constrained-allocation fba: modeling the interplay of...

Constrained-allocation FBA

Constrained-allocation FBA: modeling the interplay of regulation andmetabolism in bacteria at genome scale

Andrea De Martino

IIT Center for Life NanoScienceCNR and Dipartimento di Fisica, Sapienza Università di Roma

[email protected]

Joint work withI M Mori (Sapienza)I E Marinari (Sapienza)I T Hwa (UCSD)

See also Matteo’s POSTER

ReferenceI submitted (2014)


Growth laws

Bacterial growth laws

I Quantitative empirical relationships linkingcellular composition to growth rate[Schaechter et al., Bremer & Dennis, etc.]

I Probed at higher and higher resolutionI Proteome organization is actively regulated

9

FIGURES AND FIGURE CAPTIONS

Fig. 1: Correlation of the RNA/protein ratio (r) with growth rate (!!!!) for various strains of Escherichia coli. A, Comparison among E. coli strains grown in minimal

medium: Strain B/r (Ref. (10) , square), 15"-bar (Ref. (13), diamond), and EQ2 (this work, filled circles). The growth rate is modulated by changing the quality of nutrients as indicated in the legend. The fraction of total protein devoted to ribosome-affiliated

proteins ( R# ) is given by the RNA/protein ratio as R r# $= % ; see Table S1. B, The

RNA/protein ratio for a family of translational mutants SmR (up-triangle) and SmP (down-triangle) and their parent strain Xac (circle) (25), grown with various nutrients (see legend and Table S2). Translational inhibition of the parent Xac strain via exposure to sub-lethal doses of chloramphenicol (circled numbers; see legend table) gave similar RNA/protein ratio as those of the mutant strains grown in medium with the same nutrient, but without chloramphenicol (light blue symbols). Dashed line is a fit to Eq. [2]. Inset:

Linear correlation of &t values obtained for the Xac, SmR and SmP strains (Table S2) with the measured translation rate of the respective strains (14). (r2=0.99).

[Scott et al. Science 2010]

FIGURE 2 Amounts and synthesis rates of molecular components in bacteria growing exponentially atrates between 0.6 and 2.5 doublings per h. The values of the RNA-to-protein (R/P; panel a) and DNA-to-protein (G/P; panel b) ratios were calculated from lines 1, 2, and 3 in Table 2. The ribosome efficiency(i.e., the protein synthesis rate per average ribosome; panel c, left ordinate) was calculated from the numberof ribosomes per cell (line 15, Table 3) and the rate of protein synthesis per cell. The latter was obtainedfrom the amount of protein per cell (line 10, Table 2) using the first-order rate equation. The peptide chainelongation rates (panel c, right ordinate) are 1.25-fold higher than the ribosome efficiency values andaccount for the fact that only about 80% of the ribosomes are active at any instant. The fraction of the totalRNA synthesis rate that is stable RNA or mRNA (rs or rm; panel d) is from line 5, Table 3. The rates ofstable RNA and mRNA synthesis per amount of protein (rs/P or rm/P; panel e) were calculated from lines 9and 10, Table 3, divided by the amount of protein per cell (line 10, Table 2). The ppGpp per protein value(ppGpp/P; panel f) is from line 11, Table 3. The cell age at which chromosome replication is initiated atoriC (ai in fractions of a generation; panel g) is calculated from C and D (lines 23 and 24, Table 3) andequation 14 in Table 5. The protein (or mass) per cell at replication initiation (panel h) was calculated fromthe initiation age (ai, panel g) and the cell mass immediately after cell division (age zero; i.e., a = 0), usingequation 17 in Table 5. The latter was obtained from the average protein or mass content of cells (lines 10or 13, respectively, Table 2), using equation 16 of Table 5. The number of replication origins at the time ofreplication initiation (Oi, panel i) was obtained from the values of C and D (Table 3), using equation 15 ofTable 5. The initiation mass (panel j), given as protein (or mass) per replication origin at the time ofreplication initiation, was obtained as the quotient of the values for Pi (or Mi) and Oi shown in panels h andi.

RNA Polymerase Synthesis and Function.

RNA polymerase concentration. The instantaneous rate of transcription in the cell depends on theconcentration of RNA polymerase, αp, measured as the fraction of total protein that is RNA polymerasecore enzyme (three subunits, α2, β, and β′). The values of αp increase with the growth rate and reflect the control ofthe synthesis of the β and β′ subunits of RNA polymerase. Since the α subunit is in excess in E. coli (see Table 4),the amount of core enzyme would seem to be limited by the amount of β and β′ subunit polypeptides. Thesynthesis of these subunits is under dual transcriptional control (i) at the level of initiation at an upstream promoterand (ii) at the level of termination-antitermination at an attenuator in front of the rpoB gene (6, 7, 40–42, 52). Bothcontrols are growth rate dependent, but the mechanisms mediating these controls are poorly understood. The

[Bremer & Dennis 1996]

[Teixeira & Neijssel, J Biotech 1997]


Growth laws




9








[Klumpp & Hwa, Curr Op Biotech 2014]

[Scott & Hwa, Curr Op Biotech 2011]


Growth laws




9








maximal reaction rates, rmax) leads to constant metabolite con-centrations, cj (homeostasis). The reactions catalyzed by thePTS, 6-phosphofructokinase (PfkA), pyruvate kinase (PykF),and pyruvate dehydrogenase have high flux control coefficients

(12). Therefore, it can be expected that the respective enzymelevels are regulated. It is known that most of the glycolysisgenes that are less transcribed (Fig. 4) are repressed by theglobal regulator protein Cra; these include pfkA, fbaA, pgk,

FIG. 4. Time series of DNA microarray and metabolic-flux analyses of the central carbon metabolism in E. coli K-12 W3110 duringglucose-limited fed-batch growth with a constant feed rate. The time courses of the transcript levels are given for samples T1 to T8 relative to thereference sample in the batch phase (R) (Fig. 1). Green, mRNA level lower than in the reference state. Red, higher mRNA level. Statisticalsignificance (P ! 0.05) is indicated by asterisks. The metabolic fluxes are given for the !0.3 h (batch), 3.9 h, and 7.7 h (fed batch) (Fig. 1). Fluxesare mean values from the stoichiometric metabolite balancing of five independent cultivations and are given as molar percentages of the glucoseinflux. Notation is according to reference 27.

7008 LEMUTH ET AL. APPL. ENVIRON. MICROBIOL.

[Lemuth et al. AEM 2008]


Growth laws




9








rate at which the sole carbon source (glucose) is consumed,with acetate formation occurring only after glucose consump-tion surpasses some threshold rate. The presence of heterolo-gous NADH oxidase had the effect of increasing the criticalglucose consumption rate (qS

crit) at which acetate first ap-peared and thereby delaying the entry of E. coli into respi-rofermentative overflow metabolism (Fig. 1). This transitionbetween respiratory and respirofermentative metabolismoccurred at a qS

crit of 0.8 g/g dry cell weight (DCW) h forNOX! and 1.2 g/g DCW h for NOX". The expression ofNADH oxidase therefore increased by 50% the value of qS

crit.During respirofermentative metabolism, NOX" exhibited alower effluent acetate concentration and a lower specific ace-tate formation rate (qA) than NOX! at any given qS. Biomassyield (YX/S) from glucose (g dry cell weight/g glucose con-sumed) was 0.42 to 0.48 g/g for NOX! during respiratorymetabolism but decreased during respirofermentative metab-

olism, consistent with a portion of the glucose carbon beingdiverted from biomass synthesis to acetate formation. ForNOX", YX/S remained 0.28 g/g at glucose consumption ratesabove 0.5 g/g DCW h (Fig. 1).

The specific oxygen consumption rate (qO2) was twice asgreat for NOX" as for NOX! at any given value of qS (Fig. 2),consistent with additional oxygen being required for increasedoxidation of NADH to NAD. NOX" also yielded a specificCO2 evolution rate (qCO2) that was about 50% greater thanthat of NOX! for any qS (Fig. 2), suggesting greater fluxthrough CO2-forming pathways (e.g., the TCA cycle) forNOX". The results show that in the presence of NADH oxi-dase, cells diverted less carbon to biomass and acetate andmore carbon to CO2 at any given rate of glucose consumption.A carbon balance for NOX! was within #8% under all con-ditions, while for NOX" the carbon balance was within #15%(data not shown), assuming identical biomass composition(and thus identical expression of biosynthetic genes). The re-dox balance closed for NOX! within #9%, while for NOX"

this balance was only within #30% (data not shown).Intracellular response due to NADH oxidase overexpres-

sion. Since the expression of heterologous NADH oxidase inE. coli would be expected to influence the steady-state intra-cellular NADH and NAD concentrations, the concentrationsof each cofactor were determined at each steady state for bothstrains. For both NOX! and NOX", the intracellular concen-tration of NAD changed less than 30%, while the NADHconcentration changed more than 10-fold between the lowestand highest glucose consumption rates. Moreover, the NADHconcentration increased more quickly for NOX! at lower val-ues of qS than for NOX". For example, at a qS of about 0.10 g/gDCW h, the NADH concentration was 0.03 $mol/g DCW forboth strains, while at a qS of about 1.0 g/g DCW h, the NADHconcentration was 0.53 $mol/g DCW for NOX! but only 0.11$mol/g DCW for NOX". These changes are reflected in theNADH/NAD ratios (redox ratios) (Fig. 3). At any given valueof qS, the redox ratio was always greater for NOX! than forNOX". The redox ratio remained at 0.01 to 0.02 for bothstrains during respiratory metabolism but increased just prior

FIG. 2. Steady-state respiration for NOX! (open symbols anddashed lines) and NOX" (solid symbols and lines). The steady-stateqO2 (‚,Œ) and qCO2 (ƒ,!) values are shown as functions of qS.

FIG. 1. Steady-state physiological profiles of E. coli in the presenceof heterologous NADH oxidase. YX/S (!, }) and qA (E, F) values arecompared for NOX! (open symbols and dashed lines) and NOX"

(solid symbols and lines) as functions of the specific glucose consump-tion rate. The highest dilution rate studied was about 80% of $max forboth strains. The arrows indicate for each strain the critical specificglucose consumption rates at which acetate formation commenced.

FIG. 3. In vivo molar concentration ratio of NADH/NAD forNOX! (!) and NOX" (■) as functions of qS. The critical value of theNADH/NAD ratio at which acetate formation commences is about0.06 for both NOX! and NOX" (indicated by vertical lines). qA valuesare also shown for NOX! (E) and NOX" (F) as functions of qS.

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[Vemuri et al. AEM 2006]Figure SI-6.

Triplicate glucose-limited chemostat experiments of E. coli growth at various dilution rates (Vemuri et al., 2006).

[Vemuri et al. AEM 2006]


Growth laws




9








rate at which the sole carbon source (glucose) is consumed,with acetate formation occurring only after glucose consump-tion surpasses some threshold rate. The presence of heterolo-gous NADH oxidase had the effect of increasing the criticalglucose consumption rate (qS

crit) at which acetate first ap-peared and thereby delaying the entry of E. coli into respi-rofermentative overflow metabolism (Fig. 1). This transitionbetween respiratory and respirofermentative metabolismoccurred at a qS

crit of 0.8 g/g dry cell weight (DCW) h forNOX! and 1.2 g/g DCW h for NOX". The expression ofNADH oxidase therefore increased by 50% the value of qS

crit.During respirofermentative metabolism, NOX" exhibited alower effluent acetate concentration and a lower specific ace-tate formation rate (qA) than NOX! at any given qS. Biomassyield (YX/S) from glucose (g dry cell weight/g glucose con-sumed) was 0.42 to 0.48 g/g for NOX! during respiratorymetabolism but decreased during respirofermentative metab-

olism, consistent with a portion of the glucose carbon beingdiverted from biomass synthesis to acetate formation. ForNOX", YX/S remained 0.28 g/g at glucose consumption ratesabove 0.5 g/g DCW h (Fig. 1).

The specific oxygen consumption rate (qO2) was twice asgreat for NOX" as for NOX! at any given value of qS (Fig. 2),consistent with additional oxygen being required for increasedoxidation of NADH to NAD. NOX" also yielded a specificCO2 evolution rate (qCO2) that was about 50% greater thanthat of NOX! for any qS (Fig. 2), suggesting greater fluxthrough CO2-forming pathways (e.g., the TCA cycle) forNOX". The results show that in the presence of NADH oxi-dase, cells diverted less carbon to biomass and acetate andmore carbon to CO2 at any given rate of glucose consumption.A carbon balance for NOX! was within #8% under all con-ditions, while for NOX" the carbon balance was within #15%(data not shown), assuming identical biomass composition(and thus identical expression of biosynthetic genes). The re-dox balance closed for NOX! within #9%, while for NOX"

this balance was only within #30% (data not shown).Intracellular response due to NADH oxidase overexpres-

sion. Since the expression of heterologous NADH oxidase inE. coli would be expected to influence the steady-state intra-cellular NADH and NAD concentrations, the concentrationsof each cofactor were determined at each steady state for bothstrains. For both NOX! and NOX", the intracellular concen-tration of NAD changed less than 30%, while the NADHconcentration changed more than 10-fold between the lowestand highest glucose consumption rates. Moreover, the NADHconcentration increased more quickly for NOX! at lower val-ues of qS than for NOX". For example, at a qS of about 0.10 g/gDCW h, the NADH concentration was 0.03 $mol/g DCW forboth strains, while at a qS of about 1.0 g/g DCW h, the NADHconcentration was 0.53 $mol/g DCW for NOX! but only 0.11$mol/g DCW for NOX". These changes are reflected in theNADH/NAD ratios (redox ratios) (Fig. 3). At any given valueof qS, the redox ratio was always greater for NOX! than forNOX". The redox ratio remained at 0.01 to 0.02 for bothstrains during respiratory metabolism but increased just prior

FIG. 2. Steady-state respiration for NOX! (open symbols anddashed lines) and NOX" (solid symbols and lines). The steady-stateqO2 (‚,Œ) and qCO2 (ƒ,!) values are shown as functions of qS.

FIG. 1. Steady-state physiological profiles of E. coli in the presenceof heterologous NADH oxidase. YX/S (!, }) and qA (E, F) values arecompared for NOX! (open symbols and dashed lines) and NOX"

(solid symbols and lines) as functions of the specific glucose consump-tion rate. The highest dilution rate studied was about 80% of $max forboth strains. The arrows indicate for each strain the critical specificglucose consumption rates at which acetate formation commenced.

FIG. 3. In vivo molar concentration ratio of NADH/NAD forNOX! (!) and NOX" (■) as functions of qS. The critical value of theNADH/NAD ratio at which acetate formation commences is about0.06 for both NOX! and NOX" (indicated by vertical lines). qA valuesare also shown for NOX! (E) and NOX" (F) as functions of qS.

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I “Phenomenological” models, e.g.[Tadmor & Tlusty, PLoS CB 2008][Klumpp & Hwa, PNAS 2008][Molenaar et al. MSB 2009]

I Here: genome-scale[ME-model, O’Brien et al. MSB 2013]


Model definition

Proteome sectors

0 0.2 0.4 0.6 0.80

2

4

6

8

10

12

14

Growth Rate (h−1)

Flux

(mm

ol/g

DW

h)

Glucose uptakeCO2 excretion(x0.5)

AKGDH fluxMALS fluxAcetate outtakeEDD flux

0 0.2 0.4 0.6 0.80

0.1

0.2

0.3

0.4

0.5

Growth rate (h−1)

Prot

eom

e fra

ctio

n

!C

!E

!R

Q

RE

C

A

B

C

C = catabolic

E = enzymes

R = ribosomal

Q = housekeeping

φX = prot. fractionof sector X

φC + φE + φR + φQ = 1 → φC + φE + φR = 1− φQ︸︷︷︸λ-indep.

I R-sector : linear dep. on λ → φR = φR,0 + wRλ

wR ' 0.169 h [Scott et al. Science 10]

I C-sector : glc sensing, transport & processing → φC = φC,0 + wglcuglc

uglc =1

MDW× (MC/µC)︸︷︷︸φCMprot/µC

× kcat[glc]

[glc] + KM︸︷︷︸kcat

=φC

wglc, wglc =

MDW

Mprot

µC

kcat

I E-sector : prop. to metabolic fluxes → φE = φE,0 +∑

i wi|vi|

MM kinetics−−−−−−→ wi =MDW

Mprot

µi

kcat,i


Model definition

CAFBA: setup

maxvλ s.t. Sv = 0

vj ≤ vj ≤ vj ∀ reaction j

wglcuglc +∑

i

wi|vi|+ wRλ = const. ' 0.484 [Scott et al. Science 10]

I λ-indep. biomass composition : LP POSTER

I Control parameters : wglc, {wi}, wR → Here : C-limitation (wglc)[

R-/E-lim.: POSTER]

I Reminder : wglc ∝ 1kcat

= 1kcat

(1 + KM

[glc]

), i.e. optimal λ↗ as wglc ↘

I wR from [Scott et al. Science 10] , i.e. wR ' 0.169 h

I {wi}’s : focus on two “limiting” cases


Model definition

CAFBA: setup


Proteome sectors

0 0.2 0.4 0.6 0.80

2

4

6

8

10

12

14

Growth Rate (h−1)

Flux

(mm

ol/g

DW

h)



0 0.2 0.4 0.6 0.80

0.1

0.2

0.3

0.4

0.5

Growth rate (h−1)

Prot

eom

e fra

ctio

n

�C

�E

�R

Q

RE

C

A

B

C

C = catabolic

E = enzymes

R = ribosomal

Q = housekeeping

�X = prot. fractionof sector X

�C + �E + �R + �Q = 1 ! �C + �E + �R = 1 � �Q| {z }�-indep.

I C-sector: glc sensing, transport & processing ! �C = �C,0 + wCuglc

I E-sector: flux-dependent ! �E = �E,0 +P

i wi|vi|

wi /µi

kcat,i

I R-sector: linear dep. on � ! �R = �R,0 + wR�

0 20 40 60 80 100 120 140 160 18010−8

10−6

10−4

10−2

100

102

Reactions

µ/k

cat (g

h/m

mol

)

167 enzymes from Brenda

−8 −6 −4 −2 0 20

5

10

15

20

25

30

log10 (µ/kcat [g h/mmol])

1. wi = wE for each i, with wE such that λ ≤ 0.9/h

2. {wi} i.i.d. sampled from p(w) ∼ 1/w with given 〈w〉︸︷︷︸〈λ〉≤ 0.9/h

and δ = log10wmaxwmin

= 1

[δ-dependence : POSTER

]


Results (E. coli)

“Disorder-averaged” case

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

2

4

6

8

10

12

14

Growth Rate (h−1)

Flux

(mm

ol/g

DWh)

500 samples, delta 1

Glucose uptakeCarbon dioxide excretion (x0.5)AKGDH fluxMALS fluxAcetate outtakeEDD flux

0.6 0.8 1 1.20

20

40

60

80

Growth rate (1/h)0 10 20

0

50

100

150

Acetate Excretion (mmol/gDWh)

0 5 10 15 20

15

20

25

30

Acetate excretion (mmol/gDWh)

CO2 e

xcre

tion

(mm

ol/g

DWh)

A B C

D


Results (E. coli)

“Disorder-averaged” case

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

9

10

Growth Rate (1/h)

Ace

tate

exc

retio

n (m

mol

/gD

Wh)

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

16

18

20

22

Growth Rate (1/h)

CO

2 exc

retio

n (m

mol

/gD

Wh)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

Growth Rate (1/h)A

KG

DH

flux

(mm

ol/g

DW

h)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

Growth Rate (1/h)

MA

LS fl

ux (m

mol

/gD

Wh)

0 0.2 0.4 0.6 0.8 10

2

4

6

8

Growth Rate (1/h)

ED

D fl

ux (m

mol

/gD

Wh)

Glucose

Glucose−6P

Lactose

Sucrose

Glycerol

Mannitol

A

B

C

D

E

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

2

4

6

8

10

12

14

Growth Rate (h−1)

Flux

(mm

ol/g

DWh)



0.6 0.8 1 1.20

20

40

60

80


0

50

100

150


0 5 10 15 20

15

20

25

30


CO2 e

xcre

tion

(mm

ol/g

DWh)

A B C

D


A closer look at CAFBA

The trade-off behind CAFBA solutions

φC+φE︷︸︸︷wglcuglc +

∑

i

wi|vi| + wRλ︸︷︷︸φR

= const.

I maxλ ↔ maxφR ↔ min [φC + φE]

I Question: can we understand CAFBA solutions as a φC/φE trade-off?

I Limiting problems ( ? = CAFBA value)

1 minimize φE at given λ : min∑

i wi|vi| s.t. Sv = 0 & λ = λ?

a maximize λ with φE ≤ φ?E : maxλ s.t. Sv = 0 &∑

i wi|vi| ≤ φ?E → “crowding”

2 minimize φC at given λ : min uglc s.t. Sv = 0 & λ = λ?

b maximize λ with φC ≤ φ?C : maxλ s.t. Sv = 0 & wglcuglc ≤ φ?C → FBA




φcrowdE ≤ φ?E ≤ φFBA

E

φFBAC ≤ φ?C ≤ φcrowd

C

I CAFBA “interpolates” between the“FBA” scenario (lower λ) and the“molecular crowding” scenario of[Beg et al PNAS 07] (higher λ)

I Averaging over wi ∝ µi/kcat,ismooths out the crossover

“crowding” sol.

“FBA” sol.




φcrowdE ≤ φ?E ≤ φFBA

E

φFBAC ≤ φ?C ≤ φcrowd

C

I CAFBA “interpolates” between the“FBA” scenario (lower λ) and the“molecular crowding” scenario of[Beg et al PNAS 07] (higher λ)

I Averaging over wi ∝ µi/kcat,ismooths out the crossover

“loopy”solutions

no solutions

crowdin

gFBA

CAFBA



“Cell-to-cell” fluctuations

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

2

4

6

8

10

12

14

Growth Rate (h−1)

Flux

(mm

ol/g

DWh)



0.6 0.8 1 1.20

20

40

60

80


0

50

100

150


0 5 10 15 20

15

20

25

30


CO2 e

xcre

tion

(mm

ol/g

DWh)

A B C

D

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

2

4

6

8

10

12

14

Growth Rate (h−1)

Flux

(mm

ol/g

DWh)



0.6 0.8 1 1.20

20

40

60

80


0

50

100

150


0 5 10 15 20

15

20

25

30


CO2 e

xcre

tion

(mm

ol/g

DWh)

A B C

D



“Cell-to-cell” fluctuations



Homogeneous versus random wi’s

0 0.2 0.4 0.6 0.80

2

4

6

8

10

12

14

Growth Rate (h−1)

Flux

(mm

ol/g

DW

h)



0 0.2 0.4 0.6 0.80

0.1

0.2

0.3

0.4

0.5

Growth rate (h−1)

Prot

eom

e fra

ctio

n

!C

!E

!R

Q

RE

C

A

B

C

wi = wE ∀i

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

2

4

6

8

10

12

14

Growth Rate (h−1)

Flux

(mm

ol/g

DWh)



0.6 0.8 1 1.20

20

40

60

80


0

50

100

150


0 5 10 15 20

15

20

25

30


CO2 e

xcre

tion

(mm

ol/g

DWh)

A B C

D

“Disorder-averaged”



Homogeneous versus random wi’s

0 0.2 0.4 0.6 0.80

2

4

6

8

10

12

14

Growth Rate (h−1)

Flux

(mm

ol/g

DW

h)



0 0.2 0.4 0.6 0.80

0.1

0.2

0.3

0.4

0.5

Growth rate (h−1)

Prot

eom

e fra

ctio

n

!C

!E

!R

Q

RE

C

A

B

C

wi = wE ∀i “Disorder-averaged”


CAFBA vs ME-model

ME-model for E. coli

Genome-scale models of metabolism and geneexpression extend and refine growth phenotypeprediction

Edward J O’Brien1, Joshua A Lerman1, Roger L Chang, Daniel R Hyduke and Bernhard Ø Palsson*

Department of Bioengineering, University of California San Diego, La Jolla, CA, USA1These authors contributed equally to this work.* Corresponding author. Department of Bioengineering, University of California San Diego, 9500 Gilman Drive, Mail Code 0412, PFBH Room 419, La Jolla,CA 92093-0412, USA. Tel.: ! 1 858 534 5668; Fax: ! 1 858 822 3120; E-mail: [email protected]

Received 22.4.13; accepted 5.9.13

Growth is a fundamental process of life. Growth requirements are well-characterized experimen-tally for many microbes; however, we lack a unified model for cellular growth. Such a model must bepredictive of events at the molecular scale and capable of explaining the high-level behavior of thecell as a whole. Here, we construct an ME-Model for Escherichia coli—a genome-scale model thatseamlessly integrates metabolic and gene product expression pathways. The model computesB80% of the functional proteome (by mass), which is used by the cell to support growth under agiven condition. Metabolism and gene expression are interdependent processes that affect andconstrain each other. We formalize these constraints and apply the principle of growth optimizationto enable the accurate prediction of multi-scale phenotypes, ranging from coarse-grained (growthrate, nutrient uptake, by-product secretion) to fine-grained (metabolic fluxes, gene expressionlevels). Our results unify many existing principles developed to describe bacterial growth.Molecular Systems Biology 9: 693; published online 1 October 2013; doi:10.1038/msb.2013.52Subject Categories: metabolic and regulatory networks; computational methodsKeywords: gene expression; genome-scale; metabolism; molecular efficiency; optimality

Introduction

The genotype–phenotype relationship is fundamental tobiology. Historically, and still for most phenotypic traits, thisrelationship is described through qualitative arguments basedon observations or through statistical correlations. Under-standing the genotype–phenotype relationship demands van-tage points at multiple scales, ranging from the molecular tothe cellular. Reductionist approaches to biology have produced‘parts lists’, and successfully identified key concepts (e.g.,central dogma) and specific chemical interactions andtransformations (e.g., metabolic reactions) fundamental tolife. However, reductionist viewpoints, by definition, do notprovide a coherent understanding of whole cell functions.For this reason, modeling whole biological systems (orsubsystems) has received increased attention.

A number of modeling approaches have been developed topredict systems-level phenotypes. What distinguish thesemodels from each other are the underlying assumptions theymake, the input data they require, and the scope and precisionof their predictions (Selinger et al, 2003). The type of modelingformalism employed is influenced by all of these distinguish-ing characteristics (Machado et al, 2011). Genome-scaleoptimality models of metabolism (termed as M-Models) havemade much progress in recent years as they require only basicknowledge of reaction stoichiometry, are genome-scale in

scope, and have fairly accurate predictive power. Recently,M-Models have been extended to include the process of geneexpression (termed as ME-Models) (Lerman et al, 2012; Thieleet al, 2012), opening up completely new vistas in thedevelopment of microbial systems biology. On the heels ofthese developments, a whole-cell model (WCM) of the humanpathogen Mycoplasma genitalium appeared (Karr et al, 2012).The WCM integrates many more cellular processes and can beused to simulate dynamic cellular states; however, it dependson detailed molecular measurements of an initial state (e.g.,growth rate, biomass composition, and gene expression).While the model described by Karr et al is a major advancetoward whole-cell computation, many practical applicationsrely on the ability to compute optimal phenotypic states. TheWCM does not have this ability owing to the disparatemathematical formalisms it employs. The WCM and gen-ome-scale optimality models thus have different capabilitiesand will find use to predict and explain different biologicalphenomena.

Here, we construct an ME-Model for E. coli K-12 MG1655.The ME-Model is a microbial growth model that computes theoptimal cellular state for growth in a given steady-stateenvironment. It takes as input the availability of nutrients tothe cell and produces experimentally testable predictions for:(1) the cell’s maximum growth rate (m*) in the specified

Molecular Systems Biology 9; Article number 693; doi:10.1038/msb.2013.52Citation: Molecular Systems Biology 9:693www.molecularsystemsbiology.com

& 2013 EMBO and Macmillan Publishers Limited Molecular Systems Biology 2013 1

and Palsson, 2010); instead, expression of specific RNA andprotein molecules are free variables determined during ME-Model simulations. ‘Coupling constraints’ (Thiele et al, 2010;

Lerman et al, 2012) relate the synthesis of RNA- and protein-based molecules to their catalytic functions in the cell(Figure 1B). The coupling constraints are based on parameters

Degradationkdeg [E ]E

Ø

Dilutionµ [E ]Synthesis

µ [E ] + kdeg [E ](at steady state)

Reaction catalyzed by E!

E

EnzymestRNAsmRNAsRNAPRibosomeOther machinery

0.0

0.00

RN

A-P

rote

in ra

tio (

g g–1

)G

luco

se fl

ux (

nmol

gD

W–1

min

–1)

Effe

ctiv

e tra

nsla

tion

rate

(aa

ribos

ome–1

s–1

)

Growth rate, µ (h–1) Growth rate, µ (h–1)

Experimental growth rate, µ (h–1)

ME-Model growth rate, µ (h–1)Growth rate, µ (h–1)

••••••••••

••••••••••

••••••••••

••••••••••

•••••••••

•••••••••••••••••••••

• • • • • • • • • • • • • ••

•••••

••

0.0

0 •••••••••••••••••••••••••••••••••••••••••••••••••••••••

••••••••••••••••••••••••••••

••••••••••••••••••••••••••••••••••••••••••••••••••

•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••

1

0.5

0 1

0.5

0 1

0.5

0

0.0

0.0• • •

••

••

•

••

••

0

•

0.0

0

Growth rate-dependentdemand functions

3. ATP demand (µ)2. DNA demand (µ)

1. Cell wall demand (µ)

BA

C D

FE

G

•

! = keff (µ)[E ] " kcat [E ]

Frac

tiona

l enz

yme

satu

ratio

nk e

ffk c

at

keffkcat

keffkcat

keffkcat

µ increases through increases in effective catalytic rate (keff)

ME-Model hyperbolic translation rate ME-Model hyperbolic translation rateME-Model constant translation rateExperimental Constant translation rate

ExperimentalME-Model

0.30

0.20

0.10

0.2 0.4 0.6 0.8 1.0 1.2

20

15

10

5

0.5 1.0 1.5 2.0 2.5

300

250

200

150

100

50

0.2 0.4 0.6

1.0

0.8

0.6

0.4

0.2

0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.7

PTS activity (maximal rate)Glucose uptake rate (effective rate)

Extending and refining growth phenotype predictionEJ O’Brien et al


I Genome-scale model integrating metabolism,gene expression and proteomic data

I Key ingredients. empirically-derived, λ-dependent demand

functions. prescriptions for flux-enzyme level

relationships. 3 different solution regimes

(SNL, Janusian, Batch)I Control parameter: glc intake (vs cost of

in-taking glc in CAFBA)


CAFBA vs ME-model


and catalysis—the cell is ‘proteome-limited’—resulting in acorresponding maximal growth rate (Figure 2A). Thisfeature allows Batch culture growth to be simulated withoutspecifying nutrient uptake bounds; instead, the ME-Modelpredicts a maximum batch growth rate and optimal substrateuptake rate.

Supporting the validity of the proteomic constraints limitinggrowth in Batch culture, optimal Batch growth rates, substrateuptake rates, and biomass yields correlate with experimentaldata for growth on different carbon sources (SupplementaryTable S5). The ME-Model predicted substrate uptake andbiomass yield closely matches laboratory evolved strains

0 5 10 15

0.0

Glucose uptake rate bound (mmol gDW–1 h–1)

µmax

Janusian Batch

suropt

Metabolism-only model

Gro

wth

rate

, µ (

h–1)

Nutrient-limited

0.0

0.35

ME-Model growth rate, µ (h–1)

Gro

wth

yie

ld (

gDW

[g g

lc]–1

)

ME C-limitedME N-limitedExperimental C-limited

0


0.0

0


Ace

tate

sec

retio

n ra

te (

mm

ol g

DW

–1 h

–1)


0

0.0

0.8

1.0 Nitrogen

•

0.0

Phosphorous

0.00

Sulfur

0.000

Magnesium

Gro

wth

rate

, µ (

h–1)

Uptake rate bound (mmol gDW–1 h–1)

A B

C D

Mid-Janusian Batch

Waste

+

10

0.5

high cost,high energy

yieldpathway

low cost,low energy

yieldpathway

(0.5 ribosomes)

10

0.5

During the Janusian transition, µ increases through differential pathway expression

E

Strictly Nutrient-Limited (SNL)

Waste

1.0*

0.8

0.6

0.4

0.2

0.6

0.4

0.2

0.0

0.8

1.0

0.6

0.4

0.2

0.0

0.8

1.0

0.6

0.4

0.2

0.0

0.8

1.0

0.6

0.4

0.2

2 4 6 8 10 12 0.2 0.4 0.6 0.8 1.0

0.05

0.10

0.15

0.20

0.005

0.010

0.015

0.020

2

4

6

8

0.2 0.4 0.6 0.8 1.0 1.2

0.60

0.55

0.50

0.45

0.40

0.2 0.4 0.6 0.8 1.0 1.2

0.2 0.3 0.4 0.5 0.60.10Experimental growth rate, µ (h–1)

0.2 0.3 0.4 0.5 0.60.1

keffkcat

keffkcat

Proteome-limited



I Genome-scale model integrating metabolism,gene expression and proteomic data

I Key ingredients. empirically-derived, λ-dependent demand

functions. prescriptions for flux-enzyme level

relationships. 3 different solution regimes

(SNL, Janusian, Batch)I Control parameter: glc intake (vs cost of

in-taking glc in CAFBA)


CAFBA vs ME-model




0 5 10 15

0.0


µmax

Janusian Batch

suropt


Gro

wth

rate

, µ (

h–1)

Nutrient-limited

0.0

0.35


Gro

wth

yie

ld (

gDW

[g g

lc]–1

)


0


0.0

0


Ace

tate

sec

retio

n ra

te (

mm

ol g

DW

–1 h

–1)


0

0.0

0.8

1.0 Nitrogen

•

0.0

Phosphorous

0.00

Sulfur

0.000

Magnesium

Gro

wth

rate

, µ (

h–1)


A B

C D

Mid-Janusian Batch

Waste

+

10

0.5


yieldpathway

low cost,low energy

yieldpathway

(0.5 ribosomes)

10

0.5


E


Waste

1.0*

0.8

0.6

0.4

0.2

0.6

0.4

0.2

0.0

0.8

1.0

0.6

0.4

0.2

0.0

0.8

1.0

0.6

0.4

0.2

0.0

0.8

1.0

0.6

0.4

0.2

2 4 6 8 10 12 0.2 0.4 0.6 0.8 1.0

0.05

0.10

0.15

0.20

0.005

0.010

0.015

0.020

2

4

6

8

0.2 0.4 0.6 0.8 1.0 1.2

0.60

0.55

0.50

0.45

0.40

0.2 0.4 0.6 0.8 1.0 1.2


0.2 0.3 0.4 0.5 0.60.1

keffkcat

keffkcat

Proteome-limited



and Palsson, 2010); instead, expression of specific RNA andprotein molecules are free variables determined during ME-Model simulations. ‘Coupling constraints’ (Thiele et al, 2010;

Lerman et al, 2012) relate the synthesis of RNA- and protein-based molecules to their catalytic functions in the cell(Figure 1B). The coupling constraints are based on parameters

Degradationkdeg [E ]E

Ø

Dilutionµ [E ]Synthesis

µ [E ] + kdeg [E ](at steady state)

Reaction catalyzed by E!

E

EnzymestRNAsmRNAsRNAPRibosomeOther machinery

0.0

0.00

RN

A-P

rote

in ra

tio (

g g–1

)G

luco

se fl

ux (

nmol

gD

W–1

min

–1)

Effe

ctiv

e tra

nsla

tion

rate

(aa

ribos

ome–1

s–1

)

Growth rate, µ (h–1) Growth rate, µ (h–1)


ME-Model growth rate, µ (h–1)Growth rate, µ (h–1)

••••••••••

••••••••••

••••••••••

••••••••••

•••••••••

•••••••••••••••••••••

• • • • • • • • • • • • • ••

•••••

••

0.0

0 •••••••••••••••••••••••••••••••••••••••••••••••••••••••

••••••••••••••••••••••••••••

••••••••••••••••••••••••••••••••••••••••••••••••••

•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••

1

0.5

0 1

0.5

0 1

0.5

0

0.0

0.0• • •

••

••

•

••

••

0

•

0.0

0

Growth rate-dependentdemand functions

3. ATP demand (µ)2. DNA demand (µ)

1. Cell wall demand (µ)

BA

C D

FE

G

•

! = keff (µ)[E ] " kcat [E ]

Frac

tiona

l enz

yme

satu

ratio

nk e

ffk c

at

keffkcat

keffkcat

keffkcat

µ increases through increases in effective catalytic rate (keff)

ME-Model hyperbolic translation rate ME-Model hyperbolic translation rateME-Model constant translation rateExperimental Constant translation rate

ExperimentalME-Model

0.30

0.20

0.10

0.2 0.4 0.6 0.8 1.0 1.2

20

15

10

5

0.5 1.0 1.5 2.0 2.5

300

250

200

150

100

50

0.2 0.4 0.6

1.0

0.8

0.6

0.4

0.2

0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.7

PTS activity (maximal rate)Glucose uptake rate (effective rate)




CAFBA vs ME-model

Acetate excretion



0 5 10 15

0.0


µmax

Janusian Batch

suropt


Gro

wth

rate

, µ (

h–1)

Nutrient-limited

0.0

0.35


Gro

wth

yie

ld (

gDW

[g g

lc]–1

)


0


0.0

0


Ace

tate

sec

retio

n ra

te (

mm

ol g

DW

–1 h

–1)


0

0.0

0.8

1.0 Nitrogen

•

0.0

Phosphorous

0.00

Sulfur

0.000

Magnesium

Gro

wth

rate

, µ (

h–1)


A B

C D

Mid-Janusian Batch

Waste

+

10

0.5


yieldpathway

low cost,low energy

yieldpathway

(0.5 ribosomes)

10

0.5


E


Waste

1.0*

0.8

0.6

0.4

0.2

0.6

0.4

0.2

0.0

0.8

1.0

0.6

0.4

0.2

0.0

0.8

1.0

0.6

0.4

0.2

0.0

0.8

1.0

0.6

0.4

0.2

2 4 6 8 10 12 0.2 0.4 0.6 0.8 1.0

0.05

0.10

0.15

0.20

0.005

0.010

0.015

0.020

2

4

6

8

0.2 0.4 0.6 0.8 1.0 1.2

0.60

0.55

0.50

0.45

0.40

0.2 0.4 0.6 0.8 1.0 1.2


0.2 0.3 0.4 0.5 0.60.1

keffkcat

keffkcat

Proteome-limited



ME-Model

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

9

10

Growth Rate (1/h)A

ceta

te e

xcre

tion

(mm

ol/g

DW

h)

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

16

18

20

22

Growth Rate (1/h)

CO

2 exc

retio

n (m

mol

/gD

Wh)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

Growth Rate (1/h)

AK

GD

H fl

ux (m

mol

/gD

Wh)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

Growth Rate (1/h)

MA

LS fl

ux (m

mol

/gD

Wh)

0 0.2 0.4 0.6 0.8 10

2

4

6

8

Growth Rate (1/h)E

DD

flux

(mm

ol/g

DW

h)

Glucose

Glucose−6P

Lactose

Sucrose

Glycerol

Mannitol

A

B

C

D

E

CAFBA


CAFBA vs ME-model

Growth yield



0 5 10 15

0.0


µmax

Janusian Batch

suropt


Gro

wth

rate

, µ (

h–1)

Nutrient-limited

0.0

0.35


Gro

wth

yie

ld (

gDW

[g g

lc]–1

)


0


0.0

0


Ace

tate

sec

retio

n ra

te (

mm

ol g

DW

–1 h

–1)


0

0.0

0.8

1.0 Nitrogen

•

0.0

Phosphorous

0.00

Sulfur

0.000

Magnesium

Gro

wth

rate

, µ (

h–1)


A B

C D

Mid-Janusian Batch

Waste

+

10

0.5


yieldpathway

low cost,low energy

yieldpathway

(0.5 ribosomes)

10

0.5


E


Waste

1.0*

0.8

0.6

0.4

0.2

0.6

0.4

0.2

0.0

0.8

1.0

0.6

0.4

0.2

0.0

0.8

1.0

0.6

0.4

0.2

0.0

0.8

1.0

0.6

0.4

0.2

2 4 6 8 10 12 0.2 0.4 0.6 0.8 1.0

0.05

0.10

0.15

0.20

0.005

0.010

0.015

0.020

2

4

6

8

0.2 0.4 0.6 0.8 1.0 1.2

0.60

0.55

0.50

0.45

0.40

0.2 0.4 0.6 0.8 1.0 1.2


0.2 0.3 0.4 0.5 0.60.1

keffkcat

keffkcat

Proteome-limited



ME-Model

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.3

0.35

0.4

0.45

0.5

0.55

Growth Rate (h−1)G

row

th Y

ield

(gD

W/g

glc)

500 samples, delta=1

CAFBAFBA

CAFBA


Epilogue

Outlook

I Empirical observation: growth-rate dependent effects on gene expression suggesting a trade-offbetween different proteome sectors

I Minimal (3-sectors) phenomenological models explain observationsI CAFBA aims at incorporating the constraints that regulation imposes on metabolism in

genome-scale modelsI Effectively uses a 4-sector partition of the proteomeI Many empirical λ-dep. features are reproduced correctly, more study is under wayI Pros: LP, robust against noise in protein costs, no fine tuning, theoretical insightI Note : “disorder-averaging” of protein costs gives qualitatively correct predictionsI Extension with λ-dependent biomass


Epilogue

Outlook

0 0.2 0.4 0.6 0.80

5

10

15

Growth Rate (h−1)

Flux

(mm

ol/g

DWh)

Glucose uptakeCO2 excretion(x0.5)AKGDH fluxMALS fluxAcetate outtakeEDD flux

0 0.2 0.4 0.6 0.80

5

10

15

Growth Rate (h−1)

Flux

(mm

ol/g

DWh)

Glucose uptakeCO2 excretion(x0.5)AKGDH fluxMALS fluxAcetate outtakeEDD flux

Figure 4: Left: Constant biomass (the iJR904 default), with wE = 0.00097 gDWh/mmol. Right:variable biomass case, with wE = 0.00090 gDWh/mmol. No significative variations in the transitionsare present, although some fluxes change slightly.

6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

5

10

15

Growth Rate (h−1)Fl

ux (m

mol

/gD

Wh)



0 0.2 0.4 0.6 0.80.3

0.35

0.4

0.45

0.5500 samples, delta=1

Growth Rate (h−1)

Gro

wth

Yie

ld (g

DW

/ggl

c)

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Growth Rate (h−1)

Flux

(mm

ol/g

DW

h)


NADTRHD fluxTRD2 flux

Figure 5: Randomization with the biomass ! functions described in Table 2 (Fig. 1) and the ATPhydrolysis flux from Eq. (10) with !extra

E = 35 mmol ATP/gDW (Fig. 3). (500 samples, " = 1,wE =0.00090 gDWh/mmol).

7

constrained-allocation fba: modeling the interplay of...

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