1. Computational and Systems Biology
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Emergence and propagation of epistasis in metabolic networks

  1. Sergey Kryazhimskiy  Is a corresponding author
  1. Division of Biological Sciences, University of California, San Diego, United States
Research Article
Cite this article as: eLife 2021;10:e60200 doi: 10.7554/eLife.60200
15 figures, 6 tables and 3 additional files

Figures

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 yμ. (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 yν.

Propagation of epistasis.

Properties of Equation 6 that maps lower-level epistasis ε⁢yμ onto higher-level epistasis ε⁢yν. Slope 1/C and fixed point 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 ε¯ of this map lies between 0 and 1 and is always unstable (open circle).

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 yν 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 yν 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
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 ε¯ of the function ϕ that maps lower-level epistasis ε⁢yμ onto higher-level epistasis ε⁢yν in modules with random parameters and for mutations with positive effects on yμ (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
Distribution of the position of the fixed point in 1000 modules from the topological class ℳio with random parameters.

Notations are as in Figure 4.

Figure 4—figure supplement 2
Distribution of the position of the fixed point in 1000 modules from the topological class ℳi with random parameters.

Notations are as in Figure 4.

Figure 4—figure supplement 3
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
Epistasis in a kinetic model of Escherichia coli glycolysis.

(A) Simplified schematic of the model (see Figure 5—figure supplement 1 for details). Different shades of gray in the background highlight four modules as indicated (see text). Light blue circles represent metabolites. Reactions are shown as lines with dark gray boxes. The enzymes catalyzing reactions whose control coefficients with respect to the flux through the module are positive are named; other enzyme names are ommitted for clarity (see Table 5 and Table 6 for abbreviations). Three reactions, catalyzed by PGI, PFK, PGDH, for which the epistasis coefficients are shown in panel B are highlighted in dark blue, red, and orange, respectively. (B) Epistasis coefficients for flux through each module between mutations perturbing the respective reactions, computed at steady state (see text and Materials and methods for details). Reactions catalyzed by PGI and PGDH are strictly parallel (path g6p-f6p-fdp-gap contains only PGI, path g6p-6pg-ribu5p-gap contains only PGDH and there is no simple path in UGPP between g6p and gap that contains both PGI and PGDH). Reactions catalyzed by PGI and PFK are serial-parallel (path g6p-f6p-fdp-gap contains both reactions, path g6p-f6p-gap contains only PGI, path g6p-6pg-ribu5p-f6p-fdp-gap contains only PFK). Reactions catalyzed by PFK and PGDH are also serial-parallel (path g6p-6pg-ribu5p-f6p-fdp-gap contains both reactions, path g6p-f6p-fdp-gap contains only PFK, path g6p-6pg-ribu5p-gap contains only PGDH).

Figure 5—figure supplement 1
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
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
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.

Simple modules.

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

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.

Classification of double-marked modules.

Notations as in Figure 7.

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.

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

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

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

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

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

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

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

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

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

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

Appendix 2—figure 1
Illustration for the proof of Lemma 1.

Tables

Table 1
Classification of double-marked modules.

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

ClassabShared metab.Verbal descriptionmℳAℳEquation for f2
ℳb,io,IO(1,2)(1,3)1Bypass and i/o reactions, shared I/O metabolite2{1,2,3}Equation (34)
ℳb,i,∅(1,2)(3,4)–Bypass and internal reactions, no shared metabolies2{1,2,3,4}Equation (35)
ℳio,io,I(1,3)(2,3)3i/o reactions, shared internal metabolite1{1,2,3}Equation (36)
ℳio,io,IO(1,3)(1,4)1i/o reactions, shared I/O metabolite2{1,2,3,4}Equation (37)
ℳio,io,∅(1,3)(2,4)–i/o reactions, no shared metabolites2{1,2,3,4}Equation (38)
ℳio,i,I(1,3)(3,4)3i/o and internal reactions, shared internal metabolite2{1,2,3,4}Equation (39)
ℳio,i,∅(1,3)(4,5)–i/o and internal reactions, no shared metabolites3{1,2,3,4,5}Equation (40)
ℳi,i,I(3,4)(3,5)3Internal reactions, shared internal metabolite3{1,2,3,4,5}Equation (41)
ℳi,i,∅(3,4)(5,6)–Internal reactions, no shared metabolites4{1,2,3,4,5,6}Equation (42)
Table 2
Strictly parallel generating topologies.
Marked reactionsGenerating topology
ClassabIDExcluded reactionsFigure
ℳb,io,IO1↔21↔3b,io,IO,F∅Figure 7
ℳb,i,∅1↔23↔4b,i,∅,F∅Figure 7
ℳio,io,IO1↔31↔4io,io,IO,F∅Figure 7
ℳio,io,∅1↔32↔4io,io,∅,P{3↔4}Figure 9
ℳio,i,I1↔33↔4io,i,I,P{2↔4}Figure 10
ℳio,i,∅1↔34↔5io,i,∅,P1{3↔4,3↔5}Figure 11
io,i,∅,P2{2↔5,3↔5}
io,i,∅,P3{2↔4,2↔5}
ℳi,i,I3↔43↔5i,i,I,P1{2↔4,2↔5}Figure 12
i,i,I,P2{1↔5,2↔5}
ℳi,i,∅3↔45↔6i,i,∅,P1{3↔5,3↔6,4↔5,4↔6}Figure 13
i,i,∅,P2{1↔5,1↔6,2↔5,2↔6}
i,i,∅,P3{2↔4,2↔6,3↔6,4↔5,4↔6}
i,i,∅,P4{2↔4,2↔5,2↔6,4↔5,4↔6}
i,i,∅,P5{1↔6,2↔4,2↔5,2↔6,4↔6}
i,i,∅,P6{1↔4,1↔6,2↔4,2↔6,4↔6}
i,i,∅,P7{1↔4,1↔5,2↔3,2↔6,3↔5,4↔6}
Table 3
Strictly serial generating topologies.
Marked reactionsGenerating topologies
ClassabIDExcluded reactionsFigure
ℳio,io,I1↔32↔3io,io,I,F∅Figure 7
ℳio,io,∅1↔32↔4io,io,∅,S{2↔3}Figure 9
ℳio,i,I1↔33↔4io,i,I,S1{2↔3}Figure 10
io,i,I,S2{1↔4}
ℳio,i,∅1↔34↔5io,i,∅,S1{1↔4,1↔5}Figure 11
io,i,∅,S2{2↔3,2↔4,3↔5}
io,i,∅,S3{1↔5,2↔3,2↔5}
ℳi,i,I3↔43↔5i,i,I,S1{1↔3,2↔3}Figure 12
i,i,I,S2{2↔3,2↔5,4↔5}
ℳi,i,∅3↔45↔6i,i,∅,S1{2↔3,2↔5,2↔6,4↔5,4↔6}Figure 13
i,i,∅,S2{1↔3,1↔6,2↔3,2↔6,4↔6}
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.

ModelInternal metabolitesConcentrations of I/O metabolitesReactionsOutput flux
UGPP6 pg, dhap, e4p, f6p, fdp, rib5p, ribu5p, sed7p, xyl5p[g6p]=3.82 mM, [gap]=0.44 mMALDO, G6PDH, PFK, PGDH, PGI, Ru5P, R5PI, TA, TIS, TKa, TKbJALDO+JTIS+JTKb+JTKa-JTA
LG2 pg, 3 pg, pgp[gap]=0.44 mM, [pep]=0.08 mMENO, GAPDH, PGK, PGMJENO
GPPall in UGPP and in LG, gap[g6p]=3.82 mM, [pep]=0.08 mMall in UGPP and in LGJENO
FULLall in GPP, g6p, pep[Ext glu]=2 µM, [pyr]=10 µMall in GPP, PTS, PK, PEPCxylJPK+JPTS
Table 5
Names of metabolites used in the kinetic model of glycolysis.
2 pg2-Phosphoglycerate
3 pg3-Phosphoglycerate
6 pg6-Phosphogluconate
dhapDihydroxyacetonephosphate
e4pErythrose-4-phosphate
f6pFructose-6-phosphate
fdpFructose-1,6-bisphosphate
g6pGlucose-6-phosphate
gapGlyceraldehyde-3-phosphate
gluGlucose
pepPhosphoenolpyruvate
pgp1,3-Diphosphoglycerate
pyrPyruvate
rib5pRibose-5-phosphate
ribu5pRibulose-5-phosphate
sed7pSedoheptulose-7-phosphate
xyl5pXylulose-5-phosphate
Table 6
Names of enzymes used in the kinetic model of glycolysis.
ALDOAldolase
ENOEnolase
G6PDHGlucose-6-phosphate dehydrogenase
GAPDHGlyceraldehyde-3-phosphate dehydrogenase
PFKPhosphofructokinase
PGDH6-Phosphogluconate dehydrogenase
PGIGlucose-6-phosphateisomerase
PGKPhosphoglycerate kinase
PGMPhosphoglycerate mutase
PEPCxylPEP carboxylase
PKPyruvate kinase
PTSPhosphotransferase system
R5PIRibose-phosphateisomerase
Ru5PRibulose-phosphate epimerase
TATransaldolase
TISTriosephosphate isomerase
TKaTransketolase, reaction a
TKbTransketolase, reaction b

Data availability

All data generated or analyzed during this study are included in the manuscript and supporting files. Code is available on GitHub.

Additional files

Supplementary file 1

Mathematica notebook ‘Case i,i,emptyset,P7.nb’ for evaluating epistasis for the generating topology i,i,∅,P7.

https://cdn.elifesciences.org/articles/60200/elife-60200-supp1-v2.nb
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
Transparent reporting form
https://cdn.elifesciences.org/articles/60200/elife-60200-transrepform-v2.docx

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