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Text S1: Framework of integrative Flux Balance Analysis (iFBA)
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We have tested an alternative model framework: iFBA. The dynamic cultivation process was decomposed
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into numerous pseudo-steady-state time intervals. At each time interval, the inflow/outflow fluxes in the
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FBA were derived from the Monod equations; while the biomass increase during the time interval was
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predicted by mini-FBAs using proper objective functions. At the end of each time interval, the predicted
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biomass increase was incorporated into the Monod equations to estimate the metabolite and substrate
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concentrations at the next time interval. Then, we obtained the inflow/outflow fluxes of mini-FBA in the
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next time interval (t+∆t). To improve the model accuracy, iFBA employed a dual-objective function w(i),
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a weighted combination of “maximizing growth rate” and “minimizing overall flux”. The time-dependent
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weight in the dual-objective function and the kinetic parameters in Monod equations were determined by
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minimizing the differences between iFBA predicted MR-1 growth kinetics and the experimentally
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measured data. The iFBA was formulated as below:
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1
min R = [Y - η ( t; β)] [Y - η ( t; β)]
'
'
T
'
'
s.t.
 X( 0 ), LACT( 0 ), ACT( 0 ), PYR( 0 )
 for i  1 : num_timepo int

t  i  dt


v
(i)  f (LACT, ACT , PYR)


μ  LACT(i)

K  LACT(i)

 [
 k  LACT(i)  k  LACT(i)]  S(t  t
Y


v
(i)  f (LACT, ACT , PYR)


μ  ACT(i)

K  ACT(i)

[
 k  PYR(i)  k  LACT(i)]  S(t  t )
Y


v
(i)  f (LACT, ACT , PYR)

μ  ACT(i)


K  ACT(i)

[
 k  PYR(i)  k  LACT(i)]  S(t  t )
Y


min [w(i)   v(i)  ( 1  w(i))  μ(i)] 




 s.t.


 S  v(i)  0





 lb  v(i)  ub


 v

(i), v
(i), v
(i)




 w(i)  p  exp (p  S (t  p ) ) 



X(i  1 )  X(i)  μ (i )  X(i)  dt

LACT (i  1 )  LACT(i)  f  X(i)  dt


ACT( i  1 )  ACT(i)  f  X(i)  dt

PYR(i  1 )  PYR(i)  f  X(i)  dt

lb  , p , p  ub
lac_ inf low
1
max ,L
s,l
pl
al
X/L
act_outflow
2
max ,A
s,a
ap
al
L
ap
al
L
X/A
pyr_outflow
3
max ,A
s,a
X/A
2
i
1
lac_ inf low
i
act_outflow
1
pyr_outflow
2
3
1
2
3
1
L








)





























2
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The new symbols introduced in iFBA are as following: num_timepoint is the number of time intervals
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decomposed during the entire cultivation process, which is 408; dt is the time of each time interval, which
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is 1/12 h. p1, p2 and p3 are three parameters used to simulate the dynamic weighting factors in the dual
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objective function. The internal dFBA problem was solved using the CPLEX solver in TOMLAB
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optimization toolbox (TOMLAB optimization Inc, Seattle, WA) within MATLAB (R2009a). The
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external optimization problem (i.e. search for weight) was solved by SNOPT solver in TOMLAB
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optimization toolbox within MATLAB (R2009a). The MATLAB code of iFBA was attached in
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Dataset S1.
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Symbols
μmax,L
μmax,P
μmax,A
YX/L
YX/P
YX/A
Ks,l
Ks,p
Ks,a
kal
kpl
kap
ke
tL
p1
p2
p3
Table R1. Parameters estimated in iFBA
Notation
Unit
Maximum specific growth rate using lactate
h-1
Maximum specific growth rate using pyruvate
h-1
Maximum specific growth rate using acetate
h-1
Apparent biomass yield coefficient from lactate
g DCW/mol lactate
Apparent biomass yield coefficient from pyruvate
g DCW/mol pyruvate
Apparent biomass yield coefficient from acetate
g DCW/mol actate
Monod lactate saturation constant
mM
Monod pyruvate saturation constant
mM
Monod acetate saturation constant
mM
Acetate production coefficient from lactate
L∙ (h∙g DCW)-1
Pyruvate production coefficient from lactate
L∙ (h∙g DCW)-1
Acetate production coefficient from pyruvate
L∙ (h∙g DCW)-1
Endogenous metabolism rate constant
h-1
Lag time in growth
h
Parameters used in tradeoff objective function
dimensionless
Parameters used in tradeoff objective function
h-1
Parameters used in tradeoff objective function
h
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6
7
8
9
iFBA
0.53
0.14
0.14
17.5
15.5
10.9
19.4
19.4
10.1
0.70
0.42
0.94
0.013
7.10
5.3×10-6
0.33
26.7
Table R2. Lack-of-fit test for iFBA
Model Name
Lack-of-fit sum of
squares, SSLOF
Degree of
freedom,
df1
Pure-error sum
of squares,
SSPE
Degree of
freedom,
df1
SSLOF / df 1
SSPE / df 2
F(df1,df2)
iFBA
3.726
56
0.813
144
11.78
1.396
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Note: In order to test whether or not a model could fit the data well, we applied the lack-of-fit
test. The F-test indicates that the iFBA model should be further improved to describe the
experimental data.
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1
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Fig. R1 Growth kinetics simulated by iFBA using “maximizing growth rate” as the objective
function (red line) or using the dual-objective function (green line). The blue dots are the
measurements.
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Fig. R2 Histogram of normalized residuals in growth kinetics simulated by iFBA.
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