Constraint-based metabolic modeling using COBRA and FBA

A project using a toy model¹ of some parts of the central metabolism of Escherichia Coli to apply COBRA, and specifically FBA, and predict the active chemical reactions under default conditions (aerobic, glucose fed), how the metabolism changes when genes that encode for cytochrome oxidases (bo and putative) are KO (switch from respiration to fermentation), and how FBA can be used to identify a single point in the constrained search space, specifically to maximize ethanol production while ensuring the organism can still grow. The model includes glycolysis, the TCA or CAC or Krebs cycle, and the electron transport chain. This project uses the Julia language and two main Julia packages: COBREXA.jl and Escher.jl. The results are included as CSVs and markdown tables, and visualizations in the form of metabolic maps are also included in SVG and PDF form. Some utility functions have also been defined for a more seamless exploratory pipeline.

Metabolism

What is metabolism? It's a set of chemical reactions that can be found in organisms and are key for sustaining life. They are either reactions that break down things (compounds such as glucose) or synthesize things (compounds such as proteins, carbohydrates, lipids and nucleic acids). Usually, breaking compounds down releases energy, while synthesizing compounds consumes energy. This energy flow happens under very specific rules, as all organisms obey the laws of thermodynamics. The second law states that in an isolated system the amount of entropy cannot decrease, which seems to contradict the complexity that can be found in living systems. The key here is that organisms are, in fact, open systems, which exchange matter and energy with their environment; dissipative systems that maintain complexity by increasing the entropy of their environment. The metabolism of a cell achieves this by coupling the spontaneous processes of breaking down things, with the non-spontaneous processes of synthesizing things. Some example mechanisms are bacteriorhodopsin, or the redox loop². You can see this³ for an example.

These chemical reactions are organized into metabolic pathways, along which a chemical gets transformed into another chemical through a series of steps. Each step happens with the help of an enzyme, which is exactly what couples the reactions to other more thermodynamically favorable ones, catalyze the reactions so that they proceed faster, and regulate the rate at which they occur. The basic metabolic pathways are found among vastly different species. For example, the set of carboxylic acids that are best known as the intermediates in the citric acid cycle are present in all known organisms. Central pathways of metabolism, such as glycolysis and the citric acid cycle, are present in all three domains of living things and were present in the last universal common ancestor (LUCA). Understanding how metabolism came to be and whether it can actually kickstart life (example⁴) is another topic of great study, and while it is a very complex process with a lot of participating intertwined mechanisms, it could have originated as something much simpler⁵. Also see this⁶.

Through metabolism we can also produce biofuels (here⁷, here⁸ and here⁹) or healthy food¹⁰, biopolymers¹¹, amino acids¹² and more.

Framework

The enzymes that facilitate all of these chemical reactions are produced through a process which involves some cell genes. How much of an enzyme is produced has to do with how much the corresponding gene or genes are used (expressed). Recent advances in Synthetic Biology (such as being able to modify a single DNA base¹³, rapidly delete endogenous genes¹⁴, insert entirely new genes¹⁵, or onto something more metabolism-related, regulate enzyme expression¹⁶) mean that we now can directly intervene and change a cell to have it do our bidding. That means we now have to understand what exactly to change in the cell to have the desirable results. In other words, we need to better understand how, systematically, changes at the gene level affect an organism's metabolism.

To do this, we can use a great variety of techniques, some of which can be found here¹⁷. We use something called COBRA (COnstraint-Based Reconstruction and Analysis) which means you gather details on all of the chemical reactions that are encoded for by the genes of the organisms; the reactions are linked to specific genes¹⁸. Then, we can use an optimization technique called Flux Balance Analysis¹⁹ to predict how the enzymes work, and thus to predict how the reaction set would change if the genes were modified.

The model used in this repository is a toy model of some parts of the central metabolism of Escherichia Coli. It includes glycolysis, the TCA or CAC or Krebs cycle, and the electron transport chain:

Here, the edges are chemical reactions, and the nodes are metabolites. Glucose (in red) is converted into ATP and other metabolic precursors used to create more cells (biomass). The majority of these reactions actually exist in E. coli, but not those lumped in the bottom right. These are what's called the biomass objective function, which is a pseudo reaction that groups all the processes that weren't modeled and that convert the metabolites generated by what's been modeled into biomass. This is an assumption due to the constraint-based approaches requiring the organisms to have adapted their metabolism, through evolution, to optimize some objective. The specific assumption taken to hold here is that the metabolism has been tuned to maximize growth rate (biomass). There initiallly were some studies showing that optimizing the assumed objective function for growth (here²⁰) and for energy use (here²¹ and here²²) you could predict the metabolic fluxes. Other studies seemed to question this universality (here²³, here²⁴ and here²⁵). Today the representation of the in vivo fluxes is thought of as a Pareto surface created from combining three objective functions: maximizing biomass and ATP generation, and minimizing the fluxes across the whole network (here²⁶).

In FBA, the reaction network is represented in the compact form of a stoichiometric matrix ( $S$ ), as is common, with $m$ rows and $n$ columns, for $n$ chemical reactions and $m$ participating compounds. The column entries are the stoichiometric coefficients of the metabolites that participate in the corresponding reaction, where there's a negative coefficient if the metabolite is consumed and a positive coefficient if it's produced. If a metabolite does not participate in a chemical reaction, the corresponding stoichiometric coefficient is zero. $S$ is a sparse matrix, since the reactions usually involve only a handful of metabolites. The flux through all the reactions across the network is represented by a vector $v$, of length $n$, for the $n$ reactions. Respectively, a vector $x$ with length $m$ represents the concentrations of all the metabolites. You can get the system containing all mass balance equations at steady state by solving for $$\frac{dx}{dt} = 0$$

---

$$Sv = 0$$

Any vector $v$ that satisfies this equation is in the null space of $S$. In realistic, large-scale models, there are more reactions than there are compounds ( $n > m$ ) which means there are more unknown variables in the system than equations, so there is no single, unique solution to this system. The solution space is defined by imposing the $Sv = 0$ mass balance constraints and capacity constraints imposed by manually selected lower and upper bounds. What FBA does is it looks into specific points within that solution space. For instance, while you might have various viable configurations, you might be interested to see which point within the solution space corresponds to maximizing biomass production, or maximizing the production of a compound, given the already present collection of constraints.

FBA, as a method that seeks to identify optimal points within a constrained space, optimizes an objective function $$Z = c^Tv$$

This can generally be any (linear) combination of fluxes, with $c$ being a vector containing weights that indicate how much a chemical reaction contributes to the objective function. If optimizing only for a single reaction, e.g. biomass production, $c$ will be a vector of zeros with an entry of one only at the position of the reaction optimizing for.

Lastly, to optimize this system linear programming methods are used to solve $Sv = 0$, given the set of lower and upper bounds on $v$ and a linear combination of fluxes as an objective function. The resulting flux distribution $v$ maximizes or minimizes the selected objective function. Because FBA reduces the large-scale, complex metabolic model to a linear program, it is quite performant and scales well. The FBA computations fall into the broader category of COnstraint-Based Reconstruction and Analysis (COBRA) methods.

Implementation

This project uses COBREXA.jl (COnstraint-Based Reconstruction and EXascale Analysis), a Julia package, for the computational work; while the backbone - the optimizer - chosen is Tulip.jl, an interior-point solver for linear optimization written purely in Julia. You can use other solvers if you'd like, as long as it's supported by JuMP; see this. It also uses Escher.jl, a Julia package that is essentially a Makie.jl recipe to plot maps of metabolic models. Lastly, CSV.jl is used for importing and exporting the FBA results, DataFrames.jl are used to have a common data structure for the various processing functions, and PrettyTables.jl is used to export the results as Markdown tables.

There are some utility functions defined:

• constructpath which allows you to easily generate a valid path for your OS to play around with importing and exporting files, as well as downloads
• writeio, to save data to a file. This is used for example to write the results as markdown tables
• readcsv and writecsv, to import/export data in CSV form
• tomarkdown, to convert the results to a markdown table
• fluxes_to_df and df_to_fluxes, to help with converting back and from a structure COBREXA uses, to a more general-purpose data structure for IO, etc.
• vismetabolism, which uses Escher to generate a metabolic map with sensible defaults, and generate_flux_edge_colors to help color the reactions according to a customizable tolerance

---

Workflow oneliner examples using various util functions:

• readcsv(constructpath(["results", "csv"], "ko_genes_reactions.csv")) |> df_to_fluxes |> flux_summary
• writeio(readcsv(constructpath(["results", "csv"], "max_etoh_reactions.csv")) |> tomarkdown, constructpath(["results", "markdown"], "max_etoh_reactions.md"))

---

The rest is pretty straightforward: The toy model gets downloaded and loaded into COBREXA in StandardModel form, we simulate the growth of E. coli under some default conditions (aerobic, glucose fed) and use FBA to predict its growth rate and active chemical reactions. FBA is performed again, but with some KO genes and with additional constraints to maximize for a specific point in solution space. The resulting metabolic maps are saved by default, if not already present.

Results

# Simulate E. coli growth.
const model = load_model(StandardModel, ModelPath)
# Flux Balance Analysis (FBA).
const fluxes = flux_balance_analysis_dict(model, Tulip.Optimizer)

# Save control metabolic reactions graph.
const tolerance = 1e-3
!isfile(DefReactionsPath) && save(DefReactionsPath, vismetabolism(MapPath, generate_flux_edge_colors(fluxes, tolerance, :red)))

At first, we can load the simulated growth in default conditions (aerobic, glucose fed) and perform FBA to predict the cell's growth rate and chemical reactions. We can then generate the metabolic map with the active chemical reactions highlighted in red:

The model predicts a growth rate of 0.87 1/h when the cell metabolizes 10 mmol glucose/gDW/h. This is fairly close to the corresponding experimental results. Some of the reactions are not used (grey dashed lines), which is expected since the metabolism is actively regulated to better perform in the current environment. Specifically, the cell uses respiration to produce energy.

Click to see results table

Reaction	Flux
ACALD	-5.65944e-9
PTAr	3.41401e-9
ALCD2x	-1.4328e-9
PDH	9.28253
PYK	1.75818
CO2t	-22.8098
EX_nh4_e	-4.76532
MALt2_2	0.0
CS	6.00725
PGM	-14.7161
TKT1	1.49698
EX_mal__L_e	0.0
ACONTa	6.00725
EX_pi_e	-3.2149
GLNS	0.223462
ICL	2.89785e-7
EX_o2_e	-21.7995
FBA	7.47738
EX_gln__L_e	0.0
EX_glc__D_e	-10.0
SUCCt3	2.55048e-8
FORt2	1.14358e-7
G6PDH2r	4.95999
AKGDH	5.06438
TKT2	1.1815
FRD7	490.0
SUCOAS	-5.06438
BIOMASS_Ecoli_core_w_GAM	0.873922
FBP	1.50898e-8
ICDHyr	6.00725
AKGt2r	-3.84984e-9
GLUSy	9.04494e-9
TPI	7.47738
FORt	-1.33188e-7
ACONTb	6.00725
EX_ac_e	3.41403e-9
GLNabc	0.0
EX_akg_e	3.84984e-9
EX_fru_e	0.0
RPE	2.67848
ACKr	-3.41402e-9
THD2	3.37384e-7
PFL	1.88301e-8
RPI	-2.2815
D_LACt2	-2.39187e-9
TALA	1.49698
EX_glu__L_e	3.3699e-9
ATPM	8.39
PPCK	5.88221e-8
ACt2r	-3.41402e-9
EX_etoh_e	1.43282e-9
NH4t	4.76532
PGL	4.95999
NADTRHD	5.3881e-7
PGK	-16.0235
LDH_D	-2.39186e-9
ME1	4.81777e-8
PIt2r	3.2149
EX_h2o_e	29.1758
EX_succ_e	1.53072e-9
ATPS4r	45.514
PYRt2	-3.92523e-9
EX_acald_e	4.22665e-9
EX_h_e	17.5309
GLCpts	10.0
GLUDy	-4.54186
CYTBD	43.599
FUMt2_2	0.0
FRUpts2	0.0
GAPD	16.0235
H2Ot	-29.1758
PPC	2.50431
NADH16	38.5346
PFK	7.47738
EX_for_e	1.88301e-8
MDH	5.06438
PGI	4.86086
O2t	21.7995
ME2	1.4966e-7
EX_pyr_e	3.92523e-9
EX_co2_e	22.8098
GND	4.95999
SUCCt2_2	2.39741e-8
GLUN	9.94954e-9
EX_fum_e	0.0
ETOHt2r	-1.43281e-9
ADK1	2.86618e-8
ACALDt	-4.22664e-9
SUCDi	495.064
EX_lac__D_e	2.39187e-9
ENO	14.7161
MALS	2.89785e-7
GLUt2r	-3.36991e-9
PPS	2.86618e-8
FUM	5.06438

---

# Knockout genes b0978 and b0734
# encoding cytochrome oxidases (bo and putative).
const ko_fluxes = flux_balance_analysis_dict(
    model,
    Tulip.Optimizer;
    modifications = [knockout(["b0978", "b0734"])],
)

# Save KO cytochrome oxidase genes metabolic reactions graph.
!isfile(KOReactionsPath) && save(KOReactionsPath, vismetabolism(MapPath, generate_flux_edge_colors(ko_fluxes, tolerance, :red)))

Now, we can knockout (remove) two genes, b0978 and b0734 which encode cytochrome oxidases (bo and putative):

The metabolism has now been drastically altered, since the cell is now forced to use a different process²⁷, called fermentation, to grow. A side-effect of this change is that the cell physiology has changed significantly, and it's growing with a smaller rate (0.21 1/h), but it also produces ethanol and acetate, which can be used as biofuels²⁸, among others.

Click to see results table

Reaction	Flux
ACALD	-8.27946
PTAr	8.50359
ALCD2x	-8.27946
PDH	6.66064e-8
PYK	8.40427
CO2t	0.378178
EX_nh4_e	-1.15416
MALt2_2	0.0
CS	0.228363
PGM	-19.1207
TKT1	-0.0378665
EX_mal__L_e	0.0
ACONTa	0.228363
EX_pi_e	-0.778644
GLNS	0.0541222
ICL	1.60351e-9
EX_o2_e	0.0
FBA	9.78946
EX_gln__L_e	0.0
EX_glc__D_e	-10.0
SUCCt3	1.4142e-8
FORt2	1.85814e-8
G6PDH2r	1.01601e-7
AKGDH	1.1816e-9
TKT2	-0.114277
FRD7	492.856
SUCOAS	-1.18159e-9
BIOMASS_Ecoli_core_w_GAM	0.211663
FBP	2.38026e-9
ICDHyr	0.228363
AKGt2r	-1.90741e-9
GLUSy	8.29697e-10
TPI	9.78946
FORt	-17.8047
ACONTb	0.228363
EX_ac_e	8.50359
GLNabc	0.0
EX_akg_e	1.90734e-9
EX_fru_e	0.0
RPE	-0.152143
ACKr	-8.50359
THD2	3.62919
PFL	17.8047
RPI	-0.152143
D_LACt2	-4.91612e-9
TALA	-0.0378665
EX_glu__L_e	9.61386e-10
ATPM	8.39
PPCK	2.06388e-9
ACt2r	-8.50359
EX_etoh_e	8.27946
NH4t	1.15416
PGL	1.01601e-7
NADTRHD	1.11162e-8
PGK	-19.4373
LDH_D	-4.91614e-9
ME1	2.02658e-9
PIt2r	0.778644
EX_h2o_e	-7.1158
EX_succ_e	1.00725e-8
ATPS4r	-5.45205
PYRt2	-5.00048e-9
EX_acald_e	5.5162e-9
EX_h_e	30.5542
GLCpts	10.0
GLUDy	-1.10003
CYTBD	0.0
FUMt2_2	0.0
FRUpts2	0.0
GAPD	19.4373
H2Ot	7.1158
PPC	0.606541
NADH16	7.28745e-9
PFK	9.78946
EX_for_e	17.8047
MDH	-1.34507e-8
PGI	9.95661
O2t	0.0
ME2	5.74003e-9
EX_pyr_e	5.00046e-9
EX_co2_e	-0.378178
GND	1.01601e-7
SUCCt2_2	4.06942e-9
GLUN	7.09021e-10
EX_fum_e	0.0
ETOHt2r	-8.27946
ADK1	3.06499e-9
ACALDt	-5.51622e-9
SUCDi	492.856
EX_lac__D_e	4.91611e-9
ENO	19.1207
MALS	1.60347e-9
GLUt2r	-9.61448e-10
PPS	3.06496e-9
FUM	-7.28752e-9

---

# Set minimum constraint for cell growth so that the resulting reaction set is viable.
model_with_bounded_production = change_bound(model, "BIOMASS_Ecoli_core_w_GAM", lower = 0.1)
const max_etoh_fluxes = flux_balance_analysis_dict(
    model_with_bounded_production,
    Tulip.Optimizer;
    modifications = [
        change_objective("EX_etoh_e"), # maximze ethanol production
    ],
)

# Save maximum EtOH production metabolic reactions graph.
!isfile(MaxEtOHReactionsPath) && save(MaxEtOHReactionsPath, vismetabolism(MapPath, generate_flux_edge_colors(max_etoh_fluxes, tolerance, :red)))

But what if we wanted to take this a step further, and search for the single point where ethanol production is optimized while ensuring that the cell can still grow? No problem:

Click to see results table

Reaction	Flux
ACALD	-18.4801
PTAr	9.30969e-13
ALCD2x	-18.4801
PDH	18.6499
PYK	7.35964
CO2t	-18.4713
EX_nh4_e	-0.54528
MALt2_2	0.0
CS	0.10789
PGM	-19.5846
TKT1	-0.01789
EX_mal__L_e	0.0
ACONTa	0.10789
EX_pi_e	-0.36787
GLNS	1.48252
ICL	4.74617e-13
EX_o2_e	-0.0319401
FBA	9.90053
EX_gln__L_e	0.0
EX_glc__D_e	-10.0
SUCCt3	1.15259
FORt2	5.96336
G6PDH2r	3.20267e-11
AKGDH	3.36493e-12
TKT2	-0.05399
FRD7	491.901
SUCOAS	-3.30308e-12
BIOMASS_Ecoli_core_w_GAM	0.1
FBP	0.66076
ICDHyr	0.10789
AKGt2r	-2.2679e-12
GLUSy	0.704085
TPI	9.90053
FORt	-6.27626
ACONTb	0.10789
EX_ac_e	1.18871e-12
GLNabc	0.0
EX_akg_e	2.44418e-12
EX_fru_e	0.0
RPE	-0.07188
ACKr	-1.01656e-12
THD2	2.22216
PFL	0.3129
RPI	-0.07188
D_LACt2	-6.48811e-12
TALA	-0.01789
EX_glu__L_e	2.94048e-12
ATPM	8.83682
PPCK	0.661514
ACt2r	-1.10257e-12
EX_etoh_e	18.4801
NH4t	0.54528
PGL	3.20673e-11
NADTRHD	1.68498
PGK	-19.7342
LDH_D	-6.48915e-12
ME1	0.775277
PIt2r	0.36787
EX_h2o_e	0.71954
EX_succ_e	1.80232e-11
ATPS4r	0.561831
PYRt2	-3.47462e-12
EX_acald_e	3.6606e-12
EX_h_e	2.3189
GLCpts	10.0
GLUDy	0.184375
CYTBD	0.0638802
FUMt2_2	0.0
FRUpts2	0.0
GAPD	19.7342
H2Ot	-0.71954
PPC	2.90079
NADH16	0.0638802
PFK	10.5613
EX_for_e	0.3129
MDH	-1.95271
PGI	9.9795
O2t	0.0319401
ME2	1.17743
EX_pyr_e	3.51713e-12
EX_co2_e	18.4713
GND	3.20642e-11
SUCCt2_2	1.15259
GLUN	0.752862
EX_fum_e	0.0
ETOHt2r	-18.4801
ADK1	0.0662497
ACALDt	-3.61812e-12
SUCDi	491.901
EX_lac__D_e	6.48726e-12
ENO	19.5846
MALS	4.99625e-13
GLUt2r	-2.80789e-12
PPS	0.0662497
FUM	-1.42832e-11

Footnotes

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