Figures and data in Emergence and propagation of epistasis in metabolic networks

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Tables
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15 figures, 6 tables and 3 additional files

Figures

Figure 1

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Illustration of a hierarchical metabolic network and its coarse-graining.

(A) White rectangle represents the whole metabolic network $𝒩$ . Example subnetworks μ and ν are represented by the dark and light gray rectangles. Only metabolites and reactions that belong to these subnetworks are shown; other metabolites and reactions in $𝒩$ are not shown. Metabolites 1 and 5 may be adjacent to other metabolites in $𝒩$ ; this fact is represented by short black lines that do not terminate in metabolites. Subnetworks μ and ν are both modules because there exists a simple path connecting their I/O metabolites that lies within μ and ν and contains all their internal metabolites (dashed blue line). (B) Network $𝒩$ can be coarse-grained by replacing module μ at steady state with an effective reaction between its I/O metabolites 1 and 2, with the rate constant is $y_{μ}$ . (C) Network $𝒩$ can be coarse-grained by replacing module ν at steady state with an effective reaction between its I/O metabolites 1 and 5, with the rate constant is $y_{ν}$ .

Figure 2

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Propagation of epistasis.

Properties of Equation 6 that maps lower-level epistasis $ε y_{μ}$ onto higher-level epistasis $ε y_{ν}$ . Slope $1 / C$ and fixed point $\bar{ε}$ depend on the topology and the rate constants of the higher-level module ν, but they are bounded, as shown. Thus, the fixed point $\bar{ε}$ of this map lies between 0 and 1 and is always unstable (open circle).

Figure 3

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Emergence of epistasis and its dependence on the topological relationship between the reactions affected by mutations.

(A) An example of a simple module ν (same as in Figure 1A) where negative, weak positive and strong positive epistasis can emerge between two mutations A and B. (B) Epistasis between mutations A and B at the level of module ν depicted in (A) as a function of the rate constant $z$ of a third reaction. The values of other parameters of the network are given in Materials and Methods. (C) An example of a simple module where reactions affected by mutations are strictly parallel. In such cases, epistasis for the effective rate constant $y_{ν}$ is non-positive. Dashed blue lines highlight paths that connect the I/O metabolites and each contain only one of the affected reactions. (D) An example of a simple module where reactions affected by mutations are strictly serial. In such cases, epistasis for the effective rate constant $y_{ν}$ is equal to or greater than 1 (i.e. strongly positive). Dashed blue line highlights a path that connects the I/O metabolites and contains both affected reactions.

Figure 4 with 3 supplements

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Sensitivity of results of Theorem 1 and Theorem 2 with respect to the magnitude of mutational effects.

(A) Distribution of the position of the fixed point $\bar{ε}$ of the function $ϕ$ that maps lower-level epistasis $ε y_{μ}$ onto higher-level epistasis $ε y_{ν}$ in modules with random parameters and for mutations with positive effects on $y_{μ}$ (see text and Materials and methods for details). All cases are shown in Figure 4—figure supplement 1 and Figure 4—figure supplement 2. The effect size of both mutations is indicated on each panel. ‘No f.p'. indicates that no fixed point exists. (B) Fraction of sampled modules (averaged across generating topologies) where mutations affect strictly serial reactions but the epistasis coefficient is less than 1, contrary to the statement of Theorem 2 (see text and Materials and methods for details). All cases stratified by generating topology are shown in Figure 4—figure supplement 3.

Figure 4—figure supplement 1

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Distribution of the position of the fixed point in 1000 modules from the topological class $ℳ^{io}$ with random parameters.

Notations are as in Figure 4.

Figure 4—figure supplement 2

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Distribution of the position of the fixed point in 1000 modules from the topological class $ℳ^{i}$ with random parameters.

Notations are as in Figure 4.

Figure 4—figure supplement 3

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Fraction of sampled modules with different strictly serial generating topologies where the epistasis coefficient falls between 0.01 and 0.99.

Dark gray bars show cases with epistasis below 1 (γ = 0:01). The same data is shown in Figure 4B, averaged over 11 generating topologies. Light gray bars show additional cases where epistasis is clustered around 1 (see Materials and methods for details). The effects of both mutations are indicated on each panel.

Figure 5 with 3 supplements

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Figure 5—figure supplement 1

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Detailed schematic of the kinetic model of glycolysis.

Blue circles indicate metabolites (see Table 5 for abbreviations). Orange rectangles indicate enzymes (see Table 6 for abbreviations). Double-arrows indicate reversible reactions. Arrows with a fletch indicate irreversible reactions. Black circles indicate reactions with multiple substrates or products. Different shades of gray indicate the FULL, GPP, UGPP, and LG models defined in Table 4.

Figure 5—figure supplement 2

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Control coefficients for the output flux in the FULL module.

The first-order flux control coefficients (FCCs) of all 18 reactions with respect to the output flux in the unpperturbed FULL model are shown. Eleven out of 18 reactions have positive FCCs, five (TA, TKa, TKb, R5PI, Ru5P) have zero FCCs and two (G6PDH and PEPCxyl) have negative FCCs. The 11 reactions with positive FCCs were retained for further analysis, the others were excluded. For these 11 reactions, the FCCs and epistasis coefficients with respect to fluxes through all modules (FULL, GPP, UGPP, LG) are shown in Figure 5—figure supplement 3.

Figure 5—figure supplement 3

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Control and epistasis coefficients for the fluxes through multiple sub-modules within glycolysis.

The top row shows the FCCs of 11 reactions whose FCCs in the FULL model are positive (see Figure 5—figure supplement 2). The matrix below shows the epistasis coefficients for each pair of these reactions. The topological relationship between reactions is indicated in the lower left corner of each panel ('P', strictly parallel; 'S', strictly serial; 'SP' serial-parallel). Points are colored orange (green) if the epistasis coefficient is less than zero (greater than one). Backgrounds of different shades of gray indicate the sub-modules, as in Figure 5—figure supplement 1.

Figure 6

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Simple modules.

(A) Linear pathway. (B) Two parallel pathways.

Figure 7

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Classification of single-marked modules.

Left column shows a general module from each topological class. The right column shows a minimal fully connected module in each topological class (see text for details). Circles represent metabolites and lines represent reactions. Only the I/O metabolites and the metabolites that participate in the marked reaction are shown, all other metabolites are suppressed. Short lines that have only one terminal metabolite represent all remaining reactions in which this metabolite participates, reactions between all other metabolites are suppressed. Metabolites are labeled according to the conventions listed in the text. The marked reaction is colored orange and labeled $a$ . The module is represented by a gray rectangle, and the rest of the network is not shown.

Figure 8

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Classification of double-marked modules.

Notations as in Figure 7.

Figure 9

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A counter example illustrating that the converse to claim 2 in Proposition 4 may not be true.

Reactions $a$ and $b$ are parallel in $(μ, a, b)$ . CGP maps the double-marked module $(μ, a, b)$ onto the minimal double-marked module $(μ^{'}, a, b)$ where reactions $a$ and $b$ are not parallel.

Figure 10

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Graphical representation of strictly serial and strictly parallel generating topologies in the class $M^{io, io, \emptyset}$ .

Fully connected topology $io, io, \emptyset, F$ is shown for reference (same as in Figure 8).

Figure 11

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Graphical representation of strictly serial and strictly parallel generating topologies in class $M^{io, i, I}$ .

Fully connected topology $io, i, I, F$ is shown for reference (same as in Figure 8).

Figure 12

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Graphical representation of strictly serial and strictly parallel generating topologies in class $M^{io, i, \emptyset}$ .

Fully connected topology $io, i, \emptyset, F$ is shown for reference (same as in Figure 8).

Figure 13

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Graphical representation of strictly serial and strictly parallel generating topologies in class $M^{i, i, I}$ .

Fully connected topology $i, i, I, F$ is shown for reference (same as in Figure 8).

Figure 14

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Graphical representation of strictly serial and strictly parallel generating topologies in class $M^{i, i, \emptyset}$ .

Fully connected topology $i, i, \emptyset, F$ is shown for reference (same as in Figure 8).

Appendix 2—figure 1

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Tables

Table 1

Classification of double-marked modules.

Metabolites are labeled according to conventions described in the text. $m_{ℳ}$ is the minimum number of internal metabolites in a module from class $ℳ$ . $A_{ℳ}$ is the set of internal and I/O metabolites in all minimal modules in class $ℳ$ .

Class	$a$	$b$	Shared metab.	Verbal description	$m_{ℳ}$	$A_{ℳ}$	Equation for f₂
$ℳ^{b, io, IO}$	$(1, 2)$	$(1, 3)$	1	Bypass and i/o reactions, shared I/O metabolite	2	${1, 2, 3}$	Equation (34)
$ℳ^{b, i, \emptyset}$	$(1, 2)$	$(3, 4)$	–	Bypass and internal reactions, no shared metabolies	2	${1, 2, 3, 4}$	Equation (35)
$ℳ^{io, io, I}$	$(1, 3)$	$(2, 3)$	3	i/o reactions, shared internal metabolite	1	${1, 2, 3}$	Equation (36)
$ℳ^{io, io, IO}$	$(1, 3)$	$(1, 4)$	1	i/o reactions, shared I/O metabolite	2	${1, 2, 3, 4}$	Equation (37)
$ℳ^{io, io, \emptyset}$	$(1, 3)$	$(2, 4)$	–	i/o reactions, no shared metabolites	2	${1, 2, 3, 4}$	Equation (38)
$ℳ^{io, i, I}$	$(1, 3)$	$(3, 4)$	3	i/o and internal reactions, shared internal metabolite	2	${1, 2, 3, 4}$	Equation (39)
$ℳ^{io, i, \emptyset}$	$(1, 3)$	$(4, 5)$	–	i/o and internal reactions, no shared metabolites	3	${1, 2, 3, 4, 5}$	Equation (40)
$ℳ^{i, i, I}$	$(3, 4)$	$(3, 5)$	3	Internal reactions, shared internal metabolite	3	${1, 2, 3, 4, 5}$	Equation (41)
$ℳ^{i, i, \emptyset}$	$(3, 4)$	$(5, 6)$	–	Internal reactions, no shared metabolites	4	${1, 2, 3, 4, 5, 6}$	Equation (42)

Table 2

Strictly parallel generating topologies.

	Marked reactions		Generating topology
Class	$a$	$b$	ID	Excluded reactions	Figure
$ℳ^{b, io, IO}$	$1 \leftrightarrow 2$	$1 \leftrightarrow 3$	$b, io, IO, F$	$\emptyset$	Figure 7
$ℳ^{b, i, \emptyset}$	$1 \leftrightarrow 2$	$3 \leftrightarrow 4$	$b, i, \emptyset, F$	$\emptyset$	Figure 7
$ℳ^{io, io, IO}$	$1 \leftrightarrow 3$	$1 \leftrightarrow 4$	$io, io, IO, F$	$\emptyset$	Figure 7
$ℳ^{io, io, \emptyset}$	$1 \leftrightarrow 3$	$2 \leftrightarrow 4$	$io, io, \emptyset, P$	${3 \leftrightarrow 4}$	Figure 9
$ℳ^{io, i, I}$	$1 \leftrightarrow 3$	$3 \leftrightarrow 4$	$io, i, I, P$	${2 \leftrightarrow 4}$	Figure 10
$ℳ^{io, i, \emptyset}$	$1 \leftrightarrow 3$	$4 \leftrightarrow 5$	$io, i, \emptyset, P_{1}$	${3 \leftrightarrow 4, 3 \leftrightarrow 5}$	Figure 11
			$io, i, \emptyset, P_{2}$	${2 \leftrightarrow 5, 3 \leftrightarrow 5}$
			$io, i, \emptyset, P_{3}$	${2 \leftrightarrow 4, 2 \leftrightarrow 5}$
$ℳ^{i, i, I}$	$3 \leftrightarrow 4$	$3 \leftrightarrow 5$	$i, i, I, P_{1}$	${2 \leftrightarrow 4, 2 \leftrightarrow 5}$	Figure 12
$ℳ^{i, i, I}$	$3 \leftrightarrow 4$	$3 \leftrightarrow 5$	$i, i, I, P_{2}$	${1 \leftrightarrow 5, 2 \leftrightarrow 5}$	Figure 12
$ℳ^{i, i, \emptyset}$	$3 \leftrightarrow 4$	$5 \leftrightarrow 6$	$i, i, \emptyset, P_{1}$	${3 \leftrightarrow 5, 3 \leftrightarrow 6, 4 \leftrightarrow 5, 4 \leftrightarrow 6}$	Figure 13
			$i, i, \emptyset, P_{2}$	${1 \leftrightarrow 5, 1 \leftrightarrow 6, 2 \leftrightarrow 5, 2 \leftrightarrow 6}$
			$i, i, \emptyset, P_{3}$	${2 \leftrightarrow 4, 2 \leftrightarrow 6, 3 \leftrightarrow 6, 4 \leftrightarrow 5, 4 \leftrightarrow 6}$
			$i, i, \emptyset, P_{4}$	${2 \leftrightarrow 4, 2 \leftrightarrow 5, 2 \leftrightarrow 6, 4 \leftrightarrow 5, 4 \leftrightarrow 6}$
			$i, i, \emptyset, P_{5}$	${1 \leftrightarrow 6, 2 \leftrightarrow 4, 2 \leftrightarrow 5, 2 \leftrightarrow 6, 4 \leftrightarrow 6}$
			$i, i, \emptyset, P_{6}$	${1 \leftrightarrow 4, 1 \leftrightarrow 6, 2 \leftrightarrow 4, 2 \leftrightarrow 6, 4 \leftrightarrow 6}$
			$i, i, \emptyset, P_{7}$	${1 \leftrightarrow 4, 1 \leftrightarrow 5, 2 \leftrightarrow 3, 2 \leftrightarrow 6, 3 \leftrightarrow 5, 4 \leftrightarrow 6}$

Table 3

Strictly serial generating topologies.

	Marked reactions		Generating topologies
Class	$a$	$b$	ID	Excluded reactions	Figure
$ℳ^{io, io, I}$	$1 \leftrightarrow 3$	$2 \leftrightarrow 3$	$io, io, I, F$	$\emptyset$	Figure 7
$ℳ^{io, io, \emptyset}$	$1 \leftrightarrow 3$	$2 \leftrightarrow 4$	$io, io, \emptyset, S$	${2 \leftrightarrow 3}$	Figure 9
$ℳ^{io, i, I}$	$1 \leftrightarrow 3$	$3 \leftrightarrow 4$	$io, i, I, S_{1}$	${2 \leftrightarrow 3}$	Figure 10
$ℳ^{io, i, I}$	$1 \leftrightarrow 3$	$3 \leftrightarrow 4$	$io, i, I, S_{2}$	${1 \leftrightarrow 4}$	Figure 10
$ℳ^{io, i, \emptyset}$	$1 \leftrightarrow 3$	$4 \leftrightarrow 5$	$io, i, \emptyset, S_{1}$	${1 \leftrightarrow 4, 1 \leftrightarrow 5}$	Figure 11
			$io, i, \emptyset, S_{2}$	${2 \leftrightarrow 3, 2 \leftrightarrow 4, 3 \leftrightarrow 5}$
			$io, i, \emptyset, S_{3}$	${1 \leftrightarrow 5, 2 \leftrightarrow 3, 2 \leftrightarrow 5}$
$ℳ^{i, i, I}$	$3 \leftrightarrow 4$	$3 \leftrightarrow 5$	$i, i, I, S_{1}$	${1 \leftrightarrow 3, 2 \leftrightarrow 3}$	Figure 12
$ℳ^{i, i, I}$	$3 \leftrightarrow 4$	$3 \leftrightarrow 5$	$i, i, I, S_{2}$	${2 \leftrightarrow 3, 2 \leftrightarrow 5, 4 \leftrightarrow 5}$	Figure 12
$ℳ^{i, i, \emptyset}$	$3 \leftrightarrow 4$	$5 \leftrightarrow 6$	$i, i, \emptyset, S_{1}$	${2 \leftrightarrow 3, 2 \leftrightarrow 5, 2 \leftrightarrow 6, 4 \leftrightarrow 5, 4 \leftrightarrow 6}$	Figure 13
$ℳ^{i, i, \emptyset}$	$3 \leftrightarrow 4$	$5 \leftrightarrow 6$	$i, i, \emptyset, S_{2}$	${1 \leftrightarrow 3, 1 \leftrightarrow 6, 2 \leftrightarrow 3, 2 \leftrightarrow 6, 4 \leftrightarrow 6}$	Figure 13

Table 4

Definition of modules in the glycolysis network shown in Figure 5—figure supplement 1.

Enzyme abbreviations are listed in Table 6. Metabolite abbreviations are listed in Table 5.

Model	Internal metabolites	Concentrations of I/O metabolites	Reactions	Output flux
UGPP	6 pg, dhap, e4p, f6p, fdp, rib5p, ribu5p, sed7p, xyl5p	[g6p]=3.82 mM, [gap]=0.44 mM	ALDO, G6PDH, PFK, PGDH, PGI, Ru5P, R5PI, TA, TIS, TKa, TKb	$J_{ALDO} + J_{TIS} + J_{TKb} + J_{TKa} - J_{TA}$
LG	2 pg, 3 pg, pgp	[gap]=0.44 mM, [pep]=0.08 mM	ENO, GAPDH, PGK, PGM	$J_{ENO}$
GPP	all in UGPP and in LG, gap	[g6p]=3.82 mM, [pep]=0.08 mM	all in UGPP and in LG	$J_{ENO}$
FULL	all in GPP, g6p, pep	[Ext glu]=2 µM, [pyr]=10 µM	all in GPP, PTS, PK, PEPCxyl	$J_{PK} + J_{PTS}$

Table 5

Names of metabolites used in the kinetic model of glycolysis.

2 pg	2-Phosphoglycerate
3 pg	3-Phosphoglycerate
6 pg	6-Phosphogluconate
dhap	Dihydroxyacetonephosphate
e4p	Erythrose-4-phosphate
f6p	Fructose-6-phosphate
fdp	Fructose-1,6-bisphosphate
g6p	Glucose-6-phosphate
gap	Glyceraldehyde-3-phosphate
glu	Glucose
pep	Phosphoenolpyruvate
pgp	1,3-Diphosphoglycerate
pyr	Pyruvate
rib5p	Ribose-5-phosphate
ribu5p	Ribulose-5-phosphate
sed7p	Sedoheptulose-7-phosphate
xyl5p	Xylulose-5-phosphate

Table 6

Names of enzymes used in the kinetic model of glycolysis.

ALDO	Aldolase
ENO	Enolase
G6PDH	Glucose-6-phosphate dehydrogenase
GAPDH	Glyceraldehyde-3-phosphate dehydrogenase
PFK	Phosphofructokinase
PGDH	6-Phosphogluconate dehydrogenase
PGI	Glucose-6-phosphateisomerase
PGK	Phosphoglycerate kinase
PGM	Phosphoglycerate mutase
PEPCxyl	PEP carboxylase
PK	Pyruvate kinase
PTS	Phosphotransferase system
R5PI	Ribose-phosphateisomerase
Ru5P	Ribulose-phosphate epimerase
TA	Transaldolase
TIS	Triosephosphate isomerase
TKa	Transketolase, reaction a
TKb	Transketolase, reaction b

Additional files

Supplementary file 1 Mathematica notebook ‘Case i,i,emptyset,P7.nb’ for evaluating epistasis for the generating topology $i, i, \emptyset, P_{7}$ .: https://cdn.elifesciences.org/articles/60200/elife-60200-supp1-v2.nb
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Supplementary file 2 PDF version of the Mathematica notebook ‘Case i,i,emptyset,P7.nb’.: https://cdn.elifesciences.org/articles/60200/elife-60200-supp2-v2.pdf
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Transparent reporting form: https://cdn.elifesciences.org/articles/60200/elife-60200-transrepform-v2.docx
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Sergey Kryazhimskiy

(2021)

Emergence and propagation of epistasis in metabolic networks

eLife 10:e60200.

https://doi.org/10.7554/eLife.60200

Figures

Illustration of a hierarchical metabolic network and its coarse-graining.

Propagation of epistasis.

Emergence of epistasis and its dependence on the topological relationship between the reactions affected by mutations.

Sensitivity of results of Theorem 1 and Theorem 2 with respect to the magnitude of mutational effects.

Distribution of the position of the fixed point in 1000 modules from the topological class $ℳ^{io}$ with random parameters.

Distribution of the position of the fixed point in 1000 modules from the topological class $ℳ^{i}$ with random parameters.

Fraction of sampled modules with different strictly serial generating topologies where the epistasis coefficient falls between 0.01 and 0.99.

Epistasis in a kinetic model of Escherichia coli glycolysis.

Detailed schematic of the kinetic model of glycolysis.

Control coefficients for the output flux in the FULL module.

Control and epistasis coefficients for the fluxes through multiple sub-modules within glycolysis.

Simple modules.

Classification of single-marked modules.

Classification of double-marked modules.

A counter example illustrating that the converse to claim 2 in Proposition 4 may not be true.

Graphical representation of strictly serial and strictly parallel generating topologies in the class $M^{io, io, \emptyset}$ .

Graphical representation of strictly serial and strictly parallel generating topologies in class $M^{io, i, I}$ .

Graphical representation of strictly serial and strictly parallel generating topologies in class $M^{io, i, \emptyset}$ .

Graphical representation of strictly serial and strictly parallel generating topologies in class $M^{i, i, I}$ .

Graphical representation of strictly serial and strictly parallel generating topologies in class $M^{i, i, \emptyset}$ .

Illustration for the proof of Lemma 1.

Tables

Classification of double-marked modules.

Strictly parallel generating topologies.

Strictly serial generating topologies.

Definition of modules in the glycolysis network shown in Figure 5—figure supplement 1.

Names of metabolites used in the kinetic model of glycolysis.

Names of enzymes used in the kinetic model of glycolysis.

Additional files

Supplementary file 1

Supplementary file 2

Transparent reporting form

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Illustration of a hierarchical metabolic network and its coarse-graining.

Propagation of epistasis.

Emergence of epistasis and its dependence on the topological relationship between the reactions affected by mutations.

Sensitivity of results of Theorem 1 and Theorem 2 with respect to the magnitude of mutational effects.

Distribution of the position of the fixed point in 1000 modules from the topological class ℳio with random parameters.

Distribution of the position of the fixed point in 1000 modules from the topological class ℳi with random parameters.

Fraction of sampled modules with different strictly serial generating topologies where the epistasis coefficient falls between 0.01 and 0.99.

Epistasis in a kinetic model of Escherichia coli glycolysis.

Detailed schematic of the kinetic model of glycolysis.

Control coefficients for the output flux in the FULL module.

Control and epistasis coefficients for the fluxes through multiple sub-modules within glycolysis.

Simple modules.

Classification of single-marked modules.

Classification of double-marked modules.

A counter example illustrating that the converse to claim 2 in Proposition 4 may not be true.

Graphical representation of strictly serial and strictly parallel generating topologies in the class Mio,io,∅.

Graphical representation of strictly serial and strictly parallel generating topologies in class Mio,i,I.

Graphical representation of strictly serial and strictly parallel generating topologies in class Mio,i,∅.

Graphical representation of strictly serial and strictly parallel generating topologies in class Mi,i,I.

Graphical representation of strictly serial and strictly parallel generating topologies in class Mi,i,∅.

Illustration for the proof of Lemma 1.

Classification of double-marked modules.

Strictly parallel generating topologies.

Strictly serial generating topologies.

Definition of modules in the glycolysis network shown in Figure 5—figure supplement 1.

Names of metabolites used in the kinetic model of glycolysis.

Names of enzymes used in the kinetic model of glycolysis.

Supplementary file 1

Supplementary file 2

Transparent reporting form

Downloads (link to download the article as PDF)

Open citations (links to open the citations from this article in various online reference manager services)

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Distribution of the position of the fixed point in 1000 modules from the topological class $ℳ^{io}$ with random parameters.

Distribution of the position of the fixed point in 1000 modules from the topological class $ℳ^{i}$ with random parameters.

Graphical representation of strictly serial and strictly parallel generating topologies in the class $M^{io, io, \emptyset}$ .

Graphical representation of strictly serial and strictly parallel generating topologies in class $M^{io, i, I}$ .

Graphical representation of strictly serial and strictly parallel generating topologies in class $M^{io, i, \emptyset}$ .

Graphical representation of strictly serial and strictly parallel generating topologies in class $M^{i, i, I}$ .

Graphical representation of strictly serial and strictly parallel generating topologies in class $M^{i, i, \emptyset}$ .