1
Supporting Information
2
Article title: A coupled thermodynamic and metabolic control analysis methodology and its evaluation on
3
glycerol biosynthesis in Saccharomyces cerevisiae
4
Journal: Biotechnology Letters
5
Authors: Markus Birkenmeier, Matthias Mack, Thorsten Röder
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Institute for Chemical Process Engineering, Mannheim University of Applied Sciences, Paul-Wittsack-Straße
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10, 68163 Mannheim, Germany. E-mail: t.roeder@hs-mannheim.de; Tel: +49 621 292 6800; Fax: +49 621 292
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6555
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10
Metabolite concentration ranges for the optimization and sampling procedure of 𝚫𝐫 𝐆′
11
The metabolite concentration ranges were mainly derived from intracellular concentrations measured and
12
reported by Cronwright et al. (2002). Since Cronwright et al. (2002) did not measure the concentrations of
13
glycerol-3-phosphate (G3P) and orthophosphate (P i), the concentration ranges of G3P and Pi were inferred from
14
concentrations reported in the literature and from typical ranges. The metabolite concentration ranges are listed
15
below:
Metabolite
Min. concentration
Max. concentration
Reference
(M)
(M)
Dihydroxyacetone phosphatea
1.2 × 10-4
6.6 × 10-4
Cronwright et al. (2002)
Glycerol-3-phosphate (G3P)
1 × 10-6
1 × 10-3
Wu et al. (2005);
Smallbone et al. (2013);
Fell (1997)
Glycerola
6.46 × 10-3
2.199 × 10-2
Cronwright et al. (2002)
NADHa
2.2 × 10-4
2.11 × 10-3
Cronwright et al. (2002)
NAD+a
3.8 × 10-4
1.62 × 10-3
Cronwright et al. (2002)
Orthophosphate (Pi)
1 × 10-3
5 × 10-2
Theobald et al. (1996);
Auesukaree et al. (2004);
Jol et al. (2012)
1
16
a
17
the three growth phases (early exponential, mid-exponential and early stationary). The standard deviations in the
18
measured concentrations were subtracted (minimum concentration) or added (maximum concentration) to
19
consider the standard deviations in the range. This procedure ensured that the concentration range satisfactorily
20
covered the likely metabolite concentrations throughout the three growth phases
21
′𝟎
Calculation of 𝜟𝒓 𝑮′𝟎
𝟏 and 𝜟𝒓 𝑮𝟐 from literature data
22
′
The 𝛥𝑟 𝐺2′0 was calculated using the apparent equilibrium constant 𝐾𝐺𝑝𝑝
𝑝 of the glycerol-3-phosphatase reaction
23
at pH 7 and 308.15 K listed in Goldberg and Tewari (1994), which is based on Romero and de Meis (1989):
Range was derived from the minimum and maximum concentrations measured by Cronwright et al. (2002) in
24
𝛥𝑟 𝐺 ′0 = −𝑅 ∙ 𝑇 ∙ ln(𝐾 ′ )
25
′
−3
𝛥𝑟 𝐺2′0 = −𝑅 ∙ 𝑇 ∙ ln(𝐾𝐺𝑝𝑝
𝑝 ) = −8.31451 ∙ 10
(S1)
𝑘𝐽
𝑚𝑜𝑙 𝐾
∙ 308.15 𝐾 ∙ ln(68) = −10.8
𝑘𝐽
𝑚𝑜𝑙
26
′
The 𝛥𝑟 𝐺1′0 at pH 7 and 308.15 K was calculated from the apparent equilibrium constant 𝐾𝐺𝑝𝑑
𝑝,𝑅𝑒𝑣𝑒𝑟𝑠𝑒,𝑇1 =309.7 𝐾
27
′0
at pH 7 and 309.7 K and the standard transformed enthalpy of reaction Δ𝑟 𝐻𝐺𝑝𝑑
𝑝,𝑅𝑒𝑣𝑒𝑟𝑠𝑒 at pH 7 for the reverse
28
reaction (𝐺3𝑃 + 𝑁𝐴𝐷 → 𝐷𝐻𝐴𝑃 + 𝑁𝐴𝐷𝐻). Data are listed in Goldberg et al. (1993) and based on Young and
29
Pace (1958).
30
′
First, we adjusted 𝐾𝐺𝑝𝑑
𝑝,𝑅𝑒𝑣𝑒𝑟𝑠𝑒,𝑇1 =309.7 𝐾 to its appropriate value at 308.15 K to ensure equal reaction conditions
31
for both of the studied reactions (Gpd p and Gpp p at pH 7 and 308.15 K). The adjustment was made by
32
′0
integrating Van’t Hoff’s equation, assuming that Δ𝑟 𝐻𝐺𝑝𝑑
𝑝,𝑅𝑒𝑣𝑒𝑟𝑠𝑒 is independent of temperature:
33
ln (
34
(S2a)
35
Converting Eq. (S2a) to
′
𝐾𝐺𝑝𝑑
𝑝,𝑅𝑒𝑣𝑒𝑟𝑠𝑒,𝑇2 =308.15 𝐾
′
𝐾𝐺𝑝𝑑
𝑝,𝑅𝑒𝑣𝑒𝑟𝑠𝑒,𝑇1 =309.7 𝐾
)=
′0
Δ𝑟 𝐻𝐺𝑝𝑑
𝑝,𝑅𝑒𝑣𝑒𝑟𝑠𝑒
𝑅
36
′
′
𝐾𝐺𝑝𝑑
𝑝,𝑅𝑒𝑣𝑒𝑟𝑠𝑒,𝑇2 =308.15 𝐾 = 𝐾𝐺𝑝𝑑 𝑝,𝑅𝑒𝑣𝑒𝑟𝑠𝑒,𝑇1 =309.7 𝐾 ∙ exp (
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(S2b)
38
1
1
𝑇1
𝑇2
∙( − )
′0
Δ𝑟𝐻𝐺𝑝𝑑
𝑝,𝑅𝑒𝑣𝑒𝑟𝑠𝑒
𝑅
1
1
𝑇1
𝑇2
∙ ( − )),
we obtain
2
′
𝐾𝐺𝑝𝑑
𝑝,𝑅𝑒𝑣𝑒𝑟𝑠𝑒,𝑇2 =308.15 𝐾
39
= 3.07 ∙ 10
−5
∙ exp (
−30
𝑘𝐽
𝑚𝑜𝑙
𝑘𝐽
8.31451 ∙ 10−3
𝑚𝑜𝑙 𝐾
1
1
∙(
−
))
309.7 𝐾 308.15 𝐾
= 3.255 ∙ 10−5 .
40
41
′
Second, we calculated the apparent equilibrium constant 𝐾𝐺𝑝𝑑
𝑝,𝐹𝑜𝑟𝑤𝑎𝑟𝑑,𝑇2 =308.15 𝐾 for the forward reaction
42
(𝐷𝐻𝐴𝑃 + 𝑁𝐴𝐷𝐻 → 𝐺3𝑃 + 𝑁𝐴𝐷):
′
𝐾𝐺𝑝𝑑
𝑝,𝐹𝑜𝑟𝑤𝑎𝑟𝑑,𝑇2 =308.15 𝐾 =
43
1
′
𝐾𝐺𝑝𝑑
𝑝,𝑅𝑒𝑣𝑒𝑟𝑠𝑒,𝑇2 =308.15 𝐾
=
1
3.255∙10−5
= 3.072 ∙ 104
44
′
This value of 𝐾𝐺𝑝𝑑
𝑝,𝐹𝑜𝑟𝑤𝑎𝑟𝑑,𝑇2 =308.15 𝐾 is consistent with the apparent equilibrium constant of glycerol-3-
45
phosphate dehydrogenase reported in other works (Cai et al. 1996; Cronwright et al. 2002).
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Finally, we computed 𝛥𝑟 𝐺1′0 by Eq. (S1):
47
′
−3
𝛥𝑟 𝐺1′0 = −𝑅 ∙ 𝑇 ∙ ln(𝐾𝐺𝑝𝑑
𝑝,𝐹𝑜𝑟𝑤𝑎𝑟𝑑,𝑇2 =308.15 𝐾 ) = −8.31451 ∙ 10
48
= −26.5
(S3)
𝑘𝐽
∙ 308.15 𝐾 ∙ ln(3.072 ∙ 104 )
𝑚𝑜𝑙 𝐾
𝑘𝐽
𝑚𝑜𝑙
49
Generating 𝜟𝒓 𝑮′𝟏 and 𝜟𝒓 𝑮′𝟐 combinations by sampling the metabolite concentrations in logarithmic space
50
For compatibility with the Artificial-Centering Hit-and-Run sampler within the MATLAB function cprnd
51
(programmed by Benham (2011) based on Kaufman and Smith (1998)), the sampling task was formulated as:
𝚫𝐫 𝐆′𝟎
52
𝐍 𝐓 ∙ ln(𝐜) < −
53
ln(𝐜𝐦𝐢𝐧 ) < ln(𝐜) < ln(𝐜𝐦𝐚𝐱 )
𝑅∙𝑇
(S4)
(S5)
54
where 𝐜𝐦𝐢𝐧 and 𝐜𝐦𝐚𝐱 are the vectors of minimum and maximum metabolite concentrations, respectively. Without
55
the constraint 𝚫𝐫 𝐆′ < 𝟎, the algorithm uniformly samples the natural logarithms of the metabolite concentrations
56
over their restricted ranges (see Eq. (S5)). Under the constraint 𝚫𝐫 𝐆′ < 𝟎, we obtained 4 × 106 combinations of
57
the two transformed Gibbs energies of reaction 𝛥𝑟 𝐺1′ and 𝛥𝑟 𝐺2′ (see Fig. 3 in the main text). The natural
58
logarithms of the metabolite concentrations were sampled as follows (each logarithm was sampled 4 × 106
59
times):
3
60
61
Under the constraint 𝚫𝐫 𝐆′ < 𝟎 (Eq. (1) in the main text), the natural logarithms of glycerol-3-phosphate (G3P),
62
glycerol and orthophosphate (Pi) were not uniformly sampled over their restricted ranges. These three
4
63
metabolites are involved in the glycerol-3-phosphatase reaction (Gpp p, reaction 2). As mentioned in the main
64
text, some combinations of constrained concentration values were not allowed to fulfill 𝚫𝐫 𝐆′ < 𝟎 (e.g. high
65
product concentrations (glycerol and P i), low reactant concentrations (G3P) or a combination of both). Therefore,
66
high product concentrations (glycerol and P i) and low reactant concentrations (G3P) were less frequently
67
sampled (see histograms above).
68
Calculation of scaled elasticity values for determining the scaled flux control coefficients
69
In this study, the scaled elasticity values were calculated using two sampling approaches:
70
In case (i), the scaled elasticities were uniformly sampled and correlated within defined ranges (Steuer et al.
71
𝐯
2006; Grimbs et al. 2007). The scaled elasticities in 𝐄𝐆𝟑𝐏
were related as follows:
𝑣
72
+,1
𝐸𝐺3𝑃
∈ (0, −1)
73
−,1
+,1
𝐸𝐺3𝑃
= 𝐸𝐺3𝑃
+ 1 ∈ (0, 1)
74
+,2
𝐸𝐺3𝑃
∈ (0, 1)
75
−,2
+,2
𝐸𝐺3𝑃
= 𝐸𝐺3𝑃
− 1 ∈ (0, −1)
𝑣
(S6)
𝑣
(S7)
𝑣
𝑣
(S8)
𝑣
(S9)
𝑣
76
+,1
where (for example) 𝐸𝐺3𝑃
denotes the sensitivity of flux 𝑣+,1 to the metabolite concentration of glycerol-3-
77
+,1
+,2
phosphate (G3P). To generate the 𝐸𝐺3𝑃
and 𝐸𝐺3𝑃
values, pseudorandom numbers were drawn from the standard
78
uniform distribution on the open interval (0,-1) or (0,1), respectively, using the rand function of MATLAB
79
−,1
−,2
+,1
+,2
(version 8.1.0.604). The values of 𝐸𝐺3𝑃
and 𝐸𝐺3𝑃
immediately follow from the 𝐸𝐺3𝑃
and 𝐸𝐺3𝑃
(see Eqs. (S7)
80
and (S9)). The scaled elasticity values in case (i) were obtained by this procedure.
81
In case (ii), the scaled metabolite concentrations were calculated by uniformly sampling the degrees of saturation
82
of active sites. The scaled metabolite concentrations were inserted in scaled elasticity expressions derived from
83
𝐯
enzyme kinetic rate laws to determine the scaled elasticities in 𝐄𝐆𝟑𝐏
(Wang et al. 2004).
84
The general expression for the degree of saturation of an active site 𝜎𝐴 is given by (Wang et al. 2004;
85
Chakrabarti et al. 2013):
86
𝑣
𝑣
𝑣
𝜎𝐴 =
𝑣
[𝐴𝑆]
[𝐴𝑇 ]
=
𝑣
[𝑆]
𝐾𝑀
[𝑆]
+1
𝐾𝑀
with [𝐴 𝑇 ] = [𝐴] + [𝐴𝑆]
𝑣
(S10)
5
87
where 𝜎𝐴 denotes the ratio of the concentration of the active site-substrate complex [𝐴𝑆] to the total
88
concentration of the active site [𝐴 𝑇 ] ([𝐴] is the concentration of the free active site). Eq. (S10) can be rearranged
89
in terms of the scaled metabolite concentration
[𝑆]
90
91
𝐾𝑀
:
𝜎𝐴
1−𝜎𝐴
(S11a)
[𝑆]
92
93
=
[𝑆]
𝐾𝑀
𝐾𝑀
≔𝑠
(S11b)
94
Using Eq. (S11a), we can generate random independent samples of the scaled metabolite concentration 𝑠 by
95
uniformly sampling the degree of saturation of an active site 𝜎𝐴 between 0 and 1 (𝜎𝐴 ≈ 0, approximately non-
96
saturated; 𝜎𝐴 ≈ 1, nearly fully saturated). As above, the degrees of saturation of active sites were sampled by
97
MATLAB’s rand function. Details of the scaled metabolite concentration concept are provided in Wang et al.
98
(2004) and Chakrabarti et al. (2013).
99
Finally, the scaled elasticity values are computed using the following expressions, which depend on the scaled
100
metabolite concentrations. Our aim was to evaluate the methodology and examine the influence of the enzyme
101
kinetics on the flux control coefficients. Therefore, we assumed no specific knowledge of the enzyme kinetic
102
rate laws governing reactions 1 and 2 (cf. Cronwright et al. (2002)).
103
The scaled elasticity expressions were based on the following reaction equations and the convenience rate law of
104
Liebermeister and Klipp (2006):
105
Reaction 1: 𝐷𝐻𝐴𝑃 + 𝑁𝐴𝐷𝐻 ↔ 𝐺3𝑃 + 𝑁𝐴𝐷
106
Reaction 2: 𝐺3𝑃 ↔ 𝑃𝑖 + 𝐺𝑙𝑦𝑐𝑒𝑟𝑜𝑙
107
In reaction 2, water was disregarded as a reactant in the rate equations (as implemented in Cronwright et al.
108
(2002)). Based on these two stoichiometries and the convenience rate law of Liebermeister and Klipp (2006), we
109
derived the following rate equations for the forward and backward fluxes:
110
𝑣+,1 = 𝑣𝑚𝑎𝑥,+,1 ∙
111
𝑣−,1 = 𝑣𝑚𝑎𝑥,−,1 ∙
𝑑ℎ𝑎𝑝∙𝑛𝑎𝑑ℎ
1+𝑑ℎ𝑎𝑝+𝑛𝑎𝑑ℎ+𝑑ℎ𝑎𝑝∙𝑛𝑎𝑑ℎ+𝑔3𝑝(1) +𝑛𝑎𝑑+𝑔3𝑝(1) ∙𝑛𝑎𝑑
𝑔3𝑝(1) ∙𝑛𝑎𝑑
1+𝑑ℎ𝑎𝑝+𝑛𝑎𝑑ℎ+𝑑ℎ𝑎𝑝∙𝑛𝑎𝑑ℎ+𝑔3𝑝(1) +𝑛𝑎𝑑+𝑔3𝑝(1) ∙𝑛𝑎𝑑
(S12)
(S13)
6
112
113
𝑣+,2 = 𝑣𝑚𝑎𝑥,+,2 ∙
(S14)
114
115
𝑔3𝑝(2)
1+𝑔3𝑝(2) +𝑝𝑖 +𝑔𝑙𝑦𝑐𝑒𝑟𝑜𝑙+𝑝𝑖 ∙𝑔𝑙𝑦𝑐𝑒𝑟𝑜𝑙
𝑣−,2 = 𝑣𝑚𝑎𝑥,−,2 ∙
𝑝𝑖 ∙𝑔𝑙𝑦𝑐𝑒𝑟𝑜𝑙
1+𝑔3𝑝(2) +𝑝𝑖 +𝑔𝑙𝑦𝑐𝑒𝑟𝑜𝑙+𝑝𝑖 ∙𝑔𝑙𝑦𝑐𝑒𝑟𝑜𝑙
(S15)
116
where (e.g. in reaction 1) 𝑔3𝑝(1) denotes the scaled metabolite concentration of glycerol-3-phosphate (G3P)
117
given by Eq. (S11b) ([𝐺3𝑃]/𝐾𝑀,𝐺3𝑃,1 ) and 𝑣𝑚𝑎𝑥,+,1 and 𝑣𝑚𝑎𝑥,−,1 are the maximal forward and maximal
118
backward rates of reaction 1, respectively. In reaction 2, 𝑔3𝑝(2) denotes the scaled metabolite concentration of
119
glycerol-3-phosphate (G3P) ([𝐺3𝑃]/𝐾𝑀,𝐺3𝑃,2 ). In the rate laws (S12) and (S13), we neglected any modifiers of
120
the inhibitors ATP, ADP and F16BP (fructose 1,6-bisphosphate) because no a priori knowledge of the enzyme
121
kinetics is assumed (cf. Cronwright et al. (2002)).
122
The scaled elasticity expressions of these rate equations (Eqs. (S12)–(S15)), with respect to glycerol-3-
123
phosphate, are given by
𝑣
124
+,1
𝐸𝐺3𝑃
=
125
−,1
𝐸𝐺3𝑃
=
126
+,2
𝐸𝐺3𝑃
=
𝑣
𝑣
127
−𝑔3𝑝(1) ∙(1+𝑛𝑎𝑑)
1+𝑑ℎ𝑎𝑝+𝑛𝑎𝑑ℎ+𝑑ℎ𝑎𝑝∙𝑛𝑎𝑑ℎ+𝑔3𝑝(1) +𝑛𝑎𝑑+𝑔3𝑝(1) ∙𝑛𝑎𝑑
1+𝑑ℎ𝑎𝑝+𝑛𝑎𝑑ℎ+𝑑ℎ𝑎𝑝∙𝑛𝑎𝑑ℎ+𝑛𝑎𝑑
1+𝑑ℎ𝑎𝑝+𝑛𝑎𝑑ℎ+𝑑ℎ𝑎𝑝∙𝑛𝑎𝑑ℎ+𝑔3𝑝(1) +𝑛𝑎𝑑+𝑔3𝑝(1) ∙𝑛𝑎𝑑
(S16)
(S17)
1+𝑝𝑖 +𝑔𝑙𝑦𝑐𝑒𝑟𝑜𝑙+𝑝𝑖 ∙𝑔𝑙𝑦𝑐𝑒𝑟𝑜𝑙
1+𝑔3𝑝(2) +𝑝𝑖 +𝑔𝑙𝑦𝑐𝑒𝑟𝑜𝑙+𝑝𝑖 ∙𝑔𝑙𝑦𝑐𝑒𝑟𝑜𝑙
(S18)
𝑣
−,2
𝐸𝐺3𝑃
=
128
129
−𝑔3𝑝(2)
1+𝑔3𝑝(2) +𝑝𝑖 +𝑔𝑙𝑦𝑐𝑒𝑟𝑜𝑙+𝑝𝑖 ∙𝑔𝑙𝑦𝑐𝑒𝑟𝑜𝑙
(S19)
130
Calculation of scaled flux control coefficients related to the steady-state net flux
131
The scaled flux control coefficients related to the forward and backward fluxes are computed by applying Eq. (9)
132
in the main text. Subsequently, the scaled flux control coefficient related to the steady-state net flux is calculated
133
in terms of the scaled flux control coefficients related to the forward and backward fluxes as follows (shown for
134
reaction 1):
7
𝑣
𝑣
𝐶𝑒1𝑛𝑒𝑡 = 𝐶𝑒1+,1 ∙
135
136
𝑣+,1
𝑣𝑛𝑒𝑡
𝑣
− 𝐶𝑒1−,1 ∙
𝑣−,1
𝑣𝑛𝑒𝑡
(S20)
137
A derivation of Eq. (S20) follows:
138
Beginning with the established equation of steady-state net flux (Eq. (5) in the main text):
139
𝑣𝑛𝑒𝑡 = 𝑣+,1 − 𝑣−,1
140
(S20a)
141
we construct the partial derivative of Eq. (S20a) with respect to 𝑒1 :
𝜕𝑣𝑛𝑒𝑡
142
𝜕𝑒1
143
144
𝜕𝑒1
=
𝜕𝑣+,1
𝜕𝑒1
−
𝜕𝑣−,1
𝜕𝑒1
(S20b)
Multiplying Eq. (S20b) by 𝑒1 /𝑣𝑛𝑒𝑡 ∙ 𝑒1 /𝑣+,1 ∙ 𝑒1 /𝑣−,1 , we obtain
𝜕𝑣𝑛𝑒𝑡
145
146
𝜕(𝑣+,1 −𝑣−,1 )
=
𝜕𝑒1
∙
𝑒1
𝑣𝑛𝑒𝑡
∙
𝑒1
𝑣+,1
∙
𝑒1
𝑣−,1
=
𝜕𝑣+,1
𝜕𝑒1
∙
𝑒1
𝑣𝑛𝑒𝑡
∙
𝑒1
𝑣+,1
∙
𝑒1
𝑣−,1
−
𝜕𝑣−,1
𝜕𝑒1
∙
𝑒1
𝑣𝑛𝑒𝑡
∙
𝑒1
𝑣+,1
∙
𝑒1
𝑣−,1
(S20c)
𝑣
𝑣
𝑣
147
Next, denoting 𝜕𝑣𝑛𝑒𝑡 /𝜕𝑒1 ∙ 𝑒1 /𝑣𝑛𝑒𝑡 , 𝜕𝑣+,1 /𝜕𝑒1 ∙ 𝑒1 /𝑣+,1 and 𝜕𝑣−,1 /𝜕𝑒1 ∙ 𝑒1 /𝑣−,1 as 𝐶𝑒1𝑛𝑒𝑡 , 𝐶𝑒1+,1 and 𝐶𝑒1−,1 ,
148
respectively, we obtain
𝑣
𝐶𝑒1𝑛𝑒𝑡 ∙
149
150
𝑒1
𝑣+,1
∙
𝑒1
𝑣−,1
𝑣
= 𝐶𝑒1+,1 ∙
𝑒1
𝑣𝑛𝑒𝑡
∙
𝑒1
𝑣−,1
𝑣
− 𝐶𝑒1−,1 ∙
𝑒1
𝑣𝑛𝑒𝑡
∙
𝑒1
𝑣+,1
.
(S20d)
151
Multiplying Eq. (S20d) by 𝑣+,1 ∙ 𝑣−,1 and canceling out the 𝑒1 ∙ 𝑒1 terms, we obtain Eq. (S20).
152
Combinations of thermodynamically feasible disequilibrium ratios 𝝆𝟏 and 𝝆𝟐 (converted from the 𝜟𝒓 𝑮′𝟏
153
and 𝜟𝒓 𝑮′𝟐 values; see Fig. 3 in the main text)
8
154
155
156
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