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Metabolic Control Theory and the
genetics and evolution of metabolic
fluxes
Christine Dillmann, Julie Fievet, Sébastien Lion,
Frédéric Gabriel, Grégoire Talbot, Delphine
Sicard, Dominique de Vienne
UMR de Génétique Végétale, INRA/UPS/CNRS/INA PG
Ferme du Moulon, 91190 Gif-sur-Yvette, France
[email protected]
Metabolic Control Theory and the genetics and
evolution of metabolic fluxes
- Metabolic fluxes as model quantitative traits and the relationship
between genotype and phenotype
- Experimental validation of the Metabolic Control Theory
- The metabolic bases of dominance and heterosis
- Evolution of enzyme concentrations in natural populations
« Quantitative » genetics
Quantitative traits
Most phenotypic traits …
Growth rate, flowering date, fruit pH, behaviour traits,
morphological traits, blood pressure, metabolic flux,
enzyme activity, mRNA/protein concentrations, etc.
A
N
7
… Display a continuous variation within populations :
6
5
4
3
2
1
0
Taille (cm)
150
160
170
180
190
200
G. J. Mendel, 1865
« Recherche sur les hybrides d’autres plantes »
AaBb
aaBB
AAbb
A
A
B
B
a
a
x
A
a
B
b

b
b
N 6
5
4
3
2
1
0
Aabb
AABb
aaBb
AaBB
AABB
aabb
0
1
2
3
4
Number of
«Capital letter » alleles
F2
Continuous variation can be maintained by independent
segregation of multiple factors. George Udny Yule, 1902.
The metabolic fluxes as model quantitative traits
The “genotype” : all the genes that determine enzyme activities
(concentrations and kinetic parameters, genetically variable)
Enzymes
E1
X0
E2
S1
…
Ej
Sj1
Sj
…
En
Xn
The “phenotype” : the flux
Kacser H. and Burns J.A., 1981. The molecular basis of dominance. Genetics, 97, 639-666.
Metabolic control theory (1)
S0
E1
E2
S1
…
Ej
Sj
Ej+1
…
En-1
Sn-1
En
Sn
Michaelis-Menten enzymes
Stationary phase : v1 = v2 = … = vj = … = vn = J
Enzymes far from saturation
Enzyme concentrations are independent
vi

( Si  Si 1 / K eq )
(V max i / Km i )
1

S i / Km
i

Si 1 / Kmi 1
 (V max i / Km i )
( Si  Si 1 / K eq )
Kacser and Burns, 1973
Heinrich and Rappoport, 1974
Metabolic control theory (2)
S0
E1
S1
E2
…
Ej
Sj
Ej+1
…
En-1
Sn-1
En
Sn
At stationary phase : v1 = v2 = … = vj = … = vn = J


S
 S0  n 


K
0,n 

J
n
1
 j 1 E
j
Kacser and Burns, 1973
Heinrich and Rappoport, 1974
Metabolic control theory (3) : genetically variable
parameters


S
n
 S0 



K
0,n 

J
n
1
 j 1 E
j
Ej 
Enzyme efficiency :
Kinetic parameters :
Aj 
Enzyme cellular concentration
k cat j
KM j
Qj
Vmax j
KM j
K 0, j 1
K 0, j 1 = Aj Qj
Genetic variability of kinetic parameters
- Few in vivo data
- Slightly variable
Wang & Dykhuisen, 2001. Pathway of gluconate
metabolism in E. coli. Evolution, 55:897.
IPG
Genetic variability of
enzyme concentrations
- Highly variables
Number of molecules per cell
fba1
Fiévet et al, 2004
Relationship between enzymes and flux : independent enzymes
Jmax
Flux J


S
n
 S0 



K
0,n 

J
n
1
 j 1 E
j
Qj or Aj or Ej = Qj x Aj
• non linear relationship between the flux and the concentration of on enzyme
of the pathway
• the flux tends asymptotically towards a maximum which depends on the
concentrations of all the enzymes of the pathway
Metabolic Control Theory and the genetics and
evolution of metabolic fluxes
- Metabolic fluxes as model quantitative traits and the relationship
between genotype and phenotype
- Experimental validation of the Metabolic Control Theory
- The metabolic bases of dominance and heterosis
- Evolution of enzyme concentrations in natural populations
Experimental validation : in vivo
Kacser and Burns, 1981. Genetics 97:639
Relationship RubisCO-photosynthesis
A typical example:
dependence of carbon
assimilation flux on
rubisco levels in
transgenic tobacco
plants.
Laurer et al, Planta 190 332345 (1993).
Experimental validation : in vitro
First part of glycolysis
GAP
glucose
HK
glucose 6 P
GPI
fructose 6 P
fructose 1,6 bisP
FBA
TPI
PFK1
DHAP
ATP
ATP
NADH
ADP
ADP
NAD+
glycérol 3 P
Créatine-P + ADP Créatine kinase
Créatine + ATP
Julie Fievet et Gilles Curien
Concentration
du NADH
Temps
Etat stationnaire
Mixing enzymes, substrates, cofactors
Enzymes
HXK+PGI+PFK
+
FBA+TPI+G3D
H+CK
Substrates
Glucose+
Creatine P+
NADH+
buffer
ATP
One tube  one «genotype»
Experimental validation
Each enzyme vary at a turn, the other
being kept constant
=
Titration curves
HXK concentration (µM)
• non linear relationship between the flux and the concentration of on enzyme
of the pathway
• the flux tends asymptotically towards a maximum which depends ont the
concentrations of all the enzymes of the pathway
Estimation of kinetic parameters :
explicit modelling
Complex equations
Many parameters
Estimation of kinetic parameters :
MCT-based modelling
J
S
1
i A Q  p
i i
i

1
1
i SA Q  Sp
i i
i
Ai composite activity parameter
pi dispensability
The maximum value for the flux is estimated
from titration curves :




1
n 1
SˆAi  ref 
 Sˆpi 
1
n 1
Qi



J
J
max i
 j max j

J  J max i
ˆ
Spi 
J max i  J i0
0
i
Jmax
pi=0
pi≠0
J0
Predicting the flux
Qref
Jmax
J0
Spi
SAi
hxk
0,1
18,24
0
0
379,93
pgi
0,15
13,27
0
0
520,5
pfk
0,29
17,87
0
0
107,02
fba
1,54
18,54
0
0
18,54
tpi
0,84
12,61
10,46
61,35
59,79
Activity parameters are estimated “in systemo”. They are different
from what can be estimated on isolated enzymes
The global equation can be used to predict the flux for other enzyme
concentrations :
J prédit 
Fievet et al, submitted
1
1
i SˆA Q  Sˆp
i i
i
Flux measured in vitro (µM/s)
Testing the predictor for the flux on 122 genotubes
r = 0,94
Predicted flux (µM/s)
Fievet et al, submitted
Metabolic Control Theory and the genetics and
evolution of metabolic fluxes
(1) Based on MCT, we validated a simple model which describes the
relationship between flux and enzyme concentrations
J
1
1
i SˆA Q  Sˆp
i i
i
(2) Composite kinetic parameters can be estimated « in vivo » from titration
curves
(3) It should also work for more complex networks like …
E5
E2
S1
E1
S3
S2
S6
E4
E3
S4
E8
E7
E6
S5
S7
E9
E10
S8
… with one
stable
stationary
state
Metabolic Control Theory and the genetics and
evolution of metabolic fluxes
- Metabolic fluxes as model quantitative traits and the relationship
between genotype and phenotype
- Experimental validation of the Metabolic Control Theory
- The metabolic bases of dominance and heterosis
- Evolution of enzyme concentrations in natural populations
Genetical consequences of ~hyperbolic relationships : dominance
Three observations
- Most deleterious mutations are recessive
- There is ~ additivity between highly deleterious mutations
- There is ~ additivity between slightly deleterious mutations
Two hypothesis to explain dominance
- R. A. Fisher (1928, 1931, 1958) : «modifiers» of
dominance relationship between alleles arise due to
natural selection
- S. Wright (1934) : dominance can be explained by the
non linear genotype-phenotype relationship.
Dominance : Fisher’s model does not work
Population genetics models : mutations are eliminated before
they become recessive.
Mutations in Chlamydomonas reinhardtii (Orr, 1991).
Dominance : Fisher’s model does not work
Recessive mutations occur in Chlamydomonas as frequently
as in drosophila
Dominance : S. Wright was right
Flux
Flux
Dominance
Weak
dominance
A1A1
A1A2
A2A2
Ei
Kacser H. and Burns J.A., 1981. The molecular basis of dominance. Genetics, 97, 639-666.
Ei
Généralisation : several variables enzymes
E. Coli - Dykhuisen et al., 1987, Genetics, 115, 25
Generalization : metabolic model for heterosis
Huître
Maïs
Levure
hybrid
P1
F1 hybrid
P2
Homozygous line
Increased vigor
Homozygous line
F1 > (P1 ,P2 )
i
ii iii
Metabolic heterosis due to dominance at different loci
Line 1 x Line 2
Hybrid F1
Ej
Ej
J
Ei
Ei
Heterosis in vitro
JP2
100%
80%
TPI
60%
FBA
PFK
40%
PGI
HXK
20%
0%
dis01
Tube
1
01*12
Tube
(1+2)/2
dis12
Tube
2
Flux
9,00
8,00
7,00
6,00
5,00
4,00
3,00
2,00
1,00
0,00
dis01
01*12
dis12
JP1
Simulations
Fievet et al, in prep
Metabolic Control Theory and the genetics and
evolution of metabolic fluxes
(4) Dominance and heterosis arise as emergent properties of metabolic
systems
(5) Heterosis can be explained by antagonistic epistatic relationships
between enzymes
Metabolic Control Theory and the genetics and
evolution of metabolic fluxes
- Metabolic fluxes as model quantitative traits and the relationship
between genotype and phenotype
- Experimental validation of the Metabolic Control Theory
- The metabolic bases of dominance and heterosis
- Evolution of enzyme concentrations in natural populations
Evolution of enzyme concentration under selection for
increasing the flux
W=
Monte-Carlo
simulations
Analytical
predictions
Natural selection shapes the sharing out of the control of the flux
Talbot et al, in prep
Dominique de Vienne
Bruno Bost
Julie Fiévet
Frédéric Gabriel
Sébastien Lion
Delphine Sicard
Grégoire Talbot
Gilles Curien
Olivier Martin
Heterosis and epistasis
-A substitution at one locus changes the effects of a substitution
at another locus
-The effect of a substituion depends on the genetic background
sJ
Flux
A 2B 2
A 2B 1
A1B2
A1B1
Enzyme A
EA2
Enzyme B
EA1 EB1
EB2
Heterosis and epistasis
Synergistic epistasis in tryptophane metabolic pathway
Tryptophane
flux
Niederberger et al., 1992,
Biochem. J. 287, 473.
Heterosis and epistasis
sJ
A 2B 2
A 2B 1
H=
A1B2
Jhyb J
A1B1
Enzyme A
Enzyme B
J
Heterosis
index
4
3.5
3
2.5
2
1.5
1
0.5
0
-0.5-0.5 0
-1
-1.5
-2
0.5
1
1.5
Antagonistic
Additivity
Fievet et al, in prep
I=1
2
2.5
3
3.5
4
Synergistic
4.5
5
Epistasis
index
Autres axes de recherche
1- Les concentrations des enzymes ne sont pas
nécessairement indépendantes
Corrélations physiologiques, positives ou négatives
La concentration d’enzymes allouée à une chaîne est nécessairement finie
( « compétition »  corrélations négatives)
 Matrice n x n des Ej/Ei
 Etotal fixé, ou fonction de coût
Autres axes de recherche
2- Comment agit la sélection pour maximiser/optimiser un flux ?
J
Red curve:
No co-regulation
- Evolution des flux avec ou sans contraintes sur les
concentrations d’enzymes.
Blue curves:
Co-regulations
Ej
- Approche expérimentale : variabilité des paramètres
enzymatiques et évolution expérimentale chez la
levure (modèle : glycolyse)