thanE. oli. The amidase a tivity is inversely proportional to D in the onstitutivemutant P. aeroginosa C11 (Fig. 1a). Moreover, this de lining trend is not dueto atabolite repression, as postulated by Clarke and oworkers (cid:22) it re(cid:29)e tsthe e(cid:27)e t of dilution. Likewise, the a tivity of al ohol oxidase is inverselyproportional to D when H. polymorpha and C. boidinii are grown in glu ose-limited hemostats. In this ase, (9) implies that at su(cid:30) iently large dilutionrates e i = V e,i α i D , (18)whi h is formally similar to (17).3.1.2 Indu er ex lusionThe data shows that indu er ex lusion ertainly exists. When glu ose orglu ose-6-phosphate are added to a ulture growing exponentially on la tose,the spe i(cid:28) uptake rate of la tose de lines ∼ ∼ β -gala tosidase ∼ α i is large ompared to 1. This is true even in the ase of glu ose ( α i =5(cid:21)10), whi h is oftentreated as a substrate onsumed by onstitutive enzymes (Notley-M Robb and Feren i,2000, Table 3). Under these onditions, the basal indu tion rate is small om-pared to the fully indu ed indu tion rate. Hen e, at all but the smallest indu er( ∼ − ). Based on the detailed study of the ptsG operon at various dilution rates,the dis repan y is probably due to enhan ed expression of the starvation sigmafa tor, RpoS, at low dilution rates (Seeto et al., 2004, Fig. 3)16evels, the spe i(cid:28) indu tion rate an be approximated by the expression V e,i x i K e,i + x i , K e,i ≡ α i K x,i , (19)and the quasisteady state indu tion rate, obtained by substituting (11) in (19),is V e,i e i σ i ¯ K e,i + e i σ i , ¯ K e,i ≡ K e,i k x,i V s,i . It is useful to make this approximation be ause the equations be ome amenableto rigorous analysis and yield simple physi al insights. In the rest of the paper,we shall on(cid:28)ne our attention to these idealized perfe tly indu ible systems.We begin by onsidering the growth of ultures limited by a single substrate,say, S . Under this ondition, s = e = 0 , and the steady state values of theremaining variable satisfy the equations ds dt = D ( s f, − s ) − V s, e σ c, (20) de dt = V e, e σ ¯ K e, + e σ − ( Y V s, e σ + k e, ) e , (21) dcdt = ( Y V s, e σ − D ) c, (22) x ≈ V s, e σ k x, . (23)These equations have three types of steady states: e > , e = 0 , c > ,denoted E ; e > , e = 0 , c = 0 , denoted E ; and e = 0 , e = 0 , c = 0 ,denoted E . The (cid:28)rst two steady states orrespond to the persisten e andwashout steady states of the lassi al Monod model (Herbert et al., 1956).The third steady state, E , is a onsequen e of perfe t indu ibility (cid:22) itexists pre isely be ause the spe i(cid:28) indu tion rate is zero in the absen e ofthe indu er.Fig. 7 shows the bifur ation diagram for eqs. (20)(cid:21)(23). It an be inferred fromthe following fa ts derived in Appendix B.(1) E always exists, but it is stable if and only if V e, σ f, ¯ K e, < k e, ⇔ σ f, < σ ∗ ≡ k e, V e, / ¯ K e, . Thus, growth annot be sustained in a hemostat if the feed on entrationis below the threshold level, σ ∗ . Under this ondition r + e, = V e, e σ ¯ K e, + e σ ≤ V e, e σ f, ¯ K e, + e σ f, ≤ k e, e ,
000 100 E t,1 D EE E e ec σ f,11 σ e e c e e c c,1 DD E Figure 7. Bifur ation diagram for single-substrate growth on S . The blue urvedenotes the urve of the riti al dilution rate, D c, , and σ ∗ denotes the thresholdfeed on entration. The 3-D insets depi t the existen e and stability of the steadystates at any given σ f, and D . Stable and unstable steady states are represented byfull and open ir les, respe tively. The dashed line shows the transition dilution rate, D t, , whi h plays a ru ial role in mixed-substrate growth (Se tion 3.3). The (cid:28)gureis not drawn to s ale: Sin e k e, ∼ . h − and V e, / ¯ K e, ∼ h − , the threshold, σ ∗ ≡ k e, / ( V e, / ¯ K e, ) , is on the order of 0.05.i.e., the indu tion rate of E never ex eeds the degradation rate. Hen e,as t → ∞ , e approa hes zero, and c, σ tend toward , σ f, , respe tively.(2) E exists if and only if σ f, > σ ∗ , in whi h ase it is given by therelations, c = 0 , σ = σ f, , and e = − (cid:16) ¯ K e, + k e, Y V s, (cid:17) + r(cid:16) ¯ K e, + k e, Y V s, (cid:17) + V e, Y V s, ( σ f, − σ ∗ )2 σ f, . It is stable if and only if D ex eeds the riti al dilution rate D c, ≡ Y V s, e σ f, . (24)Evidently, D c, is an in reasing fun tion of σ f, that is zero when σ f, = σ ∗ .We show below that D c, is the maximum spe i(cid:28) growth rate that anbe sustained in the hemostat ( r g < D c, ) .At (cid:28)rst sight, this steady state poses a paradox: There are no ells inthe hemostat, and yet, these non-existent ells have positive enzymelevels. The paradox disappears on e it is re ognized that E re(cid:29)e ts thelong-run behavior of a small ino ulum introdu ed into a sterile hemostatoperating at σ f, > σ ∗ and D > D c, . As t → ∞ , the spe i(cid:28) growth rate18f the ells approa hes the maximum level, D c, . However, sin e r g < D c, , dcdt = ( r g − D ) c < ( D c, − D ) c < , so that c → , σ → σ f, , and e approa hes the positive value givenabove. Thus, in ontrast to E , the ells fail to a umulate despitesustained indu tion of the enzymes be ause the dilution rate is higherthan the maximum spe i(cid:28) growth rate that an be attained.(3) E exists if and only if σ f, > σ ∗ and D < D c, , in whi h ase it is givenby the relations x = DY k x, , (25) e = V e, D + k e, DD + Y V s, ¯ K e, , (26) σ = DY V s, e = ( D + k e, ) (cid:16) D + Y V s, ¯ K e, (cid:17) Y V s, V e, , (27) c = Y ( s f, − s ) , (28)all of whi h follow from the fa t that the spe i(cid:28) substrate uptake rateis proportional to D , i.e., r s, = DY . (29)Furthermore, E is stable whenever it exists. Evidently, this is the onlysteady state that allows positive ell densities to be sustained.The riti al dilution rate, D c, , is the maximum spe i(cid:28) growth rate that an be maintained at steady state. To see this, observe that at E , thespe i(cid:28) growth rate equals the dilution rate. Thus, (27) says that spe i(cid:28) growth rate in reases with σ , and is maximal when σ = σ f, . Letting σ = σ f, in (27) and solving for D yields (24).Fig. (7) shows that at any given dilution rate and feed on entration, exa tlyone of the three steady states is stable. It is therefore straightforward to dedu ethe variation of the steady states along any path in the σ f, , D -plane. Theexperiments are typi ally performed by varying either σ f, or D , while theother one is held onstant. We shall on(cid:28)ne our attention to these two ases.3.2.1 Variation of steady states with σ f, at (cid:28)xed D If the feed on entration is in reased at a su(cid:30) iently small (cid:28)xed D (horizontalarrow in Fig. 7), steady growth is feasible only if the feed on entration lies inthe region to the right of the blue urve, where E is stable. The variationof the steady states in this region is given by (25)(cid:21)(28), whi h imply thatas the feed on entration in reases, x , e , σ remain onstant, and c in reases19a) (b)Figure 8. Growth of E. oli ML308 in a glu ose-limited hemostat. (a) When thefeed on entration of glu ose is in reased at a (cid:28)xed dilution rate, the residual glu- ose on entration ( losed symbols) is onstant, and the ell density (open symbols)in reases linearly (Lendenmann, 1994, Appendix B). The data for D = 0 . h − and D = 0 . h − is represented by the symbols, △ , N and (cid:3) , (cid:4) , respe tively. (a) Whenthe dilution rate is in reased at a (cid:28)xed feed on entration (100 mg L − ), the residualsubstrate on entration in reases monotoni ally with D , and approa hes a thresholdat small dilution rates (Kovárová et al., 1996, Fig. 2). The data was obtained at 4temperatures.linearly. Thus, any in rease in the feed on entration is ompensated by a pro-portional in rease in the ell density (cid:22) all other variables remain un hanged.We shall appeal to this fa t later.No data is available for the indu er or enzyme levels, but experiments per-formed with glu ose-limited ultures of E. oli ML308 show that at dilutionrates of . and 0.6 h − , the residual substrate on entration is indeed inde-pendent of the feed on entration, and the ell density does in rease linearlywith the feed on entration (Fig. 8a). No threshold was observed in these ex-periments be ause the feed on entrations were relatively high ( ≥ mg L − ).As we show below, the threshold on entration for glu ose is ∼ µ g L − .3.2.2 Variation of steady states with D at (cid:28)xed σ f, If the dilution rate is in reased at any (cid:28)xed σ f, < σ ∗ , there is no growthregardless of the dilution rate at whi h the hemostat is operated. If σ f, > σ ∗ ,the variation of the steady states with D is qualitatively identi al to that of theMonod model: The persisten e steady state, E , is stable for < D < D c, ,and the washout steady state, E , is stable for D ≥ D c, (verti al arrow inFig. 7). We shall on(cid:28)ne our attention to E , sin e E is independent of D .Variation of indu er and enzyme levels It follows from (25)(cid:21)(26) thatas the dilution rate in reases over the range, (0 , D c, ) , the indu er level in- reases linearly, whereas the enzyme level passes through a maximum at20 = r(cid:16) Y V s, ¯ K e, (cid:17) k e, . The latter is onsistent with the data for systemswith indu ible enzymes (Fig. 2).In ontrast to the hypothesis of Clarke et al., the model explains the non-monotoni pro(cid:28)le of the enzyme level as the out ome of a balan e betweenindu tion and dilution/degradation (as opposed to atabolite repression). Tosee this, observe that the steady state enzyme level is given by the ratio of theindu tion and dilution/degradation rates, i.e., e = r + e, D + k e, . Furthermore, sin e x in reases with D , so does r + e, , but it saturates at su(cid:30)- iently large D . At low values of D , the enzyme level in reases with D be ausethe indu tion rate in reases with D . At high D , the steady state enzyme levelinversely proportional to D be ause the indu tion rate saturates.The validity of the above explanation rests upon the hypothesis that the indu -tion rate saturates at large D . This hypothesis is supported by the data for thela and ptsG operons. When bat h ultures of E. oli growing exponentiallyon saturating on entrations of la tose (whi h are physiologi ally identi al to ultures growing in a la tose-limited hemostat near the riti al dilution rate)are exposed to 0.4 mM IPTG, there is almost no hange in the β -gala tosidasea tivity (Hogema et al., 1998, Table 1). Similarly, the PtsG a tivity of wild-type and ptsG - onstitutive E. oli is the same in glu ose-ex ess bat h ulturesand ontinuous ultures operating at high dilution rates (Seeto et al., 2004,Fig. 3). Indu tion is therefore already saturated during growth of E. oli onla tose or glu ose, provided the dilution rate is su(cid:30) iently large. It remains tobe seen if this hypothesis is also valid for the other systems. However, at largedilution rates, the enzyme a tivity is more or less inversely proportional to D for all the systems shown Fig. 2, a result expe ted only if the indu tion ratesaturates at large D . Thus, the de line of the enzyme a tivity in both onsti-tutive mutants and wild-type ells is due to dilution, rather than ataboliterepression.Variation of the substrate on entration and ell density A ordingto the Monod model, the residual substrate on entration is proportional tothe dilution rate, i.e., σ = D/r max g, , where r max g, denotes the maximum spe i(cid:28) growth rate on S (Herbert et al., 1956). In ontrast, eq. (27) yields σ = σ ∗ + k e, + Y V s, ¯ K e, Y V s, V e, ! D + Y V s, V e, ! D , σ approa hes the threshold level, σ ∗ , and at large dilution rates, σ is proportional to D . Both these propertiesfollow from the relation, r s, ≡ V s, e σ = D/Y , and the steady state pro(cid:28)lesof the peripheral enzyme a tivity. Indeed, at low dilution rates, σ is onstantbe ause the enzyme level in reases linearly. Likewise, at high dilution rates, σ ∼ D be ause e ∼ /D .The experiments provide lear eviden e of threshold on entrations. Kovarovaet al. have shown that when E. oli ML308 is grown in a glu ose-limited hemostat, the residual glu ose on entration approa hes threshold levels of20(cid:21)40 µ g/L at small dilution rates (Fig. 8b). The authors did not identifythe spe i(cid:28) me hanism underlying the threshold. It may exist, for example,simply be ause the maintenan e requirements pre lude growth. However, weshall show below that in some ases, the threshold exists pre isely be ause theenzymes are not indu ed at su(cid:30) iently low values of the substrate on entra-tion.3.3 Mixed-substrate growthWe have shown above that in hemostats limited by a single substrate, enzymesynthesis is abolished if the feed on entration is below the threshold level,and this o urs be ause the indu tion rate of the enzyme annot mat h thedegradation rate. Our next goal is to show that in mixed-substrate ultures,enzyme synthesis is often abolished be ause the indu tion rate of the enzyme annot mat h its dilution rate. To this end, we shall begin with the ase ofmixed-substrate bat h ultures, where this phenomenon is parti ularly trans-parent. This analysis of bat h ultures will also enable us to establish theformal similarity between the lassi(cid:28) ation of the growth patterns in bat hand ontinuous ultures.3.3.1 Bat h ultures3.3.1.1 Mathemati al lassi(cid:28) ation of the growth patterns Thegrowth of mixed-substrate bat h ultures is empiri ally lassi(cid:28)ed based onthe substrate onsumption pattern (preferential or simultaneous) during the(cid:28)rst exponential growth phase. The mathemati al orrelate of the foregoingempiri al lassi(cid:28) ation is provided by the bifur ation diagram for the equations de dt = V e, e σ , ¯ K e, + e σ , − ( Y V s, e σ , + Y V s, e σ , + k e, ) e , (30) de dt = V e, e σ , ¯ K e, + e σ , − ( Y V s, e σ , + Y V s, e σ , + k e, ) e , (31)22hi h des ribe the motion of the enzymes toward the quasisteady state or-responding to the (cid:28)rst exponential growth phase.We gain physi al insight into the foregoing equations by observing that theyare formally similar to the Lotka-Volterra model for two ompeting spe ies dN dt = a N − ( b N + b N ) N , (32) dN dt = a N − ( b N + b N ) N , (33)where N i is the population density of the i -th spe ies; a i N i is the (unrestri ted)growth rate of the i -th spe ies in the absen e of any ompetion; and b ii N i , b ij N i N j represent the growth rate redu tion due to intra- and inter-spe i(cid:28) ompetition, respe tively. Evidently, the net indu tion rate, V e,i e i σ ,i / ( ¯ K e,i + e i σ ,i ) − k e,i e i , and the two dilution terms in (30)(cid:21)(31) are the orrelates ofthe unrestri ted growth rate and the two ompetition terms in the Lotka-Volterra model. Thus, the intera tion between the enzymes is ompetitive(the enzymes inhibit ea h other via the dilution terms), despite the absen eof any trans riptional repression (the indu tion rate of E i is una(cid:27)e ted bya tivity of E j , j = i ).Now, the Lotka-Volterra equations are known to yield oexisten e or extin -tion steady states, depending on the values of the parameters, b ij (Murray,1989). The formal similarity of eqs. (30)(cid:21)(31) and (32)(cid:21)(33) suggests that theenzymes will exhibit oexisten e and extin tion steady states, depending onthe values of the physiologi al parameters and the initial substrate on entra-tions. We show below that this is indeed the ase. Moreover, the oexisten eand extin tion steady states are the orrelates of simultaneous and preferentialgrowth, respe tively.Eqs. (30)(cid:21)(31) have 4 types of steady states: e = 0 , e = 0 ; e > , e = 0 ; e = 0 , e > ; and e > , e > , whi h are denoted E , E , E , and E ,respe tively. Ea h of these steady states has a simple biologi al interpretation.If the enzyme levels rea h E (resp., E ), none (resp., both) of the twosubstrates are onsumed during the (cid:28)rst exponential growth phase. If theenzyme levels rea h E (resp., E ), only S (resp., S ) is onsumed during the(cid:28)rst exponential growth phase. The bifur ation diagram shows the existen eand stability of these steady states (and hen e, the growth pattern) at anygiven σ , and σ , (Fig. 9). It an be inferred from the following fa ts derivedin Appendix A.(1) E always exists (for all σ , and σ , ). It is stable if and only if r ∗ g,i ≡ V e,i σ ,i ¯ K e,i − k e,i < ⇔ σ ,i < σ ∗ i , i = 1 , e e g,1 r g,1 r << ** r , g,2g,2 r < ee r g,2 r g,1 * < g,2 r < g,12 r < g,1 * r << g,2 * r , g,1 * r < g,12 r < g,2 r , g,1 r < g,1 r g,2 r * < r g,12 < r < * g,2 r << g,1 σ σ e e e e e e e e s s c t s s c t s s c ts s c t s s c t < * r < r < * g,2 r , g,2 r g,1 g,1 < * r g,2 , g,2 r , * g,1 r , g,1 r σ * σ * 1 Figure 9. Bifur ation diagram lassifying the growth patterns of mixed-substratebat h ultures at various initial substrate on entrations, σ , and σ , . The greenand bla k lines represent the threshold on entrations for indu tion of E and E ,respe tively. The blue and red urves represent the lo us of initial substrate on- entrations de(cid:28)ned by the equations, r g, = r ∗ g, and r g, = r ∗ g, , respe tively. Theinsets in ea h of the six regions show the dynami s at the orresponding initial sub-strate on entrations. The (cid:28)rst inset shows the motion of the enzymes during the(cid:28)rst exponential growth phase; the se ond one shows the resultant evolution of thesubstrate on entrations and ell density.i.e., the point, ( σ , , σ , ) , lies below the green line and to the left of thebla k line in Fig. 9.(2) E exists if and only if σ , > σ ∗ , in whi h ase it is unique, and givenby e = − (cid:16) ¯ K e, + k e, Y V s, (cid:17) + r(cid:16) ¯ K e, + k e, Y V s, (cid:17) + K e, r ∗ g, Y V s, σ , , e = 0 . At this steady state, the ells onsume only S , and grow exponentiallyat the spe i(cid:28) growth rate r g, ≡ Y V s, e σ , . It is stable if and only if r ∗ g, < r g, , (34)24.e., ( σ , , σ , ) lies below the blue urve in Fig. 9, de(cid:28)ned by the equation r ∗ g, = r g, . (35)Thus, E is stable under two distin t sets of onditions. Either σ , isbelow the threshold level, σ ∗ , in whi h ase synthesis of E annot besustained be ause its indu tion rate annot mat h its degradation rate.Alternatively, σ , is above the threshold level, but sustained synthesisof E remains infeasible be ause the dilution of E due to growth on S is too large. Thus, r ∗ g, an be viewed as the highest spe i(cid:28) growth ratetolerated by the indu tion ma hinery for E : Synthesis of E is abolishedwhenever r g, ex eeds r ∗ g, . We shall refer to r ∗ g, as the spe i(cid:28) growthrate for extin tion of E .For any given σ , > σ ∗ , the exponential growth rate, r g, , an be higher orlower than r ∗ g, (depending on the value of σ , ), but it annot ex eed the orresponding extin tion growth rate, r ∗ g, . This re(cid:29)e ts the biologi allyplausible fa t that e annot be zero during exponential growth on S .(3) E exists if and only if σ , > σ ∗ , in whi h ase it is (uniquely) given bythe expressions e = 0 , e = − (cid:16) ¯ K e, + k e, Y V s, (cid:17) + r(cid:16) ¯ K e, + k e, Y V s, (cid:17) + K e, r ∗ g, Y V s, σ , . At this steady state, the ells onsume only S , and grow exponentiallyat the spe i(cid:28) growth rate r g, ≡ Y V s, e σ , . It is stable if and only if r ∗ g, < r g, , (36)i.e., ( σ , , σ , ) lies to the left of the red urve in Fig. 9, de(cid:28)ned by theequation r ∗ g, = r g, . (37)Furthermore, r g, ≤ r ∗ g, with equality being attained pre isely when σ = σ ∗ , in whi h ase r g, = r ∗ g, = 0 .(4) E exists and is unique if and only if E and E are unstable, a onditionsatis(cid:28)ed pre isely when r ∗ g, > r g, , r ∗ g, > r g, , i.e., ( σ , , σ , ) lie between the blue and red urves in Fig. 9. At thissteady state, both enzymes (cid:16) oexist(cid:17) be ause neither substrate supportsa spe i(cid:28) growth rate that is large enough to annihilate the enzymes forthe other substrate. Hen e, both substrates are onsumed, and the ells25row exponentially at the spe i(cid:28) growth rate r g, ≡ Y V s, e σ , + Y V s, e σ , , where e , e are the unique positive solutions of (30)(cid:21)(31).(5) The blue urve and red urves are given by the equations r ∗ g, = k e, Y V s, ¯ K e, ! r ∗ g, + 1 Y V s, ¯ K e, (cid:16) r ∗ g, (cid:17) , (38) r ∗ g, = k e, Y V s, ¯ K e, ! r ∗ g, + 1 Y V s, ¯ K e, (cid:16) r ∗ g, (cid:17) , (39)respe tively. Both equations de(cid:28)ne graphs of in reasing fun tions passingthrough ( σ ∗ , σ ∗ ) , but the blue urve always lies below the red urve.Thus, the bifur ation diagram onsists of 6 distin t regions su h that exa tlyone of the four steady states is stable in ea h region.The bifur ation diagram implies that there are six possible growth patterns,depending on the initial substrate on entrations. Three of these growth pat-terns o ur if σ , < σ ∗ or/and σ , < σ ∗ . If both σ , and σ , are below theirrespe tive threshold on entrations, neither substrate is onsumed. If only oneof them is below its threshold level, say, σ , < σ ∗ , only S is onsumed duringthe (cid:28)rst exponential growth phase. However, growth is not diauxi be ause S is never onsumed. If σ , > σ ∗ and σ , > σ ∗ , the model predi ts the existen eof three additional growth patterns.(1) If σ , , σ , lie in the region between the blue and green urves, e ap-proa hes zero during the (cid:28)rst exponential growth phase. Importantly,sin e σ , is higher than the threshold level, E is synthesized upon ex-haustion of S at the end of the (cid:28)rst exponential growth phase. Hen e,growth is diauxi with preferential onsumption of S .(2) If σ , , σ , lie in the region between the red and bla k urves, e ap-proa hes zero during the (cid:28)rst exponential growth phase, i.e., there isdiauxi growth with preferential onsumption of S .(3) If σ , , σ , lie between the red and blue urves, e and e attain posi-tive values during the (cid:28)rst exponential growth phase. Consequently, bothsubstrates are onsumed until one of them is exhausted, and there is nodiauxi lag before the remaining substrate is onsumed.We show below that the foregoing lassi(cid:28) ation is onsistent with the data.3.3.1.2 Growth patterns at saturating substrate on entrationsPhysiologi al experiments, whi h are generally performed with saturating sub-strate on entrations ( σ , , σ , ≈ ), show that depending on the ell type,26 σ e e σ * e e σ * e e e e e e e e σ e e e σ * e e σ * e e e e e e σσ e Figure 10. Bifur ation diagrams orresponding to diauxi growth under substrate-ex- ess onditions ( σ , , σ , ≈ ). (a) Preferential onsumption of S . (b) Preferential onsumption of S .the two substrates are onsumed sequentially or simultaneously. The behav-ior of the model is onsistent with this observation. To see this, observe thatdepending on the values of the physiologi al parameters (whi h, in e(cid:27)e t, de-termine the ell type), there are three possible arrangements of the red andblue urves, ea h of whi h yields a distin t growth pattern at saturating sub-strate on entrations. Indeed, S is onsumed preferentially if (1 , lies belowthe blue urve (Fig. 10a). This o urs pre isely when the physiologi al param-eters satisfy the ondition V e, ¯ K e, > V e, ¯ K e, + 1 Y V s, ¯ K e, V e, ¯ K e, ! , (40)obtained from (38) by negle ting enzyme degradation ( k e,i = 0 ). Likewise, S is onsumed preferentially if (1 , lies above the red urve (Fig. 10b), i.e., thephysiologi al parameters satisfy the ondition V e, ¯ K e, > V e, ¯ K e, + 1 Y V s, ¯ K e, V e, ¯ K e, ! , (41)obtained from (39) upon setting k e,i = 0 . Finally, the substrates are onsumedsimultaneously if (1,1) lies above the blue urve and below the red urve(Fig. 9), i.e., both the foregoing onditions are violated.The biologi al meaning of these onditions was dis ussed in earlier work (Narang and Pilyugin,2007a). Indeed, one an he k that (40) and (41) are equivalent to the ondi-27ions α < α ∗ ≡ κ (cid:18) − κ + q κ + 4 (cid:19) ,α > α ∗ ≡ κ (cid:18) − κ + q κ + 4 (cid:19) , respe tively, where α ≡ q Y V s, V e, / q Y V s, V e, is a measure of the maximumspe i(cid:28) growth rate on S relative to that on S , and κ i ≡ ¯ K e,i / q V e,i / ( Y i V s,i ) is a measure of saturation onstant for indu tion of E i . Thus, the substratesare onsumed preferentially whenever the maximum spe i(cid:28) growth rate onone of the substrates is su(cid:30) iently large ompared to that on the other sub-strate. Just how large this ratio must be depends on the saturation onstants, κ i . Enzymes with small saturation onstants are quasi- onstitutive (cid:22) theirsynthesis annot be abolished even if the other substrate supports a largespe i(cid:28) growth rate. In ontrast, synthesis of enzymes with large saturation onstants an be abolished even if the other substrate supports a omparablespe i(cid:28) growth rate.3.3.1.3 Transitions triggered by hanges in substrate on entra-tions Figs. 9(cid:21)10 imply that the growth pattern an be hanged by alteringthe initial substrate on entrations. We show below that this on lusion is onsistent with the data.As a (cid:28)rst example, onsider a ell type that prefers to onsume S undersubstrate-ex ess onditions ( σ , , σ , ≈ ). Its growth pattern at any σ , , σ , is then des ribed by Fig. 10a, whi h implies that if σ , is de reased su(cid:30)- iently from 1, both substrates will be onsumed simultaneously. Su h behav-ior has been observed in experiments. If the initial on entrations of glu oseand gala tose are 20 mg/L and 5 mg/L, respe tively, E. oli ML308 onsumesglu ose before gala tose (Fig. 11a). If the initial on entration of glu ose is re-du ed to 5 mg/L or lower, glu ose and gala tose are onsumed simultaneously(Fig. 11b).The data in Figs. 11a,b are open to question be ause the initial ell densitiesare so large ( ∼ ells L − ) that the substrates may have been exhaustedbefore the enzymes rea hed the quasisteady state orresponding to balan edgrowth. However, similar results have also been obtained in studies performedwith very small initial ell densities ( (cid:21) ells L − ). Indeed, S hmidt andAlexander observed that (cid:16)simultaneous use of two substrates is on entrationdependent . . . Pseudomonas sp. ANL 50 mineralized glu ose and aniline simul-taneously when present at 3 µ g L − but metabolized them diauxi ally at 300 µ g L − (cid:17) (S hmidt and Alexander, 1985, Figs. 3(cid:21)4). Likewise, Pseudomonas28a) (b)( ) (d)Figure 11. Transitions of the growth patterns triggered by hanges in the initialsubstrate on entrations. Upper panel: Bat h growth of Es heri hia oli ML308on mixtures of glu ose and gala tose. (a) At high initial glu ose on entrations (20mg L − ), glu ose is onsumed before gala tose (Lendenmann, 1994). (b) At low ini-tial glu ose on entrations (5 mg L − ), glu ose and gala tose are onsumed simulta-neously (Egli et al., 1993, Fig. 1). Lower panel: Bat h growth of Pseudomonas onPNP and PNP + glu ose (S hmidt et al., 1987, Figs. 1,4). DPM denotes disintegra-tions per min of C-labelled PNP and glu ose. ( ) No PNP is onsumed if its initial on entration is 10 µ g L − . Signi(cid:28) ant onsumption o urs at initial on entrationsof 50 and 100 µ g L − . (d) PNP and glu ose are onsumed simultaneously if theirinitial on entrations are high (3 and 10 mg L − , respe tively).a idovorans onsumed a etate before phenol when the on entrations of thesesubstrates were >70 and 2 µ g L − , respe tively. If the initial on entration ofa etate was redu ed to 13 µ g L − , both substrates were onsumed simultane-ously.In terms of the model, these transitions from sequential to simultaneous sub-strate onsumption an be understood as follows. Sequential substrate on-sumption o urs at high on entrations of the preferred substrate be ause itsupports su h a large spe i(cid:28) growth rate that the enzymes asso iated withthe se ondary substrate are diluted to extin tion. Redu ing the initial on- entration of the preferred substrate de reases its ability to support growth,and thus dilute the se ondary substrate enzymes to extin tion. Consequently,both substrates are onsumed. 29he growth of Pseudomonas sp. on p-nitrophenol (PNP) and glu ose furnishesadditional examples of growth pattern transitions driven by hanges in the ini-tial substrate on entrations. S hmidt et al. found that no PNP was onsumedif its initial on entration was 10 µ g L − , and signi(cid:28) ant mineralization o - urred at the higher initial on entrations of 50 and 100 µ g L − (Fig. 11 ).It follows that the threshold on entration for PNP lies between 10 and 50 µ g L − . Importantly, the addition of 20 mg L − glu ose to a ulture ontaining10 µ g L − PNP failed to stimulate PNP onsumption. The authors on ludedthat (cid:16)the threshold was not a result of the fa t that the on entration of PNPwas too low to meet maintenan e energy requirements of the organism, butrather supports the hypothesis that the on entration was too low to indu edegradative enzymes.(cid:17) Finally, it was observed that when the initial on en-trations of PNP and glu ose were in reased to saturating levels of 3 mg L − and 10 mg L − , respe tively, both substrates were onsumed simultaneously(Fig. 11d). All these results are onsistent with the bifur ation diagram shownin Fig. 9.Figs. 9(cid:21)10 extend the results obtained in Narang and Pilyugin, 2007a. There,we onstru ted bifur ation diagrams des ribing the variation of the growthpatterns in response to hanges in the physiologi al parameters at ((cid:28)xed) satu-rating substrate on entrations. These geneti bifur ation diagrams explainedthe phenotypes of many di(cid:27)erent mutants and re ombinant ells. Figs. 9(cid:21)10 an be viewed as epigeneti bifur ation diagrams, sin e they des ribe the vari-ation of the growth pattern when a given ell type, hara terized by (cid:28)xedphysiologi al parameters, is exposed to various initial substrate on entra-tions.3.3.2 Continuous ulturesIn the experimental literature, the growth pattern of mixed-substrate ontin-uous ultures is lassi(cid:28)ed based on the manner in whi h the two substratesare onsumed when the dilution rate is in reased at a given pair of feed on- entrations. Either both substrates are onsumed at all dilution rates up towashout, or both substrates are onsumed up to an intermediate dilution ratebeyond whi h only one of the substrates is onsumed. We show below that themodel predi ts these growth patterns. Moreover, it an quantitatively apturethe variation of the steady state enzyme levels, substrate on entrations, and ell density with respe t to the dilution rate and the feed on entrations.3.3.2.1 Mathemati al lassi(cid:28) ation of the growth patterns Themathemati al orrelate of the experimentally observed growth patterns or-responds to the bifur ation diagram for eqs. (8)(cid:21)(12), whi h des ribes the30able 1Ne essary and su(cid:30) ient onditions for existen e and stability of the steady statesfor mixed-substrate growth in ontinuous ultures (see Appendix C for details).Steady De(cid:28)ning Biologi al Existen e Stabilitystate property meaning ondition(s) ondition(s) E e = 0 , e = 0 , No substrate Always D t, < ,c = 0 onsumed exists D t, < E e > , e = 0 , Only S D t, > , D > D t, c > onsumed < D < D c, E e > , e = 0 , Washout D t, > D t, < D c, ,c = 0 of E D > D c, E e = 0 , e > Only S D t, > , D > D t, c > onsumed < D < D c, E e = 0 , e > , Washout D t, > D t, < D c, ,c = 0 of E D > D c, E e > , e > , Both S , S D t, , D t, > , Whenever it c > onsumed D < D t, , D t, , D c exists E e > , e > , Washout D t, > D c, , D > D c c = 0 of E D t, > D c, existen e and stability of the steady states in the 3-dimensional σ f, , σ f, , D -spa e.Eqs. (8)(cid:21)(12) have 7 steady states, but 5 of them are (cid:16)single-substrate steadystates(cid:17) (see rows 1(cid:21)5 of Table 1). Indeed, E , E , and E (resp., E , E , and E ) are observed in hemostats limited by S (resp., S ). Theirexisten e in mixed-substrate growth implies that even if both substrates aresupplied to the hemostat, there are steady states at whi h only one of thesubstrates is onsumed. We shall see below that these steady states an be omestable under ertain onditions. Here, it su(cid:30) es to observe that only two of 7steady states, namely, E ( e > , e > , c > ) and E ( e > , e > , c > ), are uniquely asso iated with mixed-substrate growth. They orrespondto simultaneous onsumption of both substrates and its washout.The existen e and stability of the steady states are ompletely determinedby (cid:28)ve spe ial dilution rates, namely, D t,i , D c,i , and D c (Table 1, olumns 4(cid:21)5). These dilution rates are the analogs of the spe ial spe i(cid:28) growth rates onsidered in the analysis of bat h ultures. Indeed, the transition dilution31ate for S i , D t,i ≡ V e,i σ f,i ¯ K e,i − k e,i , (42)is the analog of the extin tion growth rate, r ∗ g,i . It is the maximum dilutionrate up to whi h synthesis of E i an be sustained. The riti al dilution rate, D c,i , given by the relation D c,i = Y i V s,i − (cid:16) ¯ K e,i + k e,i Y i V s,i (cid:17) + r(cid:16) ¯ K e,i + k e,i Y i V s,i (cid:17) + K e,i D t,i Y V s,i , is the analog of r g,i . It is the maximum dilution rate up to whi h growth anbe sustained in a hemostat supplied with only S i , and satis(cid:28)es the relation, D ci ≤ D t,i with equality being obtained pre isely when σ i = σ ∗ i , in whi h ase, D c,i = D t,i = 0 (see dashed and blue lines in Fig. 7). Finally, the riti aldilution rate, D c , is de(cid:28)ned as D c ≡ Y V s, e σ f, + Y V s, e σ f, , where e , e are the unique positive steady state solutions of (30)(cid:21)(31) with σ ,i repla ed by σ f,i . It is the analog of the fun tion, r g, . Thus, r ∗ g,i , r g,i , r g, ,and their analogs, D t,i , D c,i , D c , are de(cid:28)ned by formally identi al expressions,the only di(cid:27)eren e being that σ ,i is repla ed by σ f,i .The formal similarity of the expressions for D t,i , D c,i , D c and r ∗ g,i , r g,i , r g, implies that for a given set of physiologi al parameters, the relations, σ f,i = σ ∗ i , D t, = D c, and D t, = D c, , de(cid:28)ne urves on the σ f, , σ f, -plane thatare identi al to the urves on the σ , , σ , -plane de(cid:28)ned by the equations, σ ,i = σ ∗ i , r ∗ g, = r g, and r ∗ g, = r g, . It follows that the relations, σ f,i = σ ∗ i , D t, = D c, , D t, = D c, , determine a partitioning of the σ f, , σ f, -plane into6 regions, whi h is identi al to the partitioning of the bifur ation diagram forbat h ultures, the only di(cid:27)eren e being that the oordinates are σ f, , σ f, ,rather than σ , , σ , (Fig. 12).Ea h of the six regions in Fig. 12 orresponds to unique growth pattern that an be inferred from the existen e and stability properties summarized in olumns 4(cid:21)5 of Table 1. For instan e, if ( σ f, , σ f, ) lies in the region betweenthe green and blue urves in Fig. 12, both substrates are onsumed for all < D < D t, , only S is onsumed for all D t, ≤ D < D c, , and neithersubstrate is onsumed be ause the ells are washed out for all D ≥ D c, . Tosee this, observe that in the region of interest (between the green and blue urves), the spe ial dilution rates stand in the relation < D c, < D t, < D c < D c, < D t, . The existen e and stability onditions listed in Table 1 therefore imply thatas D in reases, 4 of the 7 steady states play no role be ause they are unstable32 DD c,2t,1 D < D c,1 ,, , D t,1c,1 D < < c,2 D D t,2 , < t,1 DD c D c,1 < t,2 DD c,2 , < < < s D t,2 c,1 D Dc s D t,21 s c,2 D c DD c,2c,1 D <<,< , D c,1 < t,1 D < D c D c,2 < t,2 D < < f,2 σ σ f,1 D e D e s s D c,2 D c D c,22 ee D s s c c,1 D D D c1 e e e c D e e e c D e D c 2 s D D c,12 ee Ds s s s Dc D t,1 c,2 D ee D s D c,2t,1 D c D s σσ DDDDD D c cc c D De * t,2 Figure 12. Bifur ation diagram lassifying the growth patterns of mixed-substrate ontinuous ultures at various feed on entrations, σ f, and σ f, . The green and bla klines represent the threshold on entrations for indu tion of E and E , respe tively.The blue and red urves represent the lo us of initial substrate on entrations de(cid:28)nedby the equations, D c, = D t, and D c, = D t, , respe tively. The insets in ea h ofthe six regions show the variation of the steady states with D at the orrespondingfeed on entrations.whenever they exist, or do not exist at all. Indeed:(1) E and E exist at all D > , but they are always unstable be ausethe stability onditions for these two steady states ( D t, , D t, > and D t, < D c, , respe tively) are violated in the region of interest .(2) E exists for all < D < D c, , but it is unstable whenever it existsbe ause the stability ondition, D > D t, , is not satis(cid:28)ed in the region ofinterest.(3) E does not exist at all sin e one of the existen e onditions, D t, > D c, ,is violated in the region of interest.It follows that only the remaining three steady states are relevant. One an he k by appealing to Table 1 that E is stable for < D < D t, , E isstable for D t, < D < D c, , and E is stable for D > D c, . Hen e, both33ubstrates are onsumed for all < D < D t, . At D = D t, , e be omes zero.Only S is onsumed for all D t, ≤ D < D c, until c be omes zero at D = D c, .Neither substrate is onsumed for D > D c, .Similar arguments, applied to all the regions in Fig. 12, show that there are 6distin t growth patterns. Three of these growth patterns o ur when at leastone of the feed on entrations is below its threshold level.(1) If both feed on entrations are below the threshold level, E is alwaysstable, so that neither substrate is onsumed at any dilution rate.(2) If σ f, < σ ∗ , but σ f, > σ ∗ , E is stable (i.e., only S is onsumed) forall < D < D c, , and E is stable (i.e., the ells wash out) whenever D ≥ D c, . This is qualitatively similar to the behavior observed duringsingle-substrate growth on S (Fig. 7). Thus, if the feed on entrationof S is at sub-threshold levels, it has no e(cid:27)e t on the behavior of the hemostat.(3) If σ f, < σ ∗ , and σ f, > σ , E is stable (i.e., only S is onsumed)for all < D < D c, , and E is stable (i.e., the ells wash out) for D ≥ D c, . This is qualitatively similar to the behavior of the hemostatduring single-substrate growth on S .If both feed on entrations are above their respe tive threshold levels, thereare three additional growth patterns.(1) If σ f, and σ f, lie between the red and blue urves, E (i.e., both sub-strates are onsumed) for < D < D c , and E is stable (i.e., ells washout) for D ≥ D c . Thus, both substrates are onsumed at all dilution ratesup to washout.(2) If σ f, and σ f, lie between the green and blue urves, E is stable (i.e.,both substrates are onsumed) for all < D < D t, , E is stable (i.e.,only S is onsumed) for all D t, ≤ D < D c, , and E is stable (the ellswash out) for all D > D c, .(3) If σ f, and σ f, lie between the bla k and red urves, E is stable (i.e.,both substrates are onsumed) for all < D < D t, , E is stable (i.e.,only S is onsumed) for all D t, ≤ D < D c, , and E is stable (the ellswash out) for all D > D c, .These growth patterns are the mathemati al orrelates of the growth patternsshown in Fig. 3.The model predi ts that the growth pattern in ontinuous ultures an be hanged by altering the feed on entrations. Unfortunately, there is no experi-mental data at subsaturating feed on entrations ( σ f,i . K s,i ) due to te hni aldi(cid:30) ulties asso iated with wall growth. Hen eforth, we shall on(cid:28)ne our at-tention to growth at saturating feed on entrations ( σ f,i ≈ ).34e have shown above that for a given ell type, the bifur ation diagrams forbat h and ontinuous ultures are formally identi al (cid:22) one an be generatedfrom the other by a mere relabeling of the oordinate axes. This formal iden-tity provides a pre ise mathemati al explanation for the empiri ally observed orrelation between the bat h and ontinuous growth patterns of a given elltype. To see this, suppose the ells in question onsume S preferentially insubstrate-ex ess bat h ultures ( σ ,i ≈ ). The bifur ation diagram for thissystem then has the form shown in Fig. 10a. The ontinuous growth patternsof this system are given by the very same (cid:28)gure, the only di(cid:27)eren e being thatthe oordinates, σ , , σ , , are repla ed by σ f, , σ f, , respe tively. It followsthat if the ells are grown in ontinuous ultures fed with saturating substrate on entrations ( σ f,i ≈ ), they will onsume both substrates at low dilutionrates, and only S at all high dilution rates up to washout (Fig. 3b). A similarargument shows that if the ells onsume both substrates in substrate-ex essbat h ulture (bifur ation diagram given by a relabeled Fig. 9), their growthin ontinuous ultures will be su h that both substrates are onsumed at alldilution rates up to washout (Fig. 3a).3.3.2.2 The model predi ts the observed variation of the steadystates with D and s f,i Although the model predi ts the existen e of theobserved growth patterns, it remains to determine if the variation of the steadystates with D and s f,i is in quantitative agreement with the data. Sin e thevariation of the (cid:16)single-substrate steady states(cid:17) with D and s f,i is onsistentwith the data (Se tion 3.2), it su(cid:30) es to fo us on E , the only growth-supporting steady state uniquely asso iated with mixed-substrate growth.This steady state an be observed in systems exhibiting both the simultaneousand the preferential growth patterns. However, sin e the data for simultaneoussystems is limited, we shall fo us on preferential systems. In our dis ussion ofthese systems, we shall assume, furthermore, that S is the (cid:16)preferred(cid:17) sub-strate. This entails no loss of generality sin e the equations for S and S (andthe asso iated physiologi al variables) are formally identi al: Inter hangingthe indi es does not hange the form of the equations.Two types of experiments an be found in the literature.(1) Either the dilution rate was hanged at (cid:28)xed feed on entrations.(2) Alternatively, the mass fra tion of the substrates in the feed, ψ i ≡ s f,i / ( s f, + s f, ) , was hanged at (cid:28)xed total feed on entration, s f,t ≡ s f, + s f, , and(su(cid:30) iently small) dilution rate.The model predi ts that in the (cid:28)rst ase, E is stable at all dilution ratesbelow D t, , the transition dilution rate for the (cid:16)less preferred(cid:17) substrate. In these ond ase, E is stable at all feed fra tions, ex ept those near the extremevalues, 0 and 1, where the model predi ts threshold e(cid:27)e ts. Su h threshold ef-35e ts have been observed when the feed ontained a very small fra tion of one ofthe substrates (Kovar et al., 2002; Ng and Dawes, 1973; Rudolph and Grady,2002). Nevertheless, we shall restri t our attention to intermediate values of ψ i at whi h E is the stable steady state.The steady state, E , satis(cid:28)es the equations D ( s f,i − s i ) − r s,i c, (43) V e,i σ i ¯ K e,i + e i σ i − ( D + k e,i ) , (44) Y r s, + Y r s, − D, (45) x i ≈ r s,i k x,i . (46)These equations do not yield an analyti al solution valid for all D and s f,i .However, we show below that they an be solved for the two limiting ases oflow and high D , from whi h the entire solution an be pie ed together.Before onsidering these limiting ases, it is useful to note that eqs. (43) and(45) imply that c = Y ( s f, − s ) + Y ( s f, − s ) , (47) Y i r s,i D = β i ≡ Y i ( s f,i − s i ) Y ( s f, − s ) + Y ( s f, − s ) . (48)The (cid:28)rst relation says that the steady state ell density is the sum of the ell densities derived from the two substrates. The se ond relation states that β i , the fra tion of biomass derived from S i , equals the fra tion of the totalspe i(cid:28) growth rate, D , supported by S i . Evidently, β i ≤ with equalitybeing attained only during single-substrate growth.Eq. (48) immediately yields r s,i = β i DY i , (49)whi h explains the following well-known empiri al observation: At any given D , the mixed-substrate spe i(cid:28) uptake rate of S i is always lower than thesingle-substrate spe i(cid:28) uptake rate of S i (reviewed in Egli, 1995). This or-relation is a natural onsequen e of the mass balan es for the substrates and ells, together with the onstant yield property. The mixed-substrate spe i(cid:28) uptake rate is smaller simply be ause both substrates are onsumed to sup-port the spe i(cid:28) growth rate, D , whereas only one substrate supports the verysame spe i(cid:28) growth rate during single-substrate growth.3.3.2.3 Low dilution rates The (cid:28)rst limiting ase orresponds to di-lution rates so small that both substrates are at subsaturating levels, i.e.,36a) (b)( ) (d)Figure 13. Upper panel: The spe i(cid:28) substrate uptake rate in reases linearly whenthe dilution rate is in reased at (cid:28)xed feed on entrations. Spe i(cid:28) uptake rates of(a) glu ose and (b) methanol during growth of H. polymorpha on various mixtures ofglu ose and methanol (Egli et al., 1986, Figs. 4(cid:21)5). The per entages show the per entmethanol in the feed. The total feed on entration of glu ose and methanol was 5g L − . Lower panel: If the fra tion of S i in the feed is in reased at a (cid:28)xed dilutionrate, the spe i(cid:28) substrate uptake rate in reases linearly with β i . ( ) Growth of H.polymorpha on a mixture of glu ose and methanol at D = 0 . h − (Egli et al.,1982b, Fig. 5). (d) Growth of E. oli K12 on various mixtures of la tose and glu oseat D = 0 . h − ( al ulated from Fig. 4 of Smith and Atkinson, 1980). The linesin the (cid:28)gures show the spe i(cid:28) substrate uptake rates predi ted by eq. (51) withthe experimentally measured single-substrate yields, Y GLU = 0 . , Y MET = 0 . in(a)(cid:21)( ), and Y GLU = Y LAC in (d). s i . K s,i ≪ s f,i . Sin e both substrates are almost ompletely onsumed, (47)(cid:21)(48) be ome c ≈ Y s f, + Y s f, , (50) r s,i = β i DY i , β i ≈ Y i s f,i Y s f, + Y s f, = Y i ψ i Y ψ + Y ψ . (51)Evidently, β i is ompletely determined by ψ i , the fra tion of S i in the feed.Moreover, β i ( ψ i ) is an in reasing fun tion of ψ i with β i (0) = 0 and β i (1) = 1 .It is identi al to ψ i when the yields on both substrates are the same.Eq. (51) implies that if D is in reased at (cid:28)xed feed on entrations, r s,i in reases37a) (b)( ) (d)Figure 14. Variation of the peripheral enzyme levels: (a) A tivity of β -gala tosidaseduring growth of E. oli K12 on mixtures of la tose (1mM) plus glu ose(2mM), glu ose-6-phosphate (2mM), or gly erol (4mM) (Smith and Atkinson, 1980,Fig. 3). (b) A tivity of gentisate-1,2-dioxygenase and glu ose-6-phosphate dehy-drogenase (G6PDH) during growth of P. aeruginosa on sali ylate + glu ose at D = 0 . h − (Rudolph and Grady, 2002, Fig. 3). ( ) A tivities of al ohol oxi-dase and formaldehyde dehydrogenase during growth of H. polymorpha on variousmixtures of glu ose and methanol at D = 0 . h − and s f,t = 5 g L − (Egli et al.,1982b, Figs. 7A,C). (d) A tivity of β -gala tosidase during growth of E. oli K12 onvarious mixtures of la tose and glu ose at D = 0 . h − ( al ulated from Fig 5 ofSmith and Atkinson, 1980).linearly. Likewise, if ψ i , and hen e, β i , is in reased at a (cid:28)xed dilution rate, r s,i in reases linearly with β i . Both on lusions are onsistent with the data.Fig. 13 shows that eq. (51) provides ex ellent (cid:28)ts to the data for growth ofH. polymorpha on glu ose + methanol and E. oli on glu ose + la tose. Thevalidity of (51) at low dilution rates has been demonstrated for many di(cid:27)erentsystems (reviewed in Kovarova-Kovar and Egli, 1998).The physiologi al variables, x i and e i , are ompletely determined by the spe-38i(cid:28) uptake rate of S i . Indeed, (46) and (44) imply that x i ≈ r s,i k x,i = β i DY i k x,i , (52) e i = V e,i D + k e,i β i DY i V s,i ¯ K e,i + β i D . (53)It follows from (53) that:(1) If the dilution rate is in reased at (cid:28)xed feed on entrations, the steadystate enzyme level passes through a maximum. However, at every dilutionrate, the mixed-substrate enzyme level is always lower than the single-substrate enzyme level.(2) If the fra tion of S i in the feed is in reased at a (cid:28)xed dilution rate, thea tivity of E i in reases with β i . The in rease is linear if D is small, andhyperboli if D is large.Both results are onsistent with the data (Fig. 14).Based on studies with bat h ultures of E. oli, it is widely believed thatglu ose and glu ose-6-phosphate (G6P) are strong inhibitors of β -gala tosidasesynthesis, whereas gly erol is a weak inhibitor. It is therefore remarkable thatthe β -gala tosidase a tivity at any given dilution rate is the same in threedi(cid:27)erent ontinuous ultures fed with la tose + glu ose, la tose + G6P, andla tose + gly erol (Fig. 14a). The model provides a simple explanation forthis data. To see this, let S and S denote la tose and glu ose/G6P/gly erol,respe tively. Now, (53) implies that any given dilution rate, the in(cid:29)uen e of S on the β -gala tosidase a tivity, e , is ompletely determined by the parameter, β ≈ Y s f, Y s f, + Y s f, , whi h is identi al to ψ , the mass fra tion of la tose in the feed, be ause theyields on la tose, glu ose, G6P, and gly erol are similar. It turns out thatthe feed on entrations used in the experiments were su h that ψ ≈ . inall the experiments. Consequently, the β -gala tosidase a tivity at any D isindependent of the hemi al identity of S .The steady state substrate on entrations are given by the expression σ i = r s,i V s,i e i = 1 Y i V s,i V e,i ( D + k e,i ) (cid:16) β i D + Y i V s,i ¯ K e,i (cid:17) , (54)whi h follows immediately from (51) and (53). Evidently, if the dilution rate isin reased at (cid:28)xed feed on entrations, the substrate on entrations in rease.However, omparison with (27) shows that at every dilution rate, the sub-strate on entrations during mixed-substrate growth are lower than the or-39a) (b)( ) (d)Figure 15. Upper panel: Variation of the glu ose and gala tose on entrations withthe dilution rate during single- and mixed-substrate growth of E. oli ML308 on glu- ose or/and gala tose (Lendenmann, 1994; Lendenmann et al., 1996, Appendix B).The residual on entrations of (a) glu ose and (b) gala tose during mixed-substrategrowth ( N ) are lower than the orresponding levels during single-substrate growth( △ ). Lower panel: The relationship between the substrate on entration and theperipheral enzyme level. (a) Growth of C. boidinii on various mixtures of methanoland glu ose at D = 0 . h − and s f,t = 5 . g L − (Egli et al., 1993, Fig. 7). Whenthe a tivity of al ohol oxidase in reases ( ≤ β MET . . ), the methanol on entra-tion is onstant. When the a tivity of al ohol oxidase saturates ( β MET & . ), themethanol on entration in reases. (b) Residual on entrations of gala tose (opensymbols) and glu ose ( losed symbols) during growth on E. oli ML308 on variousmixtures of gala tose and glu ose at D = 0 . h − and s f,t = 1 ( △ , N ), 10 ( ♦ , (cid:7) ), and100 ( (cid:3) ) mg L − (Lendenmann, 1994; Lendenmann et al., 1996).responding levels during single-substrate growth. This agrees with the data(Figs. 15a,b). Eq. (54) also implies that if the fra tion of S i in the feed, andhen e, β i , is in reased at a (cid:28)xed dilution rate, the substrate on entrationis onstant when β i is small, and in reases when β i is large. This o urs be- ause at small β i , the linear in rease of r s,i is driven entirely by the linearin rease of e i . Sin e the enzyme level saturates at su(cid:30) iently large β i , furtherimprovement of r s,i is obtained by a orresponding in rease of the substrate on entration. Fig 15 shows that this on lusion is onsistent with the data.To be sure, there are instan es in whi h the substrate on entrations appearto in rease for all ≤ β i ≤ (Fig. 15d). The model implies that this o ursbe ause Y i V s,i ¯ K e,i is so small that the peripheral enzymes are saturated at40elatively small values of β i D .In general, one expe ts the mixed-substrate steady states to depend on threeindependent parameters, namely, D , σ f, , and σ f, . However, the model im-plies that at su(cid:30) iently small dilution rates (su h that s i ≪ s f,i ), the steadystate spe i(cid:28) uptake rates, and hen e, the steady state enzyme levels andsubstrate on entrations, are ompletely determined by only two indepen-dent parameters, D and β i ( ψ i ) . Experiments provide dire t eviden e support-ing this on lusion. Indeed, the data in Fig. 15d were obtained by perform-ing experiments with three di(cid:27)erent values of the total feed on entration( s f,t = 1 , , mg/L). It was observed that at any given fra tion of gala -tose in the feed, the residual sugar on entrations were the same, regardless ofthe total feed on entration. Thus, the residual substrate on entrations are ompletely determined by the dilution rate and the fra tion of the substratesin the feed (rather than the absolute values of the feed on entrations). Thesame is true of the enzyme levels (Fig. 14b). Athough three di(cid:27)erent substrates(glu ose, G6P, gly erol) were used in the experiments, the β -gala tosidase a -tivity at any given D was the same regardless of the identity of the substrate,sin e the fra tion of la tose in the feed was identi al ( ∼ s approa hes saturatinglevels ( s ≫ K s, ), while s remains at subsaturating levels ( s . K s, ). Thedilution rate at whi h s swit hes from subsaturating to saturating levels,denoted D s, , an be roughly estimated from the relation ≈ Y V s, V e, ( D s, + k e, ) (cid:16) D s, β + Y V s, ¯ K e, (cid:17) , (55)obtained by letting σ ≈ in eq. (54). At dilution rates ex eeding D s, , growthis limited by S only. Based on our earlier analysis of single-substrate growth,we expe t that the steady state on entrations of S and the physiologi alvariables are ompletely determined by D , and the ell density in reases lin-early with the feed on entration of S (Fig. 8). We show below that this isindeed the ase.When s ≫ K s, , eqs. (46) and (44) yield x = r s, k x, ≈ V s, e k x, ,e = V e, D + k e, − ¯ K e, σ ≈ ¯ K e, D + k e, ( D t, − D ) , (56)41a) H h - L r s,2 H ggdw - h - L (b)( ) (d)Figure 16. Near the transition dilution rate, the spe i(cid:28) substrate uptake rates areindependent of the feed omposition. Upper panel: Model simulations. The arrowspoint in the dire tion of in reasing S in the feed. The full lines show the spe i(cid:28) uptake rates predi ted by numeri al simulations of the model. The dashed linesshow the spe i(cid:28) uptake rates predi ted by eqs. (51) and (57)(cid:21)(58). Lower panel:Spe i(cid:28) uptake rates of glu ose and methanol during mixed-substrate growth of H.polymorpha (Egli et al., 1986, Figs. 4(cid:21)5).where the se ond relation was obtained by appealing to the approximations, σ , σ f, ≈ . Evidently, r s, ≈ V s, e (57)is a fun tion of D . Sin e Y r s, + Y r s, = D , we on lude that r s, = D − Y r s, Y , (58)and x = r s, k x, , e = V e, D + k e, x K e, + x , (59)are also fun tions of D .Sin e r s, and r s, are ompletely determined by D , the urves representingthe spe i(cid:28) substrate uptake rates at various feed ompositions must ollapseinto a single urve. This on lusion is onsistent with the data for growth of H.polymorpha on glu ose + methanol (Figs. 16 ,d). Indeed, simulations of the42able 2Parameter values used to simulate the growth of H. polymorpha on mixtures ofglu ose ( S ) and methanol ( S ). The yields are based on experimental measure-ments (Egli et al, 1986). The remaining parameters, whi h have not been mea-sured experimentally, are based on order-of-magnitude estimates obtained from well- hara terized systems, su h as the la operon (Shoemaker et al, 2003, Appendix A) i Y i V s,i (h − ) K s,i (g L − ) k x,i (h − ) V e,i (h − ) K e,i k e,i (h − )1 0.55 − × − × − − × − × − D & . h − ( urves labeled (cid:3) and ♦ in Fig. 16 ).Likewise, the single-substrate spe i(cid:28) uptake rate of methanol deviates fromthe model predi tion when D & . h − ( urve labeled △ in Fig. 16d). Underthese ex eptional onditions, the residual methanol on entrations are so highthat toxi e(cid:27)e ts arise, thus redu ing the yields on glu ose and methanol. Theobserved spe i(cid:28) uptake rates are therefore higher than the predi ted rates.Eqs. (56)(cid:21)(57) imply that at su(cid:30) iently high dilution rates, e and r s, de- rease with D until they be ome zero at D = D t, (Fig. 17b). Consequently, r s, and e in rease with D to ensure that the total spe i(cid:28) growth rate re-mains equal to D (Fig. 17a). Both on lusions are onsistent with the data forgrowth of H. polymorpha on glu ose + methanol. The a tivities of the glu oseenzymes, hexokinase and 6-phosphoglu onate dehydrogenase, in rease with D (Fig. 17 ), while the a tivity of the methanol enzyme, formaldehyde dehydro-genase, de reases with D (Fig. 17d).Sin e r s, and e are fun tions of D , so is σ = r s, V s, e . (60)In fa t, the only variables that depend on the feed on entrations are the elldensity and the on entration of S . To see this, it su(cid:30) es to observe thateq. (43) yields c = D ( s f, − s ) r s, ≈ Dr s, ! s f, , (61) s f, − s = r s, cD ≈ r s, r s, ! s f, , (62)where the terms in parentheses are fun tions of D . As expe ted, c in reaseslinearly with s f, be ause growth is limited by S only. Fig. 18 shows thatnumeri al simulations of the model with the parameter values in Table 2 are43a) (b)( ) (d)Figure 17. Variation of the peripheral enzyme levels for glu ose and methanol. Up-per panel: Model simulations. The arrows point in the dire tion of in reasing S inthe feed. The full lines show the spe i(cid:28) uptake rates predi ted by numeri al simu-lations of the model. The dashed lines show the enzyme levels predi ted by eqs. (53)and (56), (59). Lower panel: A tivities of ( ) the glu ose enzymes, hexokinaseand 6-phoshoglu onate dehydrogenase (6PGDH), and (d) the methanol enzyme,formaldehyde dehydrogenase, during mixed-substrate growth of H. polymorpha on3 g L − glu ose and 2 g L − methanol (Egli et al., 1982a, Fig. 4).in good agreement with the data for growth of H. polymorpha on glu ose +methanol (Fig. 4).3.3.2.5 The pat hed limiting solutions approximate the exa t so-lution Taken together, the two approximate solutions orresponding to lowand high dilution rates oin ide with the (exa t) numeri al solution at all dilu-tion rates, ex ept in a small neighborhood of the dilution rate at whi h the twoapproximate solutions interse t ( ompare the dashed and full lines in Figs. 16(cid:21)18). This remarkable agreement between the approximate and exa t solutionobtains be ause as D in reases, s hanges from subsaturating to saturatinglevels so rapidly that the hange is, for all pra ti al purposes, dis ontinuous.The swit hing dilution rate, D s, , provides a good approximation to the dilu-tion rate at whi h this near-dis ontinuous hange o urs. The model thereforeprovides expli it formulas that approximate the spe i(cid:28) uptake rates, sub-strate on entrations, ell density, and enzyme levels at all dilution rates and44a) (b)Figure 18. The model simulations are in good agreement with the data for growthof H. polymorpha on glu ose + methanol shown in Fig. 4. The arrows point in thedire tion of in reasing S in the feed. The full lines show the substrate on entrationsand ell density predi ted by numeri al simulations of the model. The dashed linesshow the substrate on entrations and ell density predi ted by eqs. (50), (54) and(60)(cid:21)(62).saturating feed on entrations. The approximate solutions orresponding tolow and high dilution rates are valid for D < D s, and D > D s, , respe tively.3.3.2.6 Egli's transition dilution rate di(cid:27)ers from our transitiondilution rate There is an important di(cid:27)eren e between the transition di-lution rates de(cid:28)ned by Egli and us. In our model, D t, is the dilution rateat whi h e , the a tivity of methanol enzymes be omes zero. This transitiondilution rate, de(cid:28)ned by eq. (42), in reases with the feed on entration ofmethanol. However, at the saturating feed on entrations used in the exper-iments, it is essentially independent of the feed on entration. On the otherhand, Egli de(cid:28)ned the transition dilution rate empiri ally as the dilution rateat whi h the residual methanol on entration a hieved a su(cid:30) iently high level(e.g., half the feed on entration of methanol). The mathemati al orrelate ofthis empiri ally de(cid:28)ned transition dilution rate is the swit hing dilution rate, D s, (at whi h s swit hes from subsaturating to saturating levels). Consistentwith Egli's observations, D s, de reases with the fra tion of methanol in thefeed. Indeed, (55) yields β = Y V s, ¯ K e, D t, − D s, D s, ( D s, + k e, ) , where D t, is essentially onstant at saturating feed on entrations. It followsthat D s, is a de reasing fun tion of β , and, hen e, ψ . Moreover, in thelimiting ases of feeds onsisting of almost pure methanol ( β ≈ ) and pureglu ose ( β ≈ ), the swit hing dilution rate, D s, , tends to D c, and D t, ,respe tively. 45a) (b)Figure 19. Yields on the substrates during growth of H. polymorpha and C. boi-dinii on glu ose + methanol. (a) The yield on methanol is nearly onstant at al-most all feed on entrations (Egli et al., 1982b, Fig. 4). The dilution rate is (cid:28)xedat ∼ − . (b) In H. polymorpha, the yield on glu ose de reases ∼
20% in thepresen e of high methanol on entrations (Egli et al., 1982a, Fig. 3).4 Dis ussionWe have shown above that a simple model a ounting for only indu tion andgrowth aptures the experimental data under a wide variety of onditions.This does not prove that the model is orre t unless the underlying me h-anism is onsistent with experiments. Now, the model is based on following3 assumptions. (a) The yield of biomass on a substrate is onstant. (b) Theindu tion rate follows hyperboli kineti s, and regulatory me hanisms have anegligible e(cid:27)e t on the indu tion rate. ( ) Ea h substrate is transported by aunique system of lumped peripheral enzymes. In what follows, we dis uss thevalidity of these assumptions.4.1 Constant yieldsBoth dire t and indire t eviden e show that in many ases, the yield is, forthe most part, onstant.Indire t eviden e is obtained from experiments in whi h the spe i(cid:28) growthand substrate uptake rates were measured during mixed-substrate growth atvarious dilution rates and feed on entrations. Given these rates, eq. (12) anbe satis(cid:28)ed by in(cid:28)nitely many Y and Y . It is therefore striking that in all theseexperiments, (12) is satis(cid:28)ed by the single-substrate yields on the two sub-strates. These in lude growth of E. oli on various sugars (Lendenmann et al.,1996; Smith and Atkinson, 1980), organi a ids (Narang et al., 1997), glu- ose + 3-phenylpropioni a id (Kovarova et al., 1997); growth of Chelaba -ter heintzii on glu ose + nitrilotria eti a id (Bally et al., 1994); and growthof methylotrophi yeasts on methanol + glu ose (Fig. 13 ) and methanol +46orbitol (Eggeling and Sahm, 1981).Dire t eviden e is obtained from experiments in whi h at least one of theyields was measured by using radioa tively labeled substrates (Fig. 19). Thesemeasurements show that the yield on a substrate remains essentially equal tothe single-substrate yield, ex ept when toxi e(cid:27)e ts arise, or the fra tion ofthe substrate in the feed is below ∼
10% (Rudolph and Grady, 2001).4.2 Indu tion kineti sThe indu tion kineti s and regulatory me hanisms an in(cid:29)uen e the behaviorof the model. Spe i(cid:28) ally, if the indu tion kineti s are sigmoidal, multiple sta-ble steady states arise, resulting in the manifestation of the maintenan e orpre-indu tion e(cid:27)e t (Narang, 1998a; Narang and Pilyugin, 2007b). Likewise,introdu tion of regulatory e(cid:27)e ts in the model will alter its quantitative pre-di tions (although the analysis of the data for the la operon shows that regu-latory e(cid:27)e ts play a minor role). However, the essential property of the model,namely, the ompetitive (Lotka-Volterra) intera tion between the enzymes, isuna(cid:27)e ted by these hanges (cid:22) the model still predi ts the existen e of ex-tin tion and oexisten e steady states obtained with the minimal model (seeSe tion 4.1 of Narang and Pilyugin, 2007a).4.3 Transport kineti sMany substrates are imported into the ell by two (or more) transport sys-tems (Feren i, 1996, Table 1). In general, these transport systems dominatesubstrate import at di(cid:27)erent substrate on entrations. For instan e, gala toseis transported by the low-a(cid:30)nity gala tose permease under substrate-ex ess onditions, and the high-a(cid:30)nity methylgala tosidase system under substrate-limiting onditions (Wei kert and Adhya, 1993). If these transport systemsdominate in distin t regimes of the dilution rate, they are unlikely to havea signi(cid:28) ant e(cid:27)e t on the model. However, if the two regimes overlap, it isdi(cid:30) ult to predi t the out ome without analyzing a suitably extended model.This will be the fo us of future work.Lumping of peripheral enzymes has a profound e(cid:27)e t if the indu er is syn-thesized and onsumed by oordinately expressed enzymes. Following Sav-ageau, we illustrate this point by appealing to the la operon (Savageau, 2001,Fig. 10). It turns out that the on entration of the indu er, allola tose, is om-pletely determined by the on entration of intra ellular la tose. Moreover, the47ynami s of intra ellular la tose, denoted x , are given by the equation dxdt = V s e p sK s + s − k x e b x − r g x, where s denotes extra ellular la tose, e p denotes permease, and e b denotes β -gala tosidase. Sin e permease and β -gala tosidase are en oded by the sameoperon, intuition suggests that they must be oordinately expressed, i.e., e p = e b . It follows that the quasisteady state on entration of intra ellularla tose, x ≈ ( V s /k x ) s/ ( K s + s ) , is independent of the enzyme level. Thus, pos-itive feedba k, a key omponent of the model disappears if the indu er (or itspre ursor) is synthesized by oordinately expressed enzymes. Under this on-dition, the positive feedba k generated by one of the enzymes (e.g., permease)is neutralized by the other enzyme (e.g., β -gala tosidase).The relative rates of permease and β -gala tosidase expression are not known,but there is good eviden e that gala tosidase and transa etylase are not ex-pressed oordinately (reviewed in Adhya, 2003). Indeed, Ullmann and owork-ers have shown that the higher the protein synthesis rate, the higher the rela-tive rate of transa etylase synthesis (Dan hin et al., 1981). Furthermore, thisun oupling o urs at the level of trans ription, and involves the trans riptiontermination fa tor, Rho (Guidi-Rontani et al., 1984). It seems likely that per-mease and β -gala tosidase are also un oupled, but this remains a hypothesisuntil dire t eviden e is obtained.5 Con lusionsWe formulated a minimal model of mixed-substrate growth that a ounts foronly indu tion and growth (but no regulation), the two pro esses that o urin all mi robes in luding ba teria and yeasts.(1) Quantitative analysis of the data for both ba teria and yeasts shows thatthe de line of the enzyme a tivities in onstitutive and wild-type ells,typi ally attributed to spe i(cid:28) regulatory me hanisms, is almost entirelydue to dilution.(2) We analyzed the model by onstru ting the bifur ation diagrams for bothbat h and ontinuous ultures.(a) The bifur ation diagram for bat h ultures des ribes the substrate onsumption patterns at any given initial substrate on entrations.It shows that: (i) When the initial on entrations of one or moresubstrates are su(cid:30) iently small, threshold e(cid:27)e ts arise be ause theenzymes annot be indu ed. (ii) Even if the initial on entrations ofboth substrates are at supra-threshold levels, indu tion of the en-zymes for one of the substrates may not o ur be ause of dilution.48hus, diauxi growth is feasible even if there are no regulatory me h-anisms. ( ) The substrate onsumption pattern an be swit hed fromsequential to simultaneous (or vi e versa) by altering the initial sub-strate on entrations. All these on lusions are onsistent with thedata in the literature.(b) The bifur ation diagram for ontinuous ultures, whi h des ribes thesubstrate onsumption patterns at any given dilution rate and feed on entrations, is formally identi al to the bifur ation diagram forbat h ultures. This gives a pre ise mathemati al basis for the well-known empiri al orrelation between the growth patterns in bat hand ontinuous ultures. Spe i(cid:28) ally, substrates that are simultane-ously onsumed in substrate-ex ess bat h ultures are onsumed si-multaneously in ontinuous ultures at all dilution rates up to washout.However, substrates that are sequentially onsumed in substrate-ex ess bat h ultures are onsumed simultaneously in ontinuous ul-tures only if the dilution rate is su(cid:30) iently small (cid:22) at large dilutionrates, only the preferred substrate is onsumed. This swit h o urspre isely at the so- alled transition dilution rate be ause synthesis ofthe se ondary substrate enzymes is abolished by dilution. The tran-sition dilution rate is signi(cid:28) antly higher the maximum growth rateon the se ondary substrate. This (cid:16)enhan ed growth e(cid:27)e t(cid:17) o urs be- ause the enzyme levels are positive at the riti al dilution rate ofthe se ondary substrate (cid:22) they be ome vanishingly small only if the ells are subje ted to even higher dilution rates.(3) At the saturating feed on entrations typi ally used in experiments, thephysiologi al steady states are ompletely determined by only two inde-pendent parameters, the dilution rate and the mass fra tion of the sub-strate in the feed. We derived analyti al expressions for the steady statevalues of the substrates, ell density, and enzymes that provide good (cid:28)tsto the extensive data for the growth of methylotrophi yeasts on glu oseand methanol.A knowledgementsThis resear h was supported in part with funds from the National S ien eFoundation under ontra t NSF DMS-0517954. One of us (A.N.) is gratefulto Dr. Stefan Oehler (IMBB-FoRTH) for dis ussions regarding the regulationof the la operon. 49eferen esAdhya, S., Jun 2003. 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J. 11, 11(cid:21)27.A Bifur ation diagram for bat h ulturesA.1 Conditions for existen e and stability of the steady statesThe steady states that an be attained during the (cid:28)rst few hours of bat hgrowth are given by the equations de dt = V e, e σ , ¯ K e, + e σ , − ( Y V s, e σ , + Y V s, e σ , + k e, ) e , (A.1) de dt = V e, e σ , ¯ K e, + e σ , − ( Y V s, e σ , + Y V s, e σ , + k e, ) e . (A.2)54he Ja obian has the form J ( e , e ) = J − Y V s, e σ , − Y V s, e σ , J , (A.3)where J ≡ V e, ¯ K e, σ , (cid:16) ¯ K e, + e σ , (cid:17) − Y V s, e σ , − Y V s, e σ , − k e, , (A.4) J ≡ V e, ¯ K e, σ , (cid:16) ¯ K e, + e σ , (cid:17) − Y V s, e σ , − Y V s, e σ , − k e, . (A.5)In what follows, we derive the existen e and stability onditions for ea h ofthe 4 steady states.(1) E ( e = 0 , e = 0 ): It is evident from (A.1)(cid:21)(A.2) that this steadyalways exists. Sin e J ( E ) = V e, σ , ¯ K e, − k e, V e, σ , ¯ K e, − k e, ,E is stable if and only if r ∗ g,i ≡ V e,i σ ,i ¯ K e,i − k e,i < ⇔ σ ,i < σ ∗ i ≡ k e,i V e,i / ¯ K e,i for i = 1 , .(2) E ( e > , e = 0 ): This steady state exists if and only if the equation V e, σ , ¯ K e, + e σ , = Y V s, e σ , + k e, (A.6)has a positive solution, e > . Su h a solution exists pre isely when r ∗ g, > ⇔ σ , > σ ∗ , in whi h ase e | E = − (cid:16) ¯ K e, + k e, Y V s, (cid:17) + r(cid:16) ¯ K e, + k e, Y V s, (cid:17) + K e, r ∗ g, Y V s, σ , ,r g, = Y V s, − (cid:16) ¯ K e, + k e, Y V s, (cid:17) + r(cid:16) ¯ K e, + k e, Y V s, (cid:17) + K e, r ∗ g, Y V s, , (A.7)55here r g, ≡ Y V s, e σ , | E is the exponential growth rate at E .To determine the stability onditions, observe that J ( E ) = J − Y V s, e σ , J , where J = V e, ¯ K e, σ , (cid:16) ¯ K e, + e σ , (cid:17) − Y V s, e σ , − k e, ,J = V e, σ , ¯ K e, − Y V s, e σ , − k e, . Now, J is always negative be ause (A.6) implies that at E , V e, ¯ K e, σ , (cid:16) ¯ K e, + e σ , (cid:17) = ¯ K e, ¯ K e, + e σ , ( Y V s, e σ , + k e, ) , so that J = − ( Y V s, e σ , + k e, ) e σ , ¯ K e, + e σ , − Y V s, e σ , < . Hen e, E is stable if and only if J < ⇔ V e, σ , ¯ K e, < Y V s, e σ , | E + k e, , ⇔ r ∗ g, < r g, . (3) E ( e = 0 , e > ): Arguments similar to those used above for E showthat E exists if and only if r ∗ g, > ⇔ σ , > σ ∗ , in whi h ase e | E = − (cid:16) ¯ K e, + k e, Y V s, (cid:17) + r(cid:16) ¯ K e, + k e, Y V s, (cid:17) + K e, r ∗ g, Y V s, σ , ,r max g, = Y V s, − (cid:16) ¯ K e, + k e, Y V s, (cid:17) + r(cid:16) ¯ K e, + k e, Y V s, (cid:17) + K e, r ∗ g, Y V s, , (A.8)where r g, ≡ Y V s, e σ , | E is the exponential growth rate at E .564) E ( e > , e > ): This steady state exists if and only if the equations V e, σ , ¯ K e, + e σ , − ( Y V s, e σ , + Y V s, e σ , + k e, ) , (A.9) V e, σ , ¯ K e, + e σ , − ( Y V s, e σ , + Y V s, e σ , + k e, ) , (A.10)have positive solutions, e , e > . The onditions for su h solutions anbe determined by eliminating one of the variables, e , e , from the aboveequations. To this end, observe that eqs. (A.9)(cid:21)(A.10) imply the relation V e, σ , ¯ K e, + e σ , − k e, = V e, σ , ¯ K e, + e σ , − k e, . (A.11)Hen e, we an eliminate e from (A.9)(cid:21)(A.10) by solving (A.11) for e ,and substituting it in (A.9) to obtain the equation V e, σ , ¯ K e, + e σ , − k e, = Y V s, e σ , + Y V s, V e, σ , V e, σ , ¯ K e, + σ , − k e, + k e, − Y V s, ¯ K e, , whi h has a (unique) positive solution, e > , if and only if V e, σ , ¯ K e, − k e, > Y V s, V e, σ , (cid:16) V e, σ , ¯ K e, − k e, (cid:17) + k e, − Y V s, ¯ K e, . One an he k that this is equivalent to the ondition r ∗ g, ≡ V e, σ , ¯ K e, − k e, > Y V s, e σ , | E ≡ r g, , i.e., E is unstable.Similarly, eliminating e from (A.9)-(A.10) yields V e, σ , ¯ K e, + e σ , − k e, = Y V s, e σ , + Y V s, V e, σ , V e, σ , ¯ K e, + σ , − k e, + k e, − Y V s, ¯ K e, , whi h has a positive solution, e > , if and only if E is unstable. We on lude that E exists and is unique if and only if both E and E (exist and) are unstable.To determine the stability ondition, observe that (A.9) implies V e, ¯ K e, σ , (cid:16) ¯ K e, + e σ , (cid:17) = ( Y V s, e σ , + Y V s, e σ , + k e, ) ¯ K e, ¯ K e, + e σ , . J = ( Y V s, e σ , + Y V s, e σ , + k e, ) ¯ K e, ¯ K e, + e σ , − Y V s, e σ , − Y V s, e σ , − k e, , = − ( Y V s, e σ , + Y V s, e σ , + k e, ) e σ , ¯ K e, + e σ , − Y V s, e σ , , and similarly, J = − ( Y V s, e σ , + Y V s, e σ , + k e, ) e σ , ¯ K e, + e σ , − Y V s, e σ , . It follows immediately that tr
J < and det J > . Hen e, E is stablewhenever it exists.A.2 Relative magnitudes of r g,i and r ∗ g,i The exponential growth rate on S i , r g,i ( σ ,i ) , is always less than r ∗ g,i ( σ ,i ) , thespe i(cid:28) growth rate at whi h E i be omes extin t. To see this, observe that(A.7)(cid:21)(A.8) an be rewritten as r ∗ g,i = k e,i Y i V s,i ¯ K e,i ! r g,i + Y i V s,i ¯ K e,i ! r g,i , (A.12)whi h implies that r g,i ≤ r ∗ g,i , with equality being attained when σ i = σ ∗ i , in whi h ase, r g,i = r ∗ g,i = 0 .A.3 Disposition of the bifur ation urvesThe lower (blue) bifur ation urve in Fig. 9 is de(cid:28)ned by the ondition r g, = r ∗ g, . (A.13)Sin e r ∗ g, = k e, Y V s, ¯ K e, ! r g, + Y V s, ¯ K e, ! r g, , (A.14)the bifur ation urve de(cid:28)ned by (A.13) satis(cid:28)es the equation r ∗ g, = k e, Y V s, ¯ K e, ! r ∗ g, + 1 Y V s, ¯ K e, (cid:16) r ∗ g, (cid:17) , r g, with r ∗ g, . Substituting the de(cid:28)nition of r ∗ g,i into the above equation yields V e, ¯ K e, ( σ , − σ ∗ ) = k e, Y V s, ¯ K e, ! " V e, ¯ K e, ( σ , − σ ∗ ) + 1 Y V s, ¯ K e, " V e, ¯ K e, ( σ , − σ ∗ ) , whi h de(cid:28)nes an in reasing urve on the σ , , σ , -plane passing through thepoint ( σ ∗ , σ ∗ ) . Moreover, sin e the points on this urve satisfy the relation V e, ¯ K e, ( σ , − σ ∗ ) > V e, ¯ K e, ( σ , − σ ∗ ) , they never go above the line V e, ¯ K e, ( σ , − σ ∗ ) = V e, ¯ K e, ( σ , − σ ∗ ) . (A.15)A similar argument shows that the upper (red) bifur ation urve in Fig. 9,whi h is de(cid:28)ned by the equation r g, = r ∗ g, , (A.16)satis(cid:28)es the relation V e, ¯ K e, ( σ , − σ ∗ ) = k e, Y V s, ¯ K e, ! " V e, ¯ K e, ( σ , − σ ∗ ) + 1 Y V s, ¯ K e, " V e, ¯ K e, ( σ , − σ ∗ ) . This relation also de(cid:28)nes an in reasing urve passing through ( σ ∗ , σ ∗ ) , but the urve never goes below the line de(cid:28)ned by (A.15) be ause V e, ¯ K e, ( σ , − σ ∗ ) > V e, ¯ K e, ( σ , − σ ∗ ) . Hen e, the urve de(cid:28)ned by (A.16) always lies above the urve de(cid:28)ned by(A.13).Finally, we note that as the two bifur ation urves approa h the point, ( σ ∗ , σ ∗ ) ,they are approximated by the lines V e, ¯ K e, ( σ , − σ ∗ ) ≈ k e, Y V s, ¯ K e, ! " V e, ¯ K e, ( σ , − σ ∗ ) ,V e, ¯ K e, ( σ , − σ ∗ ) ≈ k e, Y V s, ¯ K e, ! " V e, ¯ K e, ( σ , − σ ∗ ) , k e,i ≪ Y i V s,i ¯ K e,i .Thus, for typi al parameter values, the urves have the quasi usp-shaped ge-ometry shown in Fig. 9.B Bifur ation diagram for ontinuous ulturesThe steady states of ontinuous ultures satisfy the equations ds dt = D ( s f, − s ) − r s, c, (B.1) ds dt = D ( s f, − s ) − r s, c, (B.2) dcdt = ( r g − D ) c, (B.3) de dt = R ≡ V e, e σ ¯ K e, + e σ − ( r g + k e, ) e , (B.4) de dt = R ≡ V e, e σ ¯ K e, + e σ − ( r g + k e, ) e , (B.5)where r g = Y r s, + Y r s, . The Ja obian is J = − D − c ∂r s, ∂s − r s, − c ∂r s, ∂e − D − c ∂r s, ∂s − r s, − c ∂r s, ∂e c ∂r g ∂s c ∂r g ∂s r g − D c ∂r g ∂e c ∂r g ∂e ∂R ∂s ∂R ∂s ∂R ∂e ∂R ∂e ∂R ∂s ∂R ∂s ∂R ∂e ∂R ∂e (B.6)We begin by onsidering the washout steady states ( c = 0 ), sin e the existen eand stability onditions for these steady states follow immediately from theforegoing analysis of bat h ultures.B.1 Existen e and stability of washout steady statesAt a washout steady state, c = 0 , s = s f, , s = s f, . Su h a steady stateexists if and only if there exist e , e > satisfying the equations de dt = V e, e σ f, ¯ K e, + e σ f, − ( Y V s, e σ f, + Y V s, e σ f, + k e, ) e , de dt = V e, e σ f, ¯ K e, + e σ f, − ( Y V s, e σ f, + Y V s, e σ f, + k e, ) e , σ ,i is repla ed by σ f,i .It follows from (B.6) that the Ja obian at a washout steady steady is J = − D − r s, − D − r s, r g − D ∂R ∂s ∂R ∂s ∂R ∂e ∂R ∂e ∂R ∂s ∂R ∂s ∂R ∂e ∂R ∂e . Thus, a washout steady state is stable if and only if at this steady state, r g < D , and the submatrix R ≡ ∂R ∂e ∂R ∂e ∂R ∂e ∂R ∂e has negative eigenvalues. This submatrix is formally similar to the matrixde(cid:28)ned by eqs. (A.3)(cid:21)(A.5), the only di(cid:27)eren e being that σ ,i is repla ed by σ f,i .Given the above results, we expe t the existen e and stability onditions forthe washout steady states, E , E , E , E , to be formally similar to theexisten e and stability onditions for the orresponding bat h ulture steadystates, namely, E , E , E , and E . We show below that this is indeed the ase.(1) E ( e = 0 , e = 0 , c = 0 ): It is evident that this steady state al-ways exists. Sin e r g | E = 0 < D , E is stable if and only if R ( E ) has negative eigenvalues. Sin e R ( E ) is formally similar to J ( E ) , we on lude that E is stable if and only if D t,i ≡ V e,i σ f,i ¯ K e,i − k e,i < ⇔ σ f,i < σ ∗ i ≡ k e,i V e,i /K e,i (B.7)for i = 1 , .(2) E ( e > , e = 0 , c = 0 ): This steady state exists if and only if V e, σ f, ¯ K e, + e σ f, = Y V s, e σ f, + k e, (B.8)whi h is formally similar to eq. (A.6). It follows that E exists if andonly if D t, > ⇔ σ f, > σ ∗ ,
61n whi h ase e | E = − (cid:16) ¯ K e, + k e, Y V s, (cid:17) + r(cid:16) ¯ K e, + k e, Y V s, (cid:17) + K e, D t, Y V s, σ f, ,D c, = Y V s, − (cid:16) ¯ K e, + k e, Y V s, (cid:17) + r(cid:16) ¯ K e, + k e, Y V s, (cid:17) + K e, D t, Y V s, , (B.9)where D c, ≡ r g | E = Y V s, e σ | E is the riti al dilution rate duringsingle-substrate growth on S . E is stable if and only if D > D c, and D t, < D c, , where the latterrelation ensures that the submatrix, R ( E ) , whi h is formally similar to J ( E ) , has negative eigenvalues. In the parti ular ase of single-substrategrowth on S , the se ond stability ondition is always satis(cid:28)ed ( D t, = − k e, < < D c, ), so that E is stable whenever D > D c, .(3) E ( e = 0 , e > , c = 0 ): Arguments similar to those used above for E show that E exists if and only if D t, > ⇔ σ f, > σ ∗ , in whi h ase e | E = − (cid:16) ¯ K e, + k e, Y V s, (cid:17) + r(cid:16) ¯ K e, + k e, Y V s, (cid:17) + K e, D t, Y V s, σ f, ,D c, = Y V s, − (cid:16) ¯ K e, + k e, Y V s, (cid:17) + r(cid:16) ¯ K e, + k e, Y V s, (cid:17) + K e, D t, Y V s, , (B.10)where D c, ≡ r g | E = Y V s, e σ | E is the riti al dilution rate duringsingle-substrate growth on S . It is stable if and only if D > D c, and D t, < D c, .(4) E ( e = 0 , e > , c > ): This steady state exists if and only if theequations V e, σ f, ¯ K e, + e σ f, − ( Y V s, e σ f, + Y V s, e σ f, + k e, ) , (B.11) V e, σ f, ¯ K e, + e σ f, − ( Y V s, e σ f, + Y V s, e σ f, + k e, ) , (B.12)have positive solutions. These equations are formally similar to eqs. (A.9)(cid:21)(A.10). Hen e, E exists and is unique if and only if < D t, < D c, , < D t, < D c, . Sin e R ( E ) is formally similar to J ( E ) , the eigenvalues of R ( E ) are always negative. It follows that E is stable if and only if D > D c ≡ r g | E = Y V s, e σ + Y V s, e σ | E .
62e annot solve expli itly for D c , but its properties an be inferred froman impli it representation, whi h an be derived as follows. At E , thesteady state enzyme balan es imply that V e,i σ f,i ¯ K e,i + e σ f,i = D c + k e, ⇔ e i σ f,i = V e,i D c + k e,i − ¯ K e,i ⇔ e i σ f,i = ¯ K e,i D c + k e,i ( D t,i − D c ) . Hen e, D c satis(cid:28)es the equation D c = Y V s, ¯ K e, D c + k e, ( D t, − D c ) + Y V s, ¯ K e, D c + k e, ( D t, − D c ) . (B.13)We shall appeal to this equation below.Before pro eeding to the persisten e steady states, we pause to onsider thegeometry and disposition of the surfa es.B.1.1 Geometry and disposition of the surfa es of D t,i ( σ f,i ) and D c,i ( σ f,i ) :The geometry and disposition of the surfa es of D c,i and D t,i follow fromarguments similar to those used above in the analysis of bat h ultures. Indeed,(B.9)(cid:21)(B.10) imply that D t,i = k e,i Y V s,i ¯ K e,i ! D c,i + Y i V s,i ¯ K e,i ! D c,i , whi h is formally similar to (A.12). It follows that D t,i ≥ D c,i , and the bifur a-tion urves de(cid:28)ned by the equations, D t, = D c, and D t, = D c, , are identi alto the bifur ation urves in Fig. 9, the only di(cid:27)eren e being that they lie onthe σ f, , σ f, -plane (as opposed to the σ , , σ , -plane).B.1.2 Geometry and disposition of the surfa e of D c ( σ f, , σ f, ) The surfa e of D t, interse ts the surfa es of both D c and D c, along the verysame urves. Indeed, it follows from (B.13) that D t, interse ts the surfa es of D c and D t, along the urve D t, = Y V s, ¯ K e, D t, + k e, ( D t, − D t, ) , whi h is identi al to the equation de(cid:28)ning the urve, D t, = D c, . A similarargument shows that the surfa e of D t, interse ts the surfa es of D c and D c, along the same urve. 63i(cid:27)erentiation of (B.13) yields ∂D c ∂σ f,i = Y i V s,i V e,i D + k e,i Y V s, ¯ K e, ( D + k e, ) + Y V s, ¯ K e, ( D + k e, ) > . This immediately implies that the magnitude of D c (relative to the magnitudesof D t,i and D c,i ) is as shown in Fig. 12.B.2 Persisten e steady statesThere are three persisten e steady states.B.2.1 E ( e > , e = 0 , c > )This steady state exists if and only if the equations D ( s f, − s ) − V s, e σ c, (B.14) V e, σ ¯ K e, + e σ − ( Y V s, e σ + k e, ) , (B.15) Y V s, e σ − D. (B.16)have positive solutions. These equations an be expli itly solved for s , e , and c . Indeed, (B.16) implies that σ = DY V s, e , whi h an be substituted in (B.14)(cid:21)(B.15) to obtain c = Y ( s f, − s ) , (B.17) e = V e, D + k e, DY V s, ¯ K e, + D , (B.18) σ = DY V s, e = 1 Y V s, V e, ( D + k e, ) (cid:16) D + Y V s, ¯ K e, (cid:17) . (B.19)Now, e > for all D > , and c > whenever s < s f, . Hen e, E exists ifand only if (B.19) has a solution satisfying < σ < σ f, . It follows from thegraph of σ (Fig. B.1a) that su h a solution exists if and only if σ f, > σ ∗ , < D < D c, . Here, D c, satis(cid:28)es the relation σ f, == 1 Y V s, V e, ( D c, + k e, ) (cid:16) D c, + Y V s, ¯ K e, (cid:17) , E satis(cid:28)es the relation, < σ < σ f, , pre isely when σ f, > σ ∗ and < D < D c, . (b) Graphi al depi tionof the onditions for existen e of E .whi h has the positive solution given by (B.9).To determine the stability of E , observe that the Ja obian is similar to thematrix − c ∂r s, ∂s − c ∂r s, ∂e − r s, Y r s, ∂R ∂s ∂R ∂e ∂R ∂e ∂R ∂s − D ∂R ∂e
00 0 0 − c ∂r s, ∂e − D , whi h represents the Ja obian of the new system of equations obtained if wemake the linear oordinate hange { s , s , e , e , c } −→ { s , e , Y s + Y s + c, e , s } . It follows that E is stable if and only if ∂R /∂e < and the submatrix − c ∂r s, ∂s − c ∂r s, ∂e ∂R ∂s ∂R ∂e has negative eigenvalues. One an he k that thi submatrix always has negativeeigenvalues, and ∂R ∂e = V e, σ f, ¯ K e, − D − k e, = D t, − D, so that E is stable if and only if D > D t, . (B.20)65n the parti ular ase of single-substrate growth on S , this ondition is alwayssatis(cid:28)ed, so that E is stable whenever it exists.B.2.2 E ( e = 0 , e > , c > )The existen e and stability onditions for E an be derived by methodsanalogous to those shown above for E .B.2.3 E ( e > , e > , c > )This steady state exists if and only if the equations D ( s f, − s ) − V s, e σ c (B.21) D ( s f, − s ) − V s, e σ c (B.22) V e, σ ¯ K e, + e σ − ( Y V s, e σ + Y V s, e σ + k e, ) (B.23) V e, σ ¯ K e, + e σ − ( Y V s, e σ + Y V s, e σ + k e, ) (B.24) Y V s, e σ + Y V s, e σ − D (B.25)have positive solutions, s i , e i , c > . It follows from (B.21)(cid:21)(B.22) that c > if and only if s i < s f,i .To prove the existen e onditions, it is useful to simplify the above equations.To this end, observe that c an be eliminated from eqs. (B.21)(cid:21)(B.22) to obtainthe relation V s, e σ ( s f, − s ) = V s, e σ ( s f, − s ) , (B.26)and eqs. (B.23)(cid:21)(B.25) immediately yield e i = V e,i D + k e,i − ¯ K e,i σ i , i = i, , (B.27)We on lude that E exists if and only there exist < s i < s f,i , e i > satisfying (B.25)(cid:21)(B.27). We show below that su h s i and e i exist if and onlyif < D < D t, , D t, , D c .To prove the su(cid:30) ien y of the ondition, < D < D t, , D t, , D c , substitute(B.27) in (B.25) and (B.26) to obtain the equations f ( σ , σ ) ≡ σ σ + σ σ = 1 , (B.28) f ( σ , σ ) ≡ V s, V e, D + k e, ( σ − σ ) ( s f, − s ) − V s, V e, D + k e, ( σ − σ ) ( s f, − s ) , (B.29)66here σ i ≡ Y i V s,i ¯ K e,i ( D + k e,i ) (cid:16) D + Y V s, ¯ K e, + Y V s, ¯ K e, (cid:17) ,σ i ≡ ( D + k e,i ) ¯ K e,i V e,i . Now, (B.28) de(cid:28)nes a line on the σ , σ -plane with the inter epts, σ and σ (full line in Fig. B.1b). As D in reases, it moves further from the origin, andpasses through the point, ( σ f, , σ f, ) , pre isely when D = D c . On the otherhand, (B.29) de(cid:28)nes a urve whi h has the geometry of the dashed urve inFig. B.1b. To see this, observe that ( σ , σ ) and ( σ f, , σ f, ) lie on the urve.Moreover, ( σ , σ ) lies to the left and below ( σ f, , σ f, ) be ause D < D t,i = V e,i σ f,i ¯ K e,i − k e,i ⇔ σ i < σ f,i , and the slope of the urve is positive sin e − ∂f /∂σ ∂f /∂σ = V s, V e, D + k e, ds dσ ( σ − σ ) + V s, V e, D + k e, ( s f, − s ) V s, V e, D + k e, ds dσ ( σ − σ ) + V s, V e, D + k e, ( s f, − s ) > for all σ i < σ i < σ f,i . Finally, ( σ , σ ) lies below the growth iso line be ause σ σ + σ σ = Y V s, ¯ K e, + Y V s, ¯ K e, D + Y V s, ¯ K e, + Y V s, ¯ K e, < for all D > . It follows that the urves de(cid:28)ned by (B.28) and (B.29) interse tat a unique point, say, ( e σ , e σ ) , su h that σ i < e σ < σ f, . Sin e σ i < e σ i ⇔ e i = V e,i D + k e,i − ¯ K e,i e σ i > E exists and is unique.The ne essity of the existen e onditions follows from the fa t that for all < σ i < σ f,i , e i < V e,i D + k e,i − ¯ K e,i σ f,i = ¯ K e,i σ i ( D + k e,i ) ( D t,i − D ) . Hen e, if D t,i ≤ or D ≥ D t,i , E does not exist be ause e i < . Likewise,if D ≥ D c , E does not exist be ause there are no < s i < s f,i satisfying(B.28).We on lude that E exists if and only if < D < D t, , D t, , D c .
67e were unable to prove the stability ondition for E111