Supplementary Material
Article title: Numerical simulation of riboflavin biosynthesis in Bacillus subtilis under production strain conditions
Journal: Biotechnology Letters
Authors: Markus Birkenmeier, Susanne Neumann, Thorsten Röder
Institute of Chemical Process Engineering, Mannheim University of Applied Sciences, Paul-Wittsack-Straße 10,
68163 Mannheim, Germany. E-mail: t.roeder@hs-mannheim.de; Tel: +49 621 292 6800; Fax: +49 621 292 6555
Validity check of the homogeneity and continuum assumption for the use of an ordinary differential equation
(ODE)-based model
Grima and Schnell (2008) described two length scales for the correct choice of a modeling approach:
(1) The Kuramoto length 𝑙𝑘,𝑗 = √𝐷𝑗 ∙ 𝜏𝑗 for a metabolite 𝑗
(2) The size of the intracellular space 𝑙𝑖,𝑗 = [𝐶𝑗 ]ST
1
3
−( )
for a metabolite 𝑗
where 𝐷𝑗 denotes the diffusion coefficient of the metabolite 𝑗, 𝜏𝑗 the characteristic lifetime of the metabolite 𝑗, and
[𝐶𝑗 ]ST the concentration of the metabolite 𝑗 in steady state (ST) expressed in particle number per volume.
In this context, we applied the simplification that all diffusion coefficients 𝐷𝑗 are equal, with a value of 50 µm2 s-1.
This value was measured for NBD-glucose (molecular mass of approx. 0.3 kDa) in the cytoplasm of Escherichia
coli by Mika et al. (2010). The molecular masses of the metabolites involved in the riboflavin biosynthesis are in the
range of 0.2–0.5 kDa. The characteristic lifetime of the metabolite 𝑗 is calculated via the quotient of the metabolite
concentration [𝐶𝑗 ]ST and the flux of the degrading reaction 𝑟ST .
The following Kuramoto length scales 𝑙𝑘,𝑗 and sizes of the intracellular space 𝑙𝑖,𝑗 can be calculated:
No. j
1
2
3
4
5
6
7
8
9
Metabolite
A
B
C
D
E
F
G
H
R
[Cj]ST
Number of
rST
τj
lk,j
li,j
(mM)
molecules per cell (-)
(mM/s)
(s)
(µm)
(µm)
0.532
0.008
0.057
2.579E-06
0.113
1.221
0.357
0.002
5.833E-04
320370
4697
34205
2
68109
735166
215046
964
351
0.0035
0.0035
0.0035
0.0035
0.0070
0.0070
0.0070
0.0035
0.0035
152.00
2.23
16.23
7.37E-04
16.16
174.40
51.01
0.46
0.17
87.18
10.56
28.49
0.19
28.42
93.38
50.50
4.78
2.89
0.015
0.060
0.031
0.864
0.024
0.011
0.017
0.101
0.142
The criteria for the use of an ODE-based model are:
(1) Homogeneity assumption: 𝑙𝑘,𝑗 ≫ 𝐿
(2) Continuum assumption: 𝑙𝑖,𝑗 ≪ 𝐿
where 𝐿 is the characteristic length of the system under consideration. In our case, we chose the characteristic length
𝐿 of 1 µm for a typical bacterial cell. Therefore, all metabolites except D fulfill the above noted criteria. The
metabolite D does not fulfill the criteria because of the fast hypothetical kinetic parameters of the phosphatase
reaction (kcat of 20 s-1, reaction 4) compared with the other reactions (see Table 1). Thus, metabolite D is quickly
degraded in the system and the stationary concentration is relatively low. This result is virtual because of our
assumption and is not crucial for the simulation.
Scaling factor for the adaptation of the kinetic parameters of the deaminase and reductase reactions from
Escherichia coli to Bacillus subtilis
The following kinetic parameters of the deaminase and reductase activities for the bifunctional Enzyme RibD of E.
coli are listed in Magalhães et al. (2008):
𝑘𝑐𝑎𝑡,𝑑𝑒𝑎𝑚𝑖𝑛𝑎𝑠𝑒,𝐵 = 370
1
1
= 6.17
min
s
𝐾𝑚,𝐵 = 1.3 mM
𝑘𝑐𝑎𝑡,𝑟𝑒𝑑𝑢𝑐𝑡𝑎𝑠𝑒,𝐶 = 19
1
1
= 0.317
min
s
𝑘𝑐𝑎𝑡,𝑟𝑒𝑑𝑢𝑐𝑡𝑎𝑠𝑒,𝑁𝐴𝐷𝑃𝐻 = 18.5
1
1
= 0.308
min
s
𝐾𝑚,𝐶 = 0.037 mM
Richter et al. (1997) measured the specific activities 𝑣 of the bifunctional enzymes RibG for B. subtilis and RibD for
E. coli:
𝑣𝑅𝑖𝑏𝐺,𝑑𝑒𝑎𝑚𝑖𝑛𝑎𝑠𝑒,𝐵 = 9.9
µmol
mg h
𝑣𝑅𝑖𝑏𝐺,𝑟𝑒𝑑𝑢𝑐𝑡𝑎𝑠𝑒,𝑁𝐴𝐷𝑃𝐻 = 0.6
𝑣𝑅𝑖𝑏𝐷,𝑑𝑒𝑎𝑚𝑖𝑛𝑎𝑠𝑒,𝐵 = 21.3
µmol
mg h
µmol
mg h
𝑣𝑅𝑖𝑏𝐷,𝑟𝑒𝑑𝑢𝑐𝑡𝑎𝑠𝑒,𝑁𝐴𝐷𝑃𝐻 = 6.7
µmol
mg h
In this study, we used the specific activities 𝑣 of Richter et al. (1997) to calculate scaling factors to adapt the
turnover numbers 𝑘𝑐𝑎𝑡 from RibD from Magalhães et al. (2008) to RibG in our model (see Table 1):
𝑘𝑐𝑎𝑡,2
𝑘𝑐𝑎𝑡,3
µmol
9.9
𝑣𝑅𝑖𝑏𝐺,𝑑𝑒𝑎𝑚𝑖𝑛𝑎𝑠𝑒,𝐵
1 min
1
mg h
= 𝑘𝑐𝑎𝑡,𝑑𝑒𝑎𝑚𝑖𝑛𝑎𝑠𝑒,𝐵 ∙ (
) = 370
∙
∙(
) = 2.87
𝑣𝑅𝑖𝑏𝐷,𝑑𝑒𝑎𝑚𝑖𝑛𝑎𝑠𝑒,𝐵
min 60 s 21.3 µmol
s
mg h
µmol
𝑣𝑅𝑖𝑏𝐺,𝑟𝑒𝑑𝑢𝑐𝑡𝑎𝑠𝑒,𝑁𝐴𝐷𝑃𝐻
1 min 0.6 mg h
1
= 𝑘𝑐𝑎𝑡,𝑟𝑒𝑑𝑢𝑐𝑡𝑎𝑠𝑒,𝐶 ∙ (
) = 19
∙
∙(
) = 0.0284
µmol
𝑣𝑅𝑖𝑏𝐷,𝑟𝑒𝑑𝑢𝑐𝑡𝑎𝑠𝑒,𝑁𝐴𝐷𝑃𝐻
min 60 s 6.7
s
mg h
We used this procedure to account for the smaller activities of RibG compared with RibD. We used the MichaelisMenten constants from RibD in our simulation because no RibG constants were available in the literature.
Full expression of the objective function for the minimization of the enzyme concentrations (Eq. (2))
𝑟3 =
[𝐸𝑅𝑖𝑏𝐺 ] ∙ 𝑘𝑐𝑎𝑡,3 ∙ [𝐶]
𝐾𝑚,𝐶 + [𝐶]
𝑟6 =
[𝐸𝑅𝑖𝑏𝐻 ] ∙ 𝑘𝑐𝑎𝑡,6 ∙ [𝐸] ∙ [𝐺]
𝐾𝑚,𝐸 ∙ 𝐾𝑚,𝐺 + 𝐾𝑚,𝐺 ∙ [𝐸] + 𝐾𝑚,𝐸 ∙ [𝐺] + [𝐸] ∙ [𝐺]
𝑟7 =
[𝐸𝑅𝑖𝑏𝐵 ] ∙ 𝑘𝑐𝑎𝑡,7 ∙ [𝐻]
𝐾𝑚,𝐻 + [𝐻]
Therefore, steady state Eq. (2) can be expressed in the following form:
min [𝐸𝑡𝑜𝑡 ] = [𝐸𝑅𝑖𝑏𝐺 ] + [𝐸𝑅𝑖𝑏𝐻 ] + [𝐸𝑅𝑖𝑏𝐵 ]
=
𝐽3
𝐽6
𝐽7
+
+
𝑘𝑐𝑎𝑡,3 ∙ [𝐶]
𝑘𝑐𝑎𝑡,6 ∙ [𝐸] ∙ [𝐺]
𝑘𝑐𝑎𝑡,7 ∙ [𝐻]
𝐾𝑚,𝐶 + [𝐶] 𝐾𝑚,𝐸 ∙ 𝐾𝑚,𝐺 + 𝐾𝑚,𝐺 ∙ [𝐸] + 𝐾𝑚,𝐸 ∙ [𝐺] + [𝐸] ∙ [𝐺] 𝐾𝑚,𝐻 + [𝐻]
Analytical expressions for the calculation of the scaled elasticities
In General: irreversible Michaelis-Menten rate law
𝑟=
[𝐸𝑛𝑧𝑦𝑚] ∙ 𝑘𝑐𝑎𝑡 ∙ [𝑆]
𝐾𝑚 + [𝑆]
𝑟
𝑒[𝑆]
=
𝜕𝑟 [𝑆]
∙
=
𝜕[𝑆] 𝑟
1
[𝑆]
1+
𝐾𝑚
Reaction 6: irreversible random bi-uni rate law (Liebermeister and Klipp 2006)
𝑟6 =
[𝐸𝑅𝑖𝑏𝐻 ] ∙ 𝑘𝑐𝑎𝑡,6 ∙ [𝐸] ∙ [𝐺]
𝐾𝑚,𝐸 ∙ 𝐾𝑚,𝐺 + 𝐾𝑚,𝐺 ∙ [𝐸] + 𝐾𝑚,𝐸 ∙ [𝐺] + [𝐸] ∙ [𝐺]
𝑟6
𝑒[𝐸]
𝜕𝑟6 [𝐸]
=
∙
=
𝜕[𝐸] 𝑟6
[𝐺]
𝐾𝑚,𝐺
[𝐸]
[𝐺]
[𝐸] [𝐺]
1+
+
+
∙
𝐾𝑚,𝐸 𝐾𝑚,𝐺 𝐾𝑚,𝐸 𝐾𝑚,𝐺
𝑟6
𝑒[𝐺]
𝜕𝑟6 [𝐺]
=
∙
=
𝜕[𝐺] 𝑟6
[𝐸]
𝐾𝑚,𝐸
[𝐸]
[𝐺]
[𝐸] [𝐺]
1+
+
+
∙
𝐾𝑚,𝐸 𝐾𝑚,𝐺 𝐾𝑚,𝐸 𝐾𝑚,𝐺
1+
1+
References – Supplementary Material
Grima R, Schnell S (2008) Modelling reaction kinetics inside cells. Essays Biochem 45:41-56
Liebermeister W, Klipp E (2006) Bringing metabolic networks to life: convenience rate law and thermodynamic
constraints. Theor Biol Med Model 3:41. doi: 10.1186/1742-4682-3-41
Mika JT, van den Bogaart G, Veenhoff L, Krasnikov V, Poolman B (2010) Molecular sieving properties of the
cytoplasm of Escherichia coli and consequences of osmotic stress. Mol Microbiol 77:200-207
Magalhães ML, Argyrou A, Cahill SM, Blanchard JS (2008) Kinetic and mechanistic analysis of the Escherichia
coli ribD-encoded bifunctional deaminase-reductase involved in riboflavin biosynthesis. Biochemistry
47:6499:6507
Richter G, Fischer M, Krieger C, Eberhardt S, Lüttgen H, Gerstenschläger I, Bacher A (1997) Biosynthesis of
riboflavin: characterization of the bifunctional deaminase-reductase of Escherichia coli and Bacillus subtilis. J
Bacteriol 179:2002-2028