Op imiza ion o me abolic s a es
And eas K emling1, Wol am Liebe meis e 2, Elad Noo 3and Meike T. Wo el4
1Sys ems Bio echnology, School o Enginee ing and Design, Technical Uni e si y o Munich, Ge many; 2Uni e si é
Pa is-Saclay, INRAE, MaIAGE, 78350 Jouy-en-Josas, F ance; 3Depa men o Plan and En i onmen al Sciences,
Weizmann Ins i u e o Science, 76100 Reho o , Is ael; 4Swamme dam Ins i u e o Li e Sciences, Uni e si y o Am-
s e dam, Ams e dam, he Ne he lands
Abs ac
Cells in a well-mixed and nu ien - ich en i onmen can be expec ed o expe ience selec ion on hei g ow h a e.
Such cells op imize hei me abolic s a e o achie e a high g ow h a e. Me abolic s a es ha lead o a high g ow h
a e a e s a es ha ealize a maximal biomass p oduc ion a e a a minimal enzyme cos . Me abolic s a es ha
op imize a speci ic lux a minimal enzyme cos a e called enzyme-e icien me abolic s a es and in his chap e we
e e o hem as op imal me abolic s a es. The calcula ion o op imal me abolic s a es is acili a ed by he esul
ha , in models wi hou u he cons ain s, he lux dis ibu ions in enzyme-e icien s a es a e Elemen a y Flux
Modes (EFMs). This esul allows o an algo i hm o ind enzyme-e icien s a es by: 1) Enume a ing he EFMs, 2)
calcula ing he minimal enzyme cos pe EFM, and 3) choosing he one wi h he lowes enzyme in es men . This
algo i hm inds op imal me abolic s a es o la ge models which canno be op imized by ’b u e o ce’, bu a e s ill
small enough o enume a e he EFMs. Finding op imal me abolic s a es unco e ed he e ec o changing ex e nal
nu ien condi ions: As g ow h condi ions a e changing, he op imal lux p o ile ei he changes con inuously (and
me aboli e and enzyme concen a ions change con inuously as well) when he same EFM emains op imal, o luxes
change discon inuously oge he wi h me aboli e and enzyme concen a ions when a di e en EFM becomes op imal.
Con ibu ions: This chap e was d a ed and w i en by And eas K emling, Wo l am Liebe meis e , Elad Noo , and
Meike Wo el. I was ini ially discussed wi h Jü gen Zanghellini and la e e iewed by Da id S. Tou igny and Hugo
Dou ado and discussed wi h S e an Mülle . We hank Coen Be ns o commen s.
Keywo ds: elemen a y lux mode (EFM) - enzyme cos minimiza ion - enzyme-e icien s a e
To ci e his chap e : A. K emling, W. Liebe meis e , E. Noo , M.T. Wo el. Op imiza ion o me abolic s a es (Ve -
sion Oc obe 2025). doi: 10.5281/zenodo.8164468. Chap e om: The Economic Cell Collec i e (2025). Economic
P inciples in Cell Biology. No comme cial publishe | Online open access book | doi: 10.5281/zenodo.8156386
The au ho s a e lis ed in alphabe ical o de .
This is a chap e om he open ex book “Economic P inciples in Cell Biology”.
F ee download om p inciplescellphysiology.o g/book-economic-p inciples/.
Lec u e slides o his chap e a e a ailable on he websi e.
c
2025 The Economic Cell Collec i e.
Licensed unde C ea i e Commons License CC-BY-SA 4.0.
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doi: 10.5281/zenodo.8156386
1
Chap e o e iew
◦Op imal me abolic s a es in his chap e e e o enzyme-e icien s a es, which a e me abolic s a es ha
ealize a gi en objec i e lux a a minimal enzyme cos .
◦In models wi hou u he lux cons ain s, lux dis ibu ions o enzyme-e icien s a es a e Elemen a y Flux
Modes (EFMs).
◦Elemen a y Flux Modes can be used o ind enzyme-e icien s a es in ne wo ks ha would be oo la ge o
op imize me abolic s a es "by b u e o ce".
◦Biomass pe enzyme e iciency can be con e ed o cell g ow h a e by simple app oxima e o mulae.
◦The Elemen a y Flux Mode ha is ealized in an enzyme-e icien s a e depends on he ex e nal condi ions.
◦As g ow h condi ions change, ei he he lux p o ile changes con inuously ( oge he wi h me aboli e and
enzyme concen a ions), o luxes change discon inuously, implying jumps also in me aboli e and enzyme
concen a ions.
7.1. In oduc ion
In a simple economic pic u e o cells, we assume ha cells adjus hei me abolic s a e in each en i onmen o
ob ain a maximal i ness ad an age. This may be impossible in eali y, bu i emains an in e es ing ques ion wha
his bes me abolic s a e would look like, acco ding o ou knowledge o cells. So wha is he bes me abolic s a e
o e all (comp ising me abolic luxes, me aboli e concen a ions and enzyme le els)? Wha pa hways should a cell
use, which enzymes should be induced o ep essed, and how should his change in a new en i onmen ? To answe
hese ques ions, we need o emembe ha all me abolic a iables ( luxes, me aboli e le els, enzyme le els, and
enzyme e iciencies) depend on each o he . Physically, luxes depend on me aboli e concen a ions h ough kine ics
and enzyme egula ion (e.g. compe i i e inhibi ion) and me aboli es a e p oduced and consumed by he luxes un il
a s eady s a e is eached. Hence, i we hink in e ms o cellula economics ( ea ing enzymes as con ol a iables),
hen all me abolic a iables mus be op imized oge he .
In he p e ious chap e s we saw some ways o p edic op imal me abolic luxes, me aboli e concen a ions and enzyme
le els sepa a ely: in Flux Balance Analysis (FBA, Chap e 5 in [1]), we op imized luxes by maximizing an objec i e
unc ion ( ypically biomass) while in Enzyme Cos Minimiza ion [2,3] (Chap e 6 in [1]) me aboli e concen a ions
we e op imized by minimizing cos (o , equi alen ly, maximizing he enzyme e iciencies). Each o hese me hods
is based on a s ong assump ion: FBA equi es measu ed lux anges and/o appa en ca aly ic a es and assumes
enzyme sa u a ion e ec s can be neglec ed, while enzyme cos minimiza ion equi es a gi en lux dis ibu ion. Bu
wha i we don’ know any o he a iables in ad ance? How can we p edic all o hem om i s p inciples?
Be o e hinking abou his, le us b ie ly s ep back: wha do we ac ually mean by an “op imal s a e”? Wha quan i y
should be maximized in me abolism? The e could be e y di e en aims (e.g. p oduc ion in bio echnology, e sus
numbe o o sp ing and su i al in a wild- ype cell). Howe e , in bo h cases an impo an aim is cell g ow h – o
a leas , a oiding s ong g ow h de ici s. Below we will see ha cell g ow h depends, o a good app oxima ion, on
biomass/enzyme e iciency, ha is, biomass p oduc ion pe o al enzyme in es ed. Hence, whene e as g ow h is
impo an , cells should maximize his e iciency.
In conclusion, we will conside he ollowing op imali y p oblem: maximize biomass/enzyme e iciency, de ined as he
p oduc ion lux pe in es ed enzyme wi h espec o all me abolic a iables (me aboli es, enzymes and luxes) and
unde all cons ain s (s eady s a e, enzyme kine ics, e c.). Solu ions o his p oblem a e conside ed op imal s a es.
2
7.2. Enzyme-e icien me abolic s a es use elemen a y lux modes
The op imiza ion p oblem in his chap e is o each maximal objec i e lux wi h minimal enzyme in es men . The
biological in e p e a ion is ha his would lead o he highes g ow h a e, because i op imizes he a io be ween gains
( luxes) and cos s (enzymes). When we sol e his op imiza ion p oblem wi h ma hema ical ools, i is con enien o
ei he ind he minimal enzyme in es men o a ce ain lux, o he maximum lux o a ixed enzyme in es men .
Al hough one could hink o di e en biological explana ions o hose wo ways o s a e he op imiza ion p oblem,
ma hema ically hey a e equi alen . Fo he ou line o he p oo ha op imal s a es a e elemen a y lux modes, i is
con enien o ix he objec i e lux o an a bi a y alue (we choose 1) and hen minimize he enzyme in es men .
This leads o he ollowing op imiza ion p oblem o e he luxes ( ), enzymes le els (e) and in e nal me aboli e
concen a ions (s):
minimize
,e,s
X
i=1
hiei(7.1)
subjec o: N· =0s eady s a e
∀i: i=eiκi(s)enzyme kine ics
e,s≥0posi i e concen a ions
= 1 ixed objec i e lux
s≤smax me aboli e bounds
whe e hia e he weigh s, Nis he s oichiome y ma ix, iis he index o he eac ions ( anging om 1 o ), wi h he
las eac ion (wi h index ) ep esen ing he objec i e. This op imiza ion p oblem s a es ha by adjus ing he luxes
( ), me aboli e concen a ions (s) and enzyme concen a ions (e), he o al cos (sum o cos s – hiei– o e e y
eac ion) is minimized, while keeping he objec i e lux cons an (any a bi a y cons an can be chosen, he e we chose
1). The weigh s (hi) can be hough o as he size o p oduc ion cos s o he enzymes (measu ed, o example, in
molecula weigh o gene leng h) We equi e ce ain cons ain s: (i) he me abolic ne wo k needs o be in s eady s a e
o a oid buil -up o in e media es, (ii) enzyme kine ics – he lux o each eac ion ( i) has o be equal o he enzyme
concen a ion (ei) imes a me aboli e dependen (e.g., sa u a ion) e m (κi(s)), (iii) all enzyme and me aboli e
concen a ions ha e o be posi i e, (i ) he objec i e lux is equal o 1, and ( ) he me aboli e concen a ions a e
wi hin hei gi en bounds. The la e cons ain is op ional and is mos ly necessa y when dealing wi h i e e sible
kine ics. Re e sible kine ics usually lead o bounded me aboli e le els because e y high concen a ions o p oduc s
inhibi he eac ion ha o ms he p oduc s.
In his sec ion, we will explain why he op imal s a e is eached a an Elemen a y Flux Mode (EFM). One impo an
s a ing poin is ha , as we ha e seen be o e in Chap e 4 in [1], con ex op imiza ion p oblems wi h only posi i i y
o equali y cons ain s (no o he inequali ies) lead o an op imal solu ion a an ex eme poin o he easible solu ion
space, and hose ex eme poin s a e Elemen a y Flux Modes. Howe e , he op imiza ion p oblem (7.1) is no con ex,
mainly due o he hype bolic dependence o eac ion a es on he concen a ions o me aboli es (κi(s)is usually no
linea in he in e nal me aboli e concen a ions).
The e a e se e al ways o p o e ha he solu ion o his op imiza ion p oblem is an EFM, o which some a e ou lined
in he pape s by Wo el e al. [4] and Mülle e al. [5]. He e we will ou line a p oo by con adic ion: assuming a
solu ion o he op imiza ion p oblem ha is no an EFM and showing ha his leads o a con adic ion.
Theo em 1. The lux dis ibu ion ha maximizes an objec i e lux o e he o al enzyme cos in a me abolic ne wo k
wi hou addi ional cons ain s is an Elemen a y Flux Mode.
3
2
3
1
E2
E3
E1
flux space
enzyme space
E2
E3
E1
enzyme space
E2
E3
E1
enzyme space
op imum
op imum
op imum
me aboli e
se I
me aboli e
se II
me aboli e
se III
inc easing
enzyme cos
inc easing
enzyme cos
inc easing
enzyme cos
Figu e 7.1: T ansla ion om lux o enzyme space e ains EFMs as ex eme ays – The op le panel shows he easible
lux space wi h he s eady s a e cons ain s, all luxes posi i e (using spli ing o luxes, as explained in he ex , i
necessa y) and a ixed objec i e lux. The ex eme poin s he e a e poin s whe e one lux becomes 0 and a e elemen a y
lux modes (see Chap e 5 in [1]). He e we show ha when we ha e assumed me aboli e concen a ions, such as when
we keep hem a an op imal solu ion, we ge a linea ans o ma ion and he ex eme ays a e main ained. Di e en
me aboli e le els, o example solu ions o di e en en i onmen al condi ions, can lead o di e en ans o ma ions
and he e o e di e en op ima (minimal o al enzyme), bu hose a e always loca ed a an EFM.
P oo . Assume we ha e some op imal s a e whe e he lux dis ibu ion is no an EFM. Any op imal solu ion is
associa ed wi h a se o luxes, enzyme concen a ions and me aboli e concen a ions. Now we se he me aboli e
concen a ions o he concen a ions o he assumed op imal s a e. Then all me aboli e-dependen e ms (κi(s))
become cons an s, and we e u n o a con ex p oblem. As explained in Chap e 10 in [1] and Figu e 7.1, he op imum
o his p oblem (now in e ms o enzyme concen a ions and luxes) is a lux dis ibu ion ha is an EFM. Bu his
con adic s ou ini ial assump ion ha he op imal s a e om which we ook he se o me aboli e concen a ions
was no an EFM. The p oo by con adic ion shows ha he op imal s a e mus be an EFM.
7.3. Enzyme-e icien s a es in an example ne wo k
To illus a e he p oo , we s udy a simple ne wo k ep esen ing g ow h on glucose and py u a e ha we ha e seen
p e iously in Chap e 5 in [1] (Figu e 7.2). We use Gand P o glucose and py u a e in he equa ions, we use he
subsc ip ex when a me aboli e is ex acellula and squa e b acke s o deno e a concen a ion. Fo he use in his
4
Box 7.A Kine ics o he example ne wo k
The de ailed kine ic equa ions o he example model (Figu e 7.2) using he ac o ized a e law (see Equa ion
(7.2) and Chap e s 3 in [1] and 6 in [1]) a e:
0=e0·k+
ca ,0·
[Gex]/KGex
1 + [G]/KG+ [Gex]/KGex
·1−e∆ G00/RT
1=e1·k+
ca ,1·
([G]/KG)([ADP]/KADP)
1 + ([P]/KP)([P]/KP)([ATP]/KATP) + ([G]/KG)([ADP]/KADP)
·1−e∆ G01/RT
2=e2·k+
ca ,2·
[P]/KP
1 + [Pex]/KPex + [P]/Kp
·1−e∆ G02/RT
3=e3·k+
ca ,3·
([P]/KP)([ADP]/KADP)([O2]/KO2)
1 + ([CO2]/KCO2)([ATP]/KATP) + ([P]/KP)([ADP]/KADP)([O2]/KO2)
·1−e∆ G03/RT
4=e4·k+
ca ,4·
[Pex]/KPex
1 + [Pex]/KPex + [P]/KP
·1−e∆ G04/RT
BM =eBM ·k+
ca ,BM ·
([P]/KP)([ATP]/KATP)
1 + ([BM]/KBM)([ADP]/KADP) + ([P]/KP)(ATP/KATP)
·1−e∆ G05/RT
(7.3)
No e ha Pis a p oduc wice in 1, as 1p oduces 2P. No e ha 2and 4a e he same eac ion, bu de ined
in he opposi e di ec ion. The s anda d se o pa ame e s we used o he oy model is all k+
ca ,i = 10 s−1excep
k+
ca ,3= 0.1s−1, all ∆ G0◦
i/RT =−440 and all KM= 1 mM. Fo he ex e nal me aboli es [Pex]=1mM,
[Gex] = 0.05 mM, [O2] = 0.1mM, [BM] = 1 mM and [CO2] = 10 mM unless men ioned o he wise.
chap e , we add enzyme kine ics o his ne wo k. We will use he ac o ized a e law as in Chap e 6 in [1], bu hen
gene alized o nssubs a es and npp oduc s (also compa e Eq. (3.10 in [1]) in Chap e 3 in [1]):
=e·k+
ca ·Qj=1
nssj/KS,j
1 + Qk=1
nppk/KP,k +Qj=1
nssj/KS,j
·1−e∆ G0/RT (7.2)
See Box 7.A o all de ailed a e laws o he example ne wo k. We can simpli y his equa ion by combining he
o wa d ca aly ic cons an , he he modynamic e iciency ac o , he sa u a ion e iciency ac o , and he egula ion
e iciency ac o (i ha exis s) in a unc ion κ(s), which only depends on he me aboli es, and no on he enzyme
concen a ions. We will below w i e κ o κ(s).
i=ei·κi(7.4)
Now we ake BM = 1 and op imize all luxes, enzyme concen a ions and me aboli e concen a ions o minimize he
enzyme cos s (e o =Piei), while sa is ying he cons ain s posed in Equa ions (7.1), o di e en le els o ex e nal
glucose and s anda d le els o he o he ex e nal me aboli es. We see ha o di e en concen a ions o ex e nal
glucose, lead o di e en op imal luxes, enzyme le els and me aboli e le els (Table 7.1).
[Gex]e o 0 1 2 3 4 BM e0e1e2e3e4eBM [G] [P] [ATP] [ADP]
0.01 156.2 5 5 0 9 0 1 54.4 4.4 0 94.4 0 2.9 0.08 15.14 0.05 20.09
0.1 91.3 50 50 99 0 0 1 61.3 11.3 14.2 0 0 4.4 0.13 4.55 0.11 20.09
1 36.2 50 50 99 0 0 1 13.0 8.0 12.5 0 0 2.7 0.60 7.65 0.11 20.09
Table 7.1: Ou comes o he op imiza ion o he example ne wo k wi h s anda d kine ics, pa ame e alues and
ex e nal concen a ions (see Box 7.A) o a ying le els o [Gex].
5
The able shows ha he o al enzyme needed o a biomass lux o one dec eases wi h inc easing glucose le els,
as we expec . In addi ion, he op imal le el o in e nal glucose inc eases wi h inc easing ex e nal glucose. This is
because a highe ex e nal glucose allows o a highe in e nal glucose while s ill main aining a s eady glucose in lux,
and a highe in e nal glucose allows ewe enzymes o d i e u he me abolism. Mo eo e , he luxes o he solu ions
ollow an EFM (see Figu e 7.2b).
We can now e o mula e he p oblem o only he lux and enzyme le els while keeping he me aboli e le els as hey
a e in he able. Wi h he me aboli e le els in he i s ow o he able, we can linea ly ela e he enzyme and lux
le els (wi h he ac o s κi ha ha e become cons an s now we ha e se he in e nal me aboli e concen a ions), and
hus he ex eme ays o he enzyme and lux space will be equal and EFMs, as poin ed ou abo e (see also Chap e
5 in [1] and Figu e 7.1). Op imiza ion in his space will lead o he op imal lux dis ibu ions ollowing an EFM (see
Box 7.B o he de ailed calcula ions). As ixing pa o he op imal solu ion should lead o he same op imal solu ion,
his equi ed he lux dis ibu ion o he op imiza ion o e all a iables o ollow an EFM, as was indeed he case.
We poin ou wo impo an aspec s, using he ne wo k (Figu e 7.2) as an example. Fi s , i is con enien o spli
e e sible eac ions such ha luxes a e always posi i e. In his case, ha means ha he e e sible eac ion om P
o Pex is spli in o he o wa d eac ion 2and he e e se eac ion 4, bo h o which can ha e only posi i e lux.
This spli ing makes su e ha EFMs a e he ex eme ays o he lux space (see Chap e 5 in [1]). This spli ing
is pu ely a ma hema ical con enience; we s ill assume his o be one eac ion in he biological sense, and he e o e
he kine ic equa ions o bo h he o wa d and he backwa d eac ions will be exac ly he same. Depending on in
which di ec ion he lux goes, ei he one o he eac ions will be posi i e and he o he ze o. Any solu ion wi h bo h
eac ions posi i e is in easible, bu minimizing enzyme le els will ne e lead o such a solu ion; he e o e we do no
need o se addi ional cons ains o a oid his. Second, he easibili y o EFMs can depend on ex e nal concen a ions.
In his ne wo k, he biomass eac ion ( BM) is he objec i e lux and he e a e h ee EFMs leading o he p oduc ion
o biomass: EFM1 consis ing o 0, 1, 2and BM, EFM2 consis ing o 0, 1, 3and BM and EFM3 consis ing
o 4, 3and BM. Howe e , i Pex is absen in he en i onmen , he up ake lux 4will always be 0 and he e o e
EFM3 will no be easible.
7.4. Calcula ion o op imal s a es
We can now use he esul ha s a es o maximal enzyme e iciency a e eached a an elemen a y lux mode o
calcula e op imal s a es in a me abolic ne wo k using he ollowing s eps:
1. Enume a e he elemen a y lux modes ha include he objec i e lux
2. Calcula e he minimal enzyme o each EFM scaled o an objec i e lux o 1
3. Compa e he EFMs and selec he one wi h minimal enzyme demands
S ep 1 is possible o ela i ely la ge ne wo ks, al hough usually no o genome scale me abolic ne wo ks. S ep 2 is
a con ex op imiza ion p oblem as we ha e seen in Chap e 6 in [1] and S ep 3 is s aigh o wa d. These h ee s eps
oge he a e called Enzyme Flux Cos Minimiza ion, because i is simila o Enzyme Cos Minimiza ion, bu while
ha is ocused on ixed luxes, Enzyme Flux Cos Minimiza ion simul aneously inds he op imal luxes, enzyme and
me aboli e le els. In his sec ion we will show he me hod on he example ne wo k o Figu e 7.2.
Fi s , we desc ibe he ne wo k wi h he s oichiome ic ma ix (N) and he concen a ion ec o (s):
N=
1−1 0 0 0 0
0 2 −1−1 1 −1
0 2 0 10 0 −100
0−2 0 −10 0 100
,s≡
[G]
[P]
[ATP]
[ADP]
(7.6)
6
Box 7.B Op imal me abolic s a es in he example ne wo k
We minimize he enzyme in es men o BM = 1 wi h [Pex] = 0 (and he e o e 4= 0 and EFM3 is no
easible) o he ne wo k in Figu e 7.2 ( he op imiza ion p oblem in Equa ion (7.1)). Assuming all hi= 1, he
objec i e unc ion P
i=1 hiei=e0+e1+e2+e3+eBM. The cons ain s BM = 1 and e,s≥0in Eq. (7.1)
a e s aigh o wa d. The s eady s a e o all in e nal me aboli es (G, P, ADP and ATP) leads o he ollowing
equali ies ( he s eady s a es o ADP and ATP lead o he same equali y):
S eady s a e ATP =⇒100 BM = 2 1+ 10 3
S eady s a e P =⇒2 1+ 4= 2+ 3+ BM
S eady s a e G =⇒ 0= 1
Subs i u ing BM = 1 and 4= 0 and sol ing his se o linea equa ions, we can w i e all luxes as unc ions
o 2: 0= 1= 5 + 5
11 2and 3= 9 −1
11 2( he e is only one independen lux in his sys em). This means
we can d aw he easible lux space on he 2line and we can exp ess he objec i e unc ion in e ms o 2:
X
i=1
hiei=e0+e1+e2+e3+eBM
= 0/κ0+ 1/κ1+ 2/κ2+ 3/κ3+ BM/κBM
= (5 + 5/11 2)/κ0+(5+5/11 2)/κ1+ 2/κ2+ (9 −1/11 2)/κ3+ 1/κBM
= (5/κ0+ 5/κ1+ 9/κ3+ 1/κBM)
| {z }
α
+ (5/(11κ0)+5/(11κ1)+1/κ2−1/(11κ3))
| {z }
β
2
=α+β 2
(7.5)
The kine ic unc ions (κi) depend on se e al pa ame e s (ex e nal me aboli e le els [Gex],O2,[CO2]and [Pex],
ca aly ic cons an s, Michaelis cons an s and Gibbs ee ene gies) and he a iables [G],[P],[ATP] and [ADP].
Tha means ha once we ha e a se o in e nal me aboli e concen a ions s, he enzyme le els in he objec i e
unc ion can be w i en as a cons an imes he lux: ei= i/κi, wi h κia cons an . Fo a se o pa ame e s,
αand βa e posi i e o nega i e depending on he choice o s. I is clea ha when we minimize his objec i e
unc ion by adjus ing 2, we will always ha e an op imum a 2= 0 (when βis posi i e) o 2= 99 (when βis
nega i e). 2= 99 is he maximum o 2because hen 3= 9 −1
11 2= 0, and highe alues o 2would lead
o nega i e alues o 3.
In conclusion, he op imum canno be a a alue o 0< 2<99. I he e would be an op imum wi h
0< 2<99, we can de e mine sand calcula e whe he β > 0 o ind a lowe objec i e alue a 2= 0 o
2= 99, con adic ing ha we s a ed wi h an op imum. Only i β= 0 he e is a ange o op ima, bu his
equi es e y p ecise pa ame e alues. 2= 0 and 2= 99 co espond o EFMs o his ne wo k (Figu e 7.2).
And wi h he s oichiome ic ma ix we can desc ibe he s eady s a e cons ain s:
d
d s=N =
1−1 0 0 0 0
0 2 −1−1 1 −1
0 2 0 10 0 −100
0−2 0 −10 0 100
0
1
2
3
4
BM
=
0
0
0
0
(7.7)
7
(A) (B)
biomass
glucose
O2
py u a e py u a eex
CO2,ex
2 ADP
2 ATP
100 ATP
100 ADP
10 ATP 10 ADP
0
3
1
BM
4
2
glucoseex
O2,ex
CO2
EFMs
glucose
espi a ion
glucose
e men a ion
py u a e
espi a ion
(C) (D)
10−210−1100101
0
200
400
600
800
1,000
de aul alue
[Gex]
e o
glc. esp.
glc. e m.
py . esp.
10−210−1100101
10−3
10−2
10−1
100
101
[Gex]
Speci ic lux
0
2
3
4
BM
Figu e 7.2: S a es o maximal e iciency in an example model – (A) Example ne wo k om Chap e 5 in [1] wi h
added s oichiome y. (B) Th ee elemen a y lux modes o his ne wo k. (C) Calcula ed enzyme in es men needed
o a biomass lux o 1. A a e y low concen a ion o ex acellula glucose ([Gex]), EFM3 has he lowes cos . Bu
as we mo e along he x-axis, a a ound [Gex] = 0.02 he e is a swi ch o EFM1 and la e , a a ound [Gex] = 0.07,
EFM2 becomes he one wi h he lowes cos . (D) Speci ic luxes ( lux di ided by o al enzyme) associa ed wi h he
op imal EFM o di e en le els o Gex. No e ha 1is no shown as i is always equal o 0. The a es show a
discon inui y when he e is a swi ch om one op imal EFM o ano he .
Now we ind he EFMs ( o example wi h EFM ool [6]). I can easily be checked ha he ollowing EFMs (deno ed
by ec o s (i)) a e in he nullspace o he s oichiome ic ma ix:
(1) =
5
5
0
9
0
1
, (2) =
50
50
99
0
0
1
, (3) =
0
0
0
10
11
1
(7.8)
8
(A) (B)
Fixed biomass ac ion
Me abolic ac ion
Ribosomal ac ion
G ow h a e
F ac ion o biomass
Biomass a e pe enzyme demand
G ow h a e
Figu e 7.3: T ansla ion o enzyme-speci ic biomass a e o g ow h a e – (A) Bo h om expe imen al da a and
a cell-op imiza ion poin o iew, he ibosomal ac ion o he p o eome inc eases wi h he g ow h a e, while he
me abolic ac ion dec eases. (B) This leads o a hype bolic dependency o he g ow h a e on he biomass p oduc ion
a e pe amoun o enzymes.
The nex s ep is o pe o m he con ex op imiza ion o e he me aboli e le els o each one o he h ee EFMs.
The e o e, we exp ess he enzyme le els as a a io o he lux and he unc ion (s), using Equa ion 7.4. Summing
o e all enzymes, we ge a unc ion o he o al enzyme cos (le el) as a unc ion o luxes, me aboli e concen a ions
and pa ame e s:
e o =X
i
ei=X
i
i
κi(s).(7.9)
We use he s anda d pa ame e s (Box 7.A) and eplace iby he alues gi en by each EFM. We a e hen le wi h
a con ex op imiza ion o e he me aboli e le els, an Enzyme Cos Minimiza ion p oblem as in Chap e 6 in [1]. Fo
[Gex] = 0.05 we ob ain a o al enzyme o 111.1 o EFM1, o 146.3 o EFM2 and 136.5 o EFM3. Tha means
ha o hese condi ions we will conclude ha EFM1 is op imal. F om he op imiza ion we ob ain he me aboli e
concen a ions: [G] = 0.08,[P] = 3.93,[ATP] = 0.11 and [ADP] = 20.09 ( ha he in e nal glucose concen a ion
is highe han he ex e nal is because we desc ibed he anspo wi h egula enzyme kine ics ins ead o anspo e
enzyme kine ics, which would ha e been mo e ealis ic). We can nex use he a e equa ions o calcula e he enzyme
le els om he luxes and me aboli e le els, using he alues o he pa ame e s and ex e nal concen a ions.
We can epea his p ocedu e o di e en le els o ex e nal concen a ions and see ha he op imal EFM can change
depending on he ex e nal concen a ion (Figu e 7.2c). When he op imum shi s o using a di e en EFM, he e is a
discon inui y in he luxes a he ex e nal me aboli e concen a ion (Figu e 7.2d). Many cells show shi s in me abolic
s a egies depending on he ex e nal condi ions and Enzyme Flux Cos Minimiza ion is one way o explaining hose
shi s.
Abo e, Enzyme Cos Flux Minimiza ion was used o ind he me abolic s a e wi h he maximum enzyme e iciency.
Al hough in ou calcula ion we ob ain he enzyme concen a ions las , i is by enzyme concen a ions ha cells
ac ually con ol me abolism. I cells p oduce enzymes a he concen a ions we calcula ed and each a s eady s a e,
his s a e will ealize he luxes and me aboli e le els ha lead o ou op imal s a e.
7.5. T ansla ing enzyme e iciency in o cell g ow h a e
In he sec ion abo e, we lea ned how o op imize me abolic s a es o a maximal o e all enzyme e iciency. Why is
his quan i y ele an ? One eason is ha o e all enzyme e iciency, acco ding o some simple easoning, de e mines
he cell’s g ow h a e. I mic obes compe e by g owing as , hei i ness is la gely de e mined by hei momen a y
g ow h a e in hei espec i e en i onmen . In such en i onmen s, he biomass/enzyme e iciency will be unde
selec ion, which makes i one o he impo an objec i e unc ions in his book. I highe enzyme e iciency means
highe g ow h a e, and i we ha e a con e sion o mula o his, we can plo he g ow h a e o he di e en EFMs
ins ead o o e all enzyme e iciency.
15
Me aboli e name Biomass s oichiome ic coe icien
AcCoA -41
ADP 547
2-oxoglu a a e -14
ATP -547
H2O -547
Pi547
CO22
CoA 41
DHAP -5
G6P -4
NAD+178
NADH -178
NH3-139
2-oxoglu a a eAce a e -24
PEP -32
Py u a e -38
E4P -5
R5P -13
Table 7.2: S oichiome y o biomass eac ion – R70
Reac ion ID EC numbe Reac ion name Fo mula
R1 2.7.1.69 p s Glucose + PEP G6P + Py u a e
R2 5.3.1.9 pgi G6P F6P
R3 2.7.1.11 p k F6P + ATP FBP + ADP
R4 3.1.3.11 bp FBP + H2OF6P + Pi
R5 4.1.2.13 ald FBP DHAP + G3P
R6 5.3.1.1 im G3P DHAP
R7 a 1.2.1.12 gap G3P + NAD++ PiBPG + NADH
R7 b 2.7.2.3 pgk BPG + ADP 3PG + ATP
R7 c 5.4.2.11 / 5.4.2.12 pgm 3PG 2PG
R8 4.2.1.11 pgh 2PG PEP
R9 2.7.1.40 pyk PEP + ADP Py u a e + ATP
RR9 2.7.9.2 pps Py u a e + 2 ATP PEP + 2 ADP + Pi
Table 7.3: Glycolysis
Reac ion ID EC numbe Reac ion name Fo mula
R10a 1.1.1.49 zw G6P + NAD+6PGL + NADH
R10b 3.1.1.31 glh 6PGL 6PGC
R10c 1.1.1.44 pgd 6PGC + NAD+NADH + CO2+ Ru5P
R11 5.1.3.1 pe Ru5P X5P
R12 5.3.1.6 pi Ru5P R5P
R13 2.2.1.1 x 1 R5P + X5P S7P + G3P
R14 2.2.1.2 al G3P + S7P E4P + F6P
R15 2.2.1.1 x 2 E4P + X5P G3P + F6P
R60 4.2.1.12 edd 6PGC KDPG
R61 4.1.2.14 eda KDPG G3P + Py u a e
Table 7.4: Pen ose Phospha e Pa hway
Reac ion ID EC numbe Reac ion name Fo mula
R20 2.3.1.54 p l Py u a e + CoA AcCoA + Fo ma e
R21 1.2.4.1 / 2.3.1.12 pdh Py u a e + NAD++ CoA AcCoA + CO2+ NADH
R22 2.3.3.1 csn 2-oxoglu a a eace a e + AcCoA Ci a e + CoA
R23 4.2.1.3 acn Ci a e iso-Ci a e
R24 1.1.1.41 icd iso-Ci a e + NAD+2-oxoglu a a e + NADH + CO2
R25 1.2.4.2 kgd 2-oxoglu a a e + NAD++ CoA NADH + Succina einyl-CoA + CO2
R26 6.2.1.5 scs Succina einyl-CoA + ADP + PiSuccina eina e + ATP + CoA
R27 1.3.5.1 sdh Succina eina e + ADP + O2[e] + PiFuma a e + ATP
R27b 1.3.5.4 d Fuma a e + NADH Succina eina e + NAD+
R28 4.2.1.2 um Fuma a e Mala e
R29 1.1.1.37 mdh Mala e + NAD+2-oxoglu a a eace a e + NADH
Table 7.5: TCA Cycle
16
Reac ion ID EC numbe Reac ion name Fo mula
R40 4.1.1.31 ppc PEP + CO22-oxoglu a a eace a e + Pi
R41 1.1.1.38 me Mala e + NAD+Py u a e + NADH + CO2
R42 4.1.1.49 ppck 2-oxoglu a a eace a e + ATP PEP + ADP + CO2
Table 7.6: Anaple o ic Reac ions
Reac ion ID EC numbe Reac ion name Fo mula
R53 1.1.1.27 ldh Py u a e + NADH Lac a e + NAD+
R54 a 1.2.1.10 ada AcCoA + NADH Ace aldehyde + NAD++ CoA
R54 b 1.1.1.1 adh Ace aldehyde + NADH ETOH + NAD+
R55a 2.3.1.8 p a AcCoA + PiAce yl-P + CoA
R55b 2.7.2.1 ack Ace yl-P + ADP Ace a e + ATP
Table 7.7: Redox-associa ed eac ions
Reac ion ID Reac ion name Fo mula
R80 oxphos NADH + 2 ADP + 0.5 O2[e] + 2 PiNAD++ 2 ATP + 3 H2O
R82 a pmain ATP + H2OADP + Pi+ ATPmain
Table 7.8: Oxida i e phospho yla ion
Reac ion ID Reac ion name Fo mula
R90 exe oh ETOH ETOH[e]
R91 exace Ace a e Ace a e[e]
R93 exNH3NH3[e] NH3
R94 exlac Lac a e Lac a e[e]
R95 exsuc Succina eina e Succina eina e[e]
R96 ex o Fo ma e Fo ma e[e]
R97 exCO2CO2CO2[e]
Table 7.9: Memb ane T anspo Reac ions
17
Reac ion ID k+
ca [1/s] Keq [uni less] Enzyme molecula weigh [Da]
R1 100 N/A 2.6·105
R10a 240 N/A 5.6·104
R10b 410 N/A 3.6·104
R10c 110 N/A 1.0·105
R11 130 2.3 2.5·104
R12 1400 2.3 1.9·104
R13 46 3.7 7.3·104
R14 17 0.9 3.5·104
R15 75 38 7.3·104
R20 4800 N/A 8.5·104
R21 38 N/A 2.8·105
R22 360 N/A 9.6·104
R23 33 0.074 9.6·104
R24 110 N/A 4.6·104
R25 150 N/A 1.2·106
R26 89 0.52 7.1·104
R27 78 N/A 7.9·105
R27b 180 N/A 1.8·105
R28 280 4.7 6.0·104
R29 210 6.1·10−53.2·104
R2 320 0.51 6.2·104
R3 110 N/A 1.4·105
R4 25 N/A 3.7·104
R40 120 N/A 2.0·105
R41 76 N/A 6.3·104
R42 51 N/A 6.0·104
R53 140 2.1·1043.7·104
R54 a 0.35 2.3·10−39.6·104
R54 b 320 2.8·1039.6·104
R55a 91 N/A 7.7·104
R55b 59 N/A 4.3·104
R5 8.0 3.0·10−43.9·104
R60 250 N/A 6.5·104
R61 80 9.6·10−32.2·104
R6 7800 11 5.4·104
R70 99 N/A 6.0·104
R7 a 230 0.088 3.6·104
R7 b 390 730 4.1·104
R7 c 53 0.16 2.9·104
R80 4.0·106N/A 9.1·105
R82 180 N/A 6.0·104
R8 210 3.5 4.6·104
R9 510 N/A 5.0·104
R90 100 N/A N/A
R91 100 N/A 5.9·104
R93 100 N/A 4.5·104
R94 100 N/A 5.9·104
R95 100 N/A 4.5·104
R96 100 N/A 3.1·104
R97 100 N/A N/A
RR9 13 N/A 8.7·104
Table 7.10: Kine ic pa ame e s associa ed wi h eac ions
18
Reac ion ID Me aboli e name KM[mM]
R1 G6P 0.102
R1 Glucose 0.116
R1 PEP 0.0983
R1 Py u a e 0.102
R10a G6P 0.314
R10a 6PGL 0.129
R10a NAD+0.863
R10a NADH 0.129
R10b 6PGL 0.168
R10b 6PGC 0.0594
R10c CO20.0626
R10c 6PGC 0.101
R10c Ru5P 0.0626
R10c NAD+0.0591
R10c NADH 0.0626
R11 Ru5P 0.0878
R11 X5P 0.114
R12 R5P 1.25
R12 Ru5P 0.558
R13 G3P 1.23
R13 R5P 0.972
R13 S7P 2.11
R13 X5P 0.157
R14 E4P 0.175
R14 F6P 0.888
R14 G3P 0.578
R14 S7P 0.206
R15 E4P 0.0934
R15 F6P 0.737
R15 G3P 1.27
R15 X5P 0.152
R20 AcCoA 0.0352
R20 CoA 0.0168
R20 Fo ma e 6.35
R20 Py u a e 2.18
R21 AcCoA 0.159
R21 CO20.159
R21 CoA 0.0629
R21 Py u a e 0.291
R21 NAD+0.0629
R21 NADH 0.159
R22 AcCoA 0.0867
R22 Ci a e 0.0756
R22 CoA 0.0756
R22 2-oxoglu a a e 0.0287
R23 Ci a e 3.49
R23 iso-Ci a e 2.42
R24 2-oxoglu a a e 0.483
R24 CO22.02
R24 iso-Ci a e 0.0227
Reac ion ID Me aboli e name KM[mM]
R24 NAD+1.06
R24 NADH 0.0119
R25 2-oxoglu a a e 0.0670
R25 CO20.108
R25 CoA 0.0927
R25 Succinyl-CoA 0.108
R25 NAD+0.0927
R25 NADH 0.108
R26 CoA 0.00731
R26 Succina e 0.237
R26 Succinyl-CoA 0.0105
R26 ADP 0.0560
R26 ATP 0.0812
R27 Fuma a e 0.0812
R27 O2[e] 0.371
R27 Succina e 0.0756
R27 ADP 0.371
R27 ATP 0.0270
R27b Fuma a e 0.0201
R27b Succina e 0.205
R27b NAD+0.0431
R27b NADH 0.232
R28 Fuma a e 0.314
R28 Mala e 0.615
R29 Mala e 3.19
R29 2-oxoglu a a e 0.0283
R29 NAD+0.460
R29 NADH 0.0321
R2 F6P 0.162
R2 G6P 0.273
R3 F6P 0.116
R3 FBP 0.113
R3 ADP 0.113
R3 ATP 0.141
R4 F6P 0.171
R4 FBP 0.0161
R40 CO20.115
R40 2-oxoglu a a e 0.0426
R40 PEP 0.364
R41 CO20.0885
R41 Mala e 0.361
R41 Py u a e 0.0885
R41 NAD+0.0691
R41 NADH 0.0885
R42 CO25.21
R42 2-oxoglu a a e 0.571
R42 PEP 0.0643
R42 ADP 0.0484
R42 ATP 0.0750
R53 LACTATE 0.517
Michaelis cons an s – pa I
19
Reac ion ID Me aboli e name KM[mM]
R53 Py u a e 0.0193
R53 NAD+0.517
R53 NADH 0.0193
R54 a AcCoA 0.0242
R54 a Ace aldehyde 1.80
R54 a CoA 0.00786
R54 a NAD+0.0415
R54 a NADH 0.113
R54 b Ace aldehyde 0.0593
R54 b ETOH 5.49
R54 b NAD+0.169
R54 b NADH 0.0593
R55a AcCoA 0.0424
R55a Ace yl-P 0.313
R55a CoA 0.0860
R55b Ace a e 3.44
R55b Ace yl-P 0.154
R55b ADP 0.402
R55b ATP 0.0714
R5 DHAP 0.0782
R5 FBP 0.204
R5 G3P 0.0782
R60 6PGC 0.0434
R60 KDPG 0.150
R61 G3P 0.00146
R61 KDPG 0.561
R61 Py u a e 0.00146
R6 DHAP 0.0750
R6 G3P 0.745
R70 AcCoA 0.462
R70 2-oxoglu a a e 0.352
R70 BIOMASS 0.0998
R70 CO20.0996
R70 CoA 0.891
R70 E4P 0.0144
R70 G6P 4.31
R70 NH30.0151
R70 2-oxoglu a a e 0.00672
R70 PEP 0.169
R70 Py u a e 0.319
R70 R5P 0.881
R70 ADP 0.0293
R70 ATP 0.342
R70 NAD+1.43
Reac ion ID Me aboli e name KM[mM]
R70 NADH 0.0913
R7 a DPG 0.0576
R7 a G3P 0.687
R7 a NAD+0.0558
R7 a NADH 0.0576
R7 b DPG 0.0426
R7 b 3PG 0.235
R7 b ADP 0.0426
R7 b ATP 0.235
R7 c 3PG 0.132
R7 c 2PG 0.0755
R80 O2[e] 0.116
R80 ADP 0.136
R80 ATP 0.0737
R80 NAD+0.0859
R80 NADH 0.116
R82 ATPmain 0.130
R82 ADP 0.130
R82 ATP 0.0769
R8 PEP 0.131
R8 2PG 0.108
R9 PEP 0.291
R9 Py u a e 0.0476
R9 ADP 0.218
R9 ATP 8.45
R90 ETOH 0.100
R90 ETOH[e] 0.100
R91 Ace a e 0.100
R91 Ace a e[e] 0.100
R93 NH30.0999
R93 NH3[e] 0.100
R94 Lac a e 0.100
R94 Lac a e[e] 0.100
R95 Succina e 0.100
R95 Succina e[e] 0.100
R96 Fo ma e 0.0999
R96 Fo ma e[e] 0.100
R97 CO20.0999
R97 CO2[e] 0.100
RR9 PEP 0.0934
RR9 Py u a e 0.0864
RR9 ADP 0.0873
RR9 ATP 0.0350
Michaelis cons an s – pa II
20
(A) (B)
Figu e 7.9: Me abolic s a egies in he E. coli model, depending on ex e nal glucose and oxygen concen a ions,
In he EFM pahse diag am. Each egion ep esen s he winning EFM as explained in Figu e 7.8. He e, he colo s
ep esen he lux in one speci ic eac ion based on he winning EFM in ha egion. (A) The lac a e sec e ion lux is
s ikingly equal o 0in mos egions. The only condi ions whe e lac a e is sec e ed is a low oxygen and medium/high
glucose concen a ions. (B) The biomass yield is, in gene al, high i and only i lac a e is no sec e ed. This makes
sense because he ca bon coming om he glucose is o en he limi ing nu ien o g ow h, and he e is a ade-o
be ween using i o biomass e sus e men a ion p oduc s such as lac a e. In e es ingly, he egion wi h high glucose
and high oxygen le els (uppe igh quad an ) is occupied by an EFM ha doesn’ achie e he highes possible yield
(i.e. max-g ). In low glucose and high oxygen, o in medium oxygen le els, he winning EFMs a e he ones wi h
ela i ely highe biomass yields.
7.9. Mo e esul s o he E. coli cen al me abolism model
This sec ion con ains addi ional esul s o he E. coli cen al me abolism model om Chap e 7 in [1], in pa icula ,
luxes plo ed in he EFM phase diag am and in lux space, as well as ideal and eal enzyme cos s o all EFMs.
Each poin in he Monod landscape in Figu e 7.8 (A) co esponds o a s a e o he model, and he calcula ions ha lead
o he g ow h a e and he "winning EFM" shown yield a ull desc ip ion o his s a e, including all luxes, me aboli e
concen a ions and enzyme le els. These da a can be explo ed and isualized in many ways. Fo illus a ion, Figu e
7.10 shows a a ian o Figu e 7.8 (A) in he ho izon al axes do no desc ibe he ex e nal concen a ions o glucose
and oxygen, bu hei up ake a es, and he e ical axes shows he biomass p oduc ion a e. Since up ake a es a e a
unc ion o ex e nal concen a ions, and he g ow h a e di ec ly depends on he biomass p oduc ion a e, we migh
ha e expec ed ha his yields he same pic u e, jus a bi s e ched along each o he axes. Howe e , he pic u e
looks e y di e en : ins ead o o ming a con inuous su ace, he poin s now all on disconnec ed ays, appa en ly
wi h one ay o each colo ed egion o he su ace. In ac , when looking a he pic u e closely, we can see ha each
o he winning EFMs gi es ise o exac ly one ay. Bu his, a e all, is logical. In he new plo , all axes e e o
eac ion a es, and o each EFM all a es come in ixed a ios, gi ing ise o a ay. So, i ou solu ions a e EFMs, his
pic u e canno be con inuous – in line wi h he ac ha , in he o iginal Monod landscape 7.8 (A), when mo ing om
one egion o he o he one, one would no ice a disc e e jump o he eac ion a es. Bu why is he new pic u e no a
con inuos su ace, i up ake a es depends smoo hly on ex e nal me aboli e concen a ions? In ac , hey do no only
depend on hese concen a ions, bu also on esou ce alloca ion o he anspo e . I his alloca ion shows a disc e e
jump (again, when mo ing om one egion o ano he one), hen also he a e shows a jump. The compa ison
be ween he wo plo s shows us wha we gain by conside ing enzyme kine ics as compa ed o a pu e s oichiome ic
model. Wi h he biomass a e as a p oxy o cell g ow h, each EFM de ines ixed a ios be ween his g ow h a e and
each o he me abolic luxes, including he up ake a es. When con inuously scaling an EFM, he glucose up ake,
21
Figu e 7.10: P opo ional scaling o luxes wi hin each EFM – In he diag am wi h glucose up ake, oxygen up ake, and
biomass p oduc ion a e on he axes, each colo ed line co esponds o one EFM, and shows he possible combina ions
o luxes ob ained om he model behind Figu e 7.8 (which also sha es he EFM colo s wi h his igu e). Impo an ly,
he e he xand yaxes ep esen up ake a es and no subs a e concen a ions. The e o e, as expec ed, each EFM
yields a s aigh line (because o he p opo ional scaling o di e en luxes o each EFM). Since – acco ding o ou
easoning – op imal lux dis ibu ions mus be EFMs, only hese combina ions o luxes a e ac ually possible. When
glucose and oxygen concen a ions a e a ied smoo hly in Figu e 7.8, he co esponding mo emen in his plo would
be along he lines and some imes, jumps be ween di e en lines (when he sys em mo es om one egion o ano he
one in Figu e 7.8).
oxygen up ake, and biomass p oduc ion will scale p opo ionally. So he ays in Figu e 7.10 e lec wha we can know
abou possible me abolic luxes based on ne wo k s uc u e alone; bu o ge o he Monod landscape, as a unc ion
o concen a ions, we had o use kine ic in o ma ion and a ex a p inciple o economical enzyme usage.
The phase diag am o “winning EFMs” can also be used o isualize o he (op imized) quan i ies as unc ions o
glucose and oxygen concen a ions. Figu e 7.9 shows as example he (biomass-speci ic) lac a e sec e ion (showing
ha also a numbe o o he winning EFMs, apa om ana-lac, sec e e lac a e) and he biomass yield on glucose.
Finally, a s a is ics o e all EFMs shows ha he ange o possible enzyme demands pe biomass p oduc ion a e
is qui e la ge: as shown in Figu e 7.11, hey a y o e mo e han wo o de s o magni ude, making some EFMs a
hund ed- old mo e enzyme-expensi e han o he s. The same plo also shows how enzyme cos s depend on he ac
ha enzymes do no ope a e a hei ull capaci y ( eaching hei kca alue), bu a bes a he enzyme e iciencies
p edic ed by enzyme cos minimiza ion. Fo ou E. coli model and he ae obic glucose condi ions s udied, i all
enzymes could ope a e a hei kca alues, his would dec ease o o e all enzyme demand by a ac o o a leas 1.4,
o maximally 4.7, depending on he EFM in ques ion. Bu s ill, in his case, o de e mining enzyme cos s he choice
o he igh EFM (e en assuming "ideal" enzymes) is much mo e impo an han conside ing he ac ual, "non-ideal"
way in which enzymes ope a e. Bu his may no always hold: unde low-oxygen condi ions, he enzyme demands o
some EFMs may inc ease much mo e d as ically.
Solu ions o p oblems
P oblem 7.1 (E ec o oxygen concen a ion)
The oxygen concen a ion a ec s only he a e 3, and an inc ease in oxygen inc eases his a e o he same enzyme
concen a ion. Since EFM2 does no con ain 3, he enzyme cos o his EFM will no change. EFM1 and EFM3
22
10−210−1100101
10−2
10−1
100
101
y = 4.7 x
y = 1.4 x
min. ideal cos = 0.039 [h−1]
min. eal cos = 0.083 [h−1]
ideal cos [g enz / g dw h−1]
eal cos [g enz / g dw h−1]
Figu e 7.11: Ideal and eal enzyme cos s o elemen a y lux modes – Fo each EFM (shown as a cyan do ), he ideal
enzyme cos pe biomass p oduc ion a e (i.e. assuming ha all he enzymes a e sa u a ed) is compa ed o he ac ual
cos (calcula ed using Enzyme Cos Minimiza ion, assuming s anda d ae obic glucose condi ions). The cos s span a
wide ange om he mos enzyme-e icien EFMs on he lowe le o he leas enzyme-e icien ones on he uppe
igh . Fo di e en EFMs, he a io o ac ual and ideal cos s a ies be ween 1.4 and 4.7. He e, he EFM wi h he
minimal ac ual cos is among he op 5 in e ms o ideal cos .
bene i om an inc ease o he oxygen concen a ion, because hey will ha e o in es less enzyme in 3 o ob ain
he same a e. The e o e, hose EFMs can become mo e bene icial and he op imal EFM could shi o one o hose
EFMs.
P oblem 7.2 (E ec o ex e nal me aboli es)
Inc easing [Pex]will bene i EFM3 and dec ease he bene i o EFM2 (because o EFM2 [Pex]will inhibi eac ion
2and he e o e mo e enzyme is needed o eac ion 2and he enzyme cos o EFM2 will inc ease. Quali a i ely,
wi h inc easing [Pex], EFM2 migh become mo e expensi e, and ei he EFM1 o EFM3 will become bene icial, o
bo h a di e en Pex concen a ions, depending on he kine ics o he eac ions.
P oblem 7.3 (S a es o maximal g ow h a e)
(a) 1=e1(k+
1s1−k−
1X), 2=e2(k+
2s2−k−
2x)and 3=e3(k+
3x−k−
3p)
(b) e o = 1
(2s1−x)+ 2
(30−x)+ 3
x
(c) When e1= 0, 1is also 0 and o achie e s eady s a e 2= 3and using he a e equa ions and illing in he
pa ame e s we ge e o = 3
(30−x)+ 3
x. We now se e o = 1 o ob ain 3= (1− 3
30−x)xwhich we can ew i e o
3=x
1+ x
30−x. We can ake he de i a i e o xand se i equal o 0 o ind he op imum, which leads o x= 15
and 3/e o =15
2. No e ha in his speci ic case we did no need o se e o = 1 and could ha e maximized
3/e o di ec ly, bu in gene al his does no always wo k. In he es o he answe s we assume we se e o = 1
and he e o e 3= 3/e o .
(d) e2= 0 implies 2= 0 and illing in s1= 10 leads o 3= (1− 3
20−x)x, which can be ew i en o 3=x
1+ x
20−x.
This is op imal when x= 10 and 3/e o = 5.
(e) e1=e2implies 1
20−x= 2
30−x. F om he s eady s a e we know ha 2= 3− 1. Filling his in and sol ing o
1leads o 1= 3x−20
x−50 . F om he o al enzyme and by eplacing e1by e2we ge e3= 1 −2 1
20−x. Pu ing
his in he equa ion o 3and using he p e ious equali y o eplace 1leads o: 3= (1 + 3
x−25 )x. Sol ing
his o 3gi es 3=x−x2
25 , which is op imal o x= 12.5wi h 3/e o =25
4.
( ) I was op imal o in es all enzyme in e2and none in e1, because ha lead o he highes speci ic lux 3(namely
3= 7.5).
(g) Since 1= 0,S1is no in ol ed in any eac ion and he solu ion is he same as abo e, x= 15 and 3/e o =15
2.
23
(h) Simila calcula ions as abo e bu now wi h s1= 50 lead o 3=x
1+ x
100−x, which is op imal when x= 50 and
gi es 3/e o = 25.
(i) Simila calcula ions as abo e bu now wi h s1= 50 lead o 3=x−x2
65 . 3/e o is maximal a x= 32.5and
akes he alue 16.25.
(j) Now s1inc eased he op imal s a egy would be o in es all enzymes in e1, and ha e e2= 0, because ha
leads o he highes speci ic lux o 3, namely 3/e o = 25.
(k) The wo EFMs in his pa hway ha p oduce Pa e 1= 3wi h 2= 0 and 2= 3wi h 1= 0. In he p oblem
we saw when we op imize he speci ic lux, we always ob ained one o hose EFMs as he bes solu ion, om
he op ions ha we es ed. The e o e, hese esul s a e in ag eemen wi h he p oo ou lined in his chap e .
Bibliog aphy
[1] The Economic Cell Collec i e, edi o . Economic P inciples in Cell Biology. F ee online book, 2023. doi: 10.5281/
zenodo.8156386.
[2] A i Flamholz, Elad Noo , A en Ba -E en, Wol am Liebe meis e , and Ron Milo. Glycoly ic s a egy as a adeo
be ween ene gy yield and p o ein cos . P oc. Na l. Acad. Sci. U. S. A., 110(24):10039–10044, June 2013. doi:
10.1073/pnas.1215283110.
[3] Elad Noo , A i Flamholz, A en Ba -E en, Dan Da idi, Ron Milo, and Wol am Liebe meis e . The p o ein cos
o me abolic luxes: P edic ion om enzyma ic a e laws and cos minimiza ion. PLoS Compu . Biol., 12(11):
e1005167, No embe 2016. doi: 10.1371/jou nal.pcbi.1005167.
[4] M.T. Wo el, H. Pe e s, J. Hulsho , B. Teusink, and F.J. B uggeman. Me abolic s a es wi h maximal speci ic
a e ca y lux h ough an elemen a y lux mode. FEBS Jou nal, 281(6):1547–1555, 2014.
[5] S e an Mülle , Geo g Regensbu ge , and Ral S eue . Enzyme alloca ion p oblems in kine ic me abolic ne wo ks:
Op imal solu ions a e elemen a y lux modes. Jou nal o Theo e ical Biology, 347:182–190, 2014.
[6] Ma co Te ze and Jö g S elling. La ge-scale compu a ion o elemen a y lux modes wi h bi pa e n ees.
Bioin o ma ics, 24(19):2229–2235, 2008. ISSN 1367-4803. doi: 10.1093/bioin o ma ics/b n401.
[7] Ma hew Sco , Ca l W Gunde son, Edua d M Ma eescu, Zhongge Zhang, and Te ence Hwa. In e dependence o
cell g ow h and gene exp ession: o igins and consequences. Science, 330(6007):1099–1102, 2010. doi: 10.1126/
science.1192588.
[8] Meike T Wo el, Elad Noo , Michael Fe is, F ank J B uggeman, and Wol am Liebe meis e . Me abolic enzyme
cos explains a iable ade-o s be ween mic obial g ow h a e and yield. PLoS Compu . Biol., 14(2):e1006010,
Feb ua y 2018. doi: 10.1371/jou nal.pcbi.1006010.
