Bio Systems Biology Flux Balance Analysis

Version Compatibility

Reference examples tested with: COBRApy 0.29+, Python 3.10+

Before using code patterns, verify installed versions match. If versions differ:

• Python: pip show <package> then help(module.function) to check signatures

If code throws ImportError, AttributeError, or TypeError, introspect the installed package and adapt the example to match the actual API rather than retrying.

Note: the objective LP is solved by a backend solver. GLPK (the COBRApy default) can return degenerate or marginally-infeasible answers on large or ill-conditioned models; prefer HiGHS (if available in the installed cobra/optlang build) or a free academic CPLEX/Gurobi for genome-scale work and any MILP/QP method (loopless, MOMA). Set with model.solver = 'glpk'|'highs'|'gurobi'|'cplex'.

Flux Balance Analysis

"Predict growth rate and metabolic fluxes for my organism" -> Solve a linear program over a genome-scale metabolic model that maximizes a biomass (or custom) objective subject to steady-state mass balance and exchange bounds, then quantify how much of that solution is actually determined.

• Python: model.optimize(), cobra.flux_analysis.flux_variability_analysis(), pfba(), cobra.sampling.sample() (COBRApy)

The governing principle: FBA is an under-determined LP, so one solution is not "the" flux distribution

FBA imposes steady state (S*v = 0) plus bounds and maximizes an objective. The steady-state constraint is a pseudo-steady-state on fast-turnover metabolite POOLS, not on the organism. Critically, the optimal objective is usually reached on a whole FACE of the solution polytope, not a single point: many different internal flux vectors give the identical maximal growth. model.optimize() returns ONE arbitrary vertex of that face. Consequences that govern every downstream decision:

• The biomass/objective VALUE is typically robust and reproducible; individual internal FLUXES are often NOT. Never report a single solution.fluxes[rxn] as "the" flux without FVA (the range) or pFBA (a parsimonious representative) or sampling (the distribution).
• A nonzero growth rate is only as meaningful as the MEDIUM and the biomass reaction. An open or under-constrained exchange set inflates growth; a copied/generic biomass reaction encodes another organism's composition. "It grows" is often an artifact of an open exchange, not biology (see Common Errors).
• FBA predicts YIELDS and growth/essentiality well but internal flux magnitudes poorly. To validate actual intracellular fluxes, 13C metabolic flux analysis MEASURES them (metabolomics/isotope-tracing) - FBA does not.

Decision: which analysis for which question

Load Models

import cobra

model = cobra.io.load_model('textbook')   # E. coli core (e_coli_core), 95 reactions, ships with cobra
model = cobra.io.load_model('iJO1366')    # genome-scale E. coli, 2583 reactions

model = cobra.io.read_sbml_model('model.xml')   # SBML (the standard exchange format)
model = cobra.io.load_json_model('model.json')  # COBRA JSON

# Curated genome-scale models: http://bigg.ucsd.edu/models (King 2016). Record the model
# version; predictions are only comparable within the same model release.

Basic FBA and honest growth interpretation

Goal: Predict the maximum growth rate and a flux distribution under a defined medium, and judge whether the number is biological.

Approach: Load a model, confirm the objective is the intended biomass reaction, solve the LP, then read the objective value while treating individual fluxes as provisional until FVA/sampling confirms them.

import cobra

model = cobra.io.load_model('textbook')

solution = model.optimize()
print(f'Objective (growth): {solution.objective_value:.4f} /h  status: {solution.status}')
print('Objective reaction:', str(model.objective.expression).split('*')[1].split()[0])

# A growth rate is interpretable ONLY against a stated medium and biomass reaction.
# Compare to a measured doubling time (mu = ln2 / t_double) rather than to a fixed
# "fast/slow" scale; absolute values are organism- and biomass-definition-specific.

Set Medium (exchange bounds; uptake is a NEGATIVE lower bound)

Goal: Impose a defined nutrient environment so growth reflects the intended condition, not leftover open exchanges.

Approach: Close every exchange, then open only the intended uptakes. By COBRApy convention an exchange EX_x_e has flux < 0 for uptake and > 0 for secretion, so uptake is set through the lower bound. The model.medium dict is the concise idiom (its values are uptake magnitudes, positive).

def set_minimal_medium(model, carbon_source='EX_glc__D_e', carbon_uptake=10):
    '''Close all uptake, then open a defined minimal medium.

    carbon_uptake in mmol/gDW/h; 10 is the standard E. coli aerobic glucose rate (iJO1366).
    '''
    for rxn in model.exchanges:
        rxn.lower_bound = 0

    minimal = {'EX_o2_e': 1000, 'EX_h2o_e': 1000, 'EX_h_e': 1000, 'EX_nh4_e': 1000,
               'EX_pi_e': 1000, 'EX_so4_e': 1000, 'EX_k_e': 1000, 'EX_mg2_e': 1000}
    for ex_id, uptake in minimal.items():
        if ex_id in model.reactions:
            model.reactions.get_by_id(ex_id).lower_bound = -uptake
    if carbon_source in model.reactions:
        model.reactions.get_by_id(carbon_source).lower_bound = -carbon_uptake
    return model

# The with-block reverts all bound changes on exit, so comparisons never leak state.
for cs in ['EX_glc__D_e', 'EX_ac_e', 'EX_succ_e']:
    with model:
        set_minimal_medium(model, carbon_source=cs)
        print(f'{cs}: growth = {model.slim_optimize():.4f}')   # slim_optimize returns the objective float only (fast)

Flux Variability Analysis (FVA): expose alternate optima

from cobra.flux_analysis import flux_variability_analysis

# Range each reaction can carry while holding the objective at (fraction_of_optimum) of the max.
fva = flux_variability_analysis(model, fraction_of_optimum=1.0)

# fraction_of_optimum < 1 relaxes the objective and reveals the alternative-optima span.
fva90 = flux_variability_analysis(model, fraction_of_optimum=0.9)

# loopless=True removes thermodynamically infeasible internal loops from the ranges (slower).
fva_ll = flux_variability_analysis(model, loopless=True)

# A reaction with maximum == minimum is fully determined; a wide range means the single
# FBA value for it was arbitrary. Blocked reactions have min == max == 0.
fva['range'] = fva['maximum'] - fva['minimum']
fva['blocked'] = fva['range'].abs() < 1e-9

Parsimonious FBA (pFBA): one realistic representative

from cobra.flux_analysis import pfba

# Among all optima, pFBA returns the flux vector minimizing the sum of absolute fluxes,
# an Occam's-razor proxy for minimal total enzyme cost (Lewis 2010). It is a principled
# single representative of the optimal face; it is NOT more "true" than the face itself,
# and it hugs the polytope boundary (a min-flux vertex), so report the FVA range alongside it
# when the internal flux values matter.
pfba_solution = pfba(model)
print(f'FBA total flux : {model.optimize().fluxes.abs().sum():.1f}')
print(f'pFBA total flux: {pfba_solution.fluxes.abs().sum():.1f}')

Loopless FBA

from cobra.flux_analysis import loopless_solution

# Projects an FBA solution onto a loopless one: no net flux around a closed cycle that
# lacks a thermodynamic driving force (Schellenberger 2011). Internal cycles are a common
# artifact of reversible reactions and gap-filling and inflate apparent flux magnitudes.
loopless = loopless_solution(model)

Flux Sampling: the distribution, without choosing an objective

Goal: Characterize the whole space of feasible steady-state fluxes rather than one optimum, giving each reaction a distribution and confidence interval.

Approach: Uniformly sample the (optionally objective-constrained) solution polytope with a Markov-chain sampler (OptGP or ACHR). Use when there is no clear objective, when the objective face is large, or when uncertainty on internal fluxes matters more than an optimum.

from cobra.sampling import sample

# n samples; method 'optgp' (parallel) or 'achr'. Thinning reduces autocorrelation.
samples = sample(model, n=1000, method='optgp', thinning=100, seed=1)
print(samples['PFK'].describe())   # per-reaction distribution, e.g. median and IQR

# To sample only high-growth states, constrain the objective first (e.g. biomass >= 0.9*max)
# inside a `with model:` block, then sample. Check mixing before trusting the distribution
# (multiple chains/seeds should agree); short chains give correlated, misleading samples.

Production Envelope (growth vs product tradeoff)

from cobra.flux_analysis import production_envelope

# Pareto frontier of growth against a secreted product; the design space for strain engineering.
# objective defaults to the model's objective (the biomass reaction) when omitted.
env = production_envelope(model, reactions=['EX_ac_e'])
# To actually DESIGN knockouts that couple product to growth, see systems-biology/strain-design.

Common Errors

Related Skills

• systems-biology/gene-essentiality - Knockout screens, MOMA/ROOM for non-re-optimized mutants
• systems-biology/context-specific-models - Constrain FBA with expression data
• systems-biology/strain-design - Design knockouts from the production envelope
• systems-biology/community-metabolic-modeling - FBA over multi-species communities
• metabolomics/isotope-tracing - 13C-MFA to measure (not predict) intracellular fluxes
• metabolomics/pathway-mapping - Map measured metabolites onto model reactions

References

• Orth JD, Thiele I, Palsson BO. 2010. What is flux balance analysis? Nat Biotechnol 28(3):245-248.
• Ebrahim A, Lerman JA, Palsson BO, Hyduke DR. 2013. COBRApy: constraint-based reconstruction and analysis for Python. BMC Syst Biol 7:74.
• Mahadevan R, Schilling CH. 2003. The effects of alternate optimal solutions in constraint-based genome-scale metabolic models. Metab Eng 5(4):264-276.
• Lewis NE, Hixson KK, Conrad TM, et al. 2010. Omic data from evolved E. coli are consistent with computed optimal growth from genome-scale models. Mol Syst Biol 6:390. (pFBA)
• Schellenberger J, Lewis NE, Palsson BO. 2011. Elimination of thermodynamically infeasible loops in steady-state metabolic models. Biophys J 100(3):544-553. (loopless)
• Megchelenbrink W, Huynen M, Marchiori E. 2014. optGpSampler: an improved tool for uniformly sampling the solution-space of genome-scale metabolic networks. PLoS One 9(2):e86587.
• Schuetz R, Kuepfer L, Sauer U. 2007. Systematic evaluation of objective functions for predicting intracellular fluxes in E. coli. Mol Syst Biol 3:119. (objective choice)
• Ibarra RU, Edwards JS, Palsson BO. 2002. Escherichia coli K-12 undergoes adaptive evolution to achieve in silico predicted optimal growth. Nature 420(6912):186-189.
• King ZA, Lu JS, Drager A, et al. 2016. BiGG Models: a platform for integrating, standardizing and sharing genome-scale models. Nucleic Acids Res 44(D1):D515-D522.