Carbon-13 metabolic flux inference

Turn reviewed carbon maps, explicit tracer mixtures, and corrected labeling measurements
into feasible flux estimates and evidence about which fluxes the experiment constrains.
Use the bundled solver rather than reconstructing isotope balances or fitting each
reaction independently. It runs mfapy's EMU forward simulator and fits fluxes in the
mass-balanced feasible space with SciPy. It does not use an FBA objective.

Scope and required evidence

This implementation supports metabolic and isotopic steady state, a single shared
flux state across one or more tracer experiments, nonnegative one-way reaction fluxes,
and carbon-subset mass distributions. Reversible reactions are two separately mapped
directions. Measurement error is Gaussian with a supplied covariance or a disclosed
diagonal approximation.

Before fitting, obtain:

• The carbon network and the source of each atom assignment. Stoichiometry alone does

not specify where labeled atoms go. Record compartments as separate metabolite IDs.

• Evidence for both steady-state assumptions. Stable metabolite abundance does not

establish isotopic steady state. Time-course labeling requires INST-MFA with pool
sizes and initial labeling; do not average it into this solver.

• Every carbon input's positional isotopomer distribution, including unlabeled

supplements, bicarbonate/CO2 when assimilated, and tracer impurity.

• Fragment carbon assignments, natural-abundance correction history, and uncertainty

of the reported mean. Raw peak intensities, derivatized spectra, and MS/MS
transitions require validated preprocessing before these inputs can be constructed.

• Flux units, extracellular rate measurements or a stated relative-flux reference,

and biologically justified bounds. Label fractions alone cannot set an absolute rate.

If necessary information is missing, name it and prepare the input template; do not
invent a fragment assignment, atom map, isotope correction, or measurement error.
Read references/input-contract.md when preparing inputs.
Read references/inference.md before interpreting an actual fit.

Install the tested engine

Run in the user's analysis directory. Set SKILL_DIR to this skill's installed directory,
using the actual resolved path. Keep environments and generated results outside the skill.

uv venv --python 3.12 .venv-mfa
uv pip install --python .venv-mfa/bin/python -r "$SKILL_DIR/assets/requirements.txt"

The following commands use .venv-mfa/bin/python; on Windows use the environment's
Scripts/python.exe. mfapy is installed from an immutable Git revision because it is
not distributed on PyPI. Installation executes dependency build code; model inputs
are data, not user-supplied Python. The adapter restricts identifiers and atom-map
syntax before they reach mfapy's internally generated numerical functions.

The pinned commit matched upstream master on 2026-09-30. Its README labels the
latest change "064", but its installed distribution still reports 0.6.3; retain
the Git commit alongside the package version in an analysis record. The refreshed
NumPy/SciPy pins require Python 3.12 or later; the commands above use the tested 3.12
environment. See the reviewed forward-model contract in
references/inference.md.

Workflow

1. Prepare explicit inputs. Copy a relevant model asset into the analysis directory,

then replace its scientific content only from reviewed evidence. The bundled models
are demonstrations, not validated organism-specific reconstructions. Use a separate
dataset for each biological condition; jointly fit tracer replicates only when their
biological flux state is defensibly shared.

1. Check the contract and feasibility.

   .venv-mfa/bin/python "$SKILL_DIR/scripts/mfa.py" check \
     --model model.json --data measurements.json --output input-check.json

This checks atom counts and conservation, fragments, tracer sums, uncertainty
matrices, bounds, and steady-state mass-balance feasibility. It cannot verify that a
chemically consistent atom map is biologically correct or that a sample reached steady state.

1. Exercise the forward model. Supply one mass-balanced flux vector in the declared

units. Compare predicted labeling with a reference or independently derived limits.

   .venv-mfa/bin/python "$SKILL_DIR/scripts/mfa.py" simulate \
     --model model.json --data measurements.json --fluxes fluxes.json \
     --output simulated-mdvs.json

1. Fit and profile the fluxes relevant to the question.

   .venv-mfa/bin/python "$SKILL_DIR/scripts/mfa.py" fit \
     --model model.json --data measurements.json --starts 12 --seed 2026 \
     --profile v3 --profile v7 --profile-points 31 --profile-starts 6 \
     --output fit.json

Replace v3 and v7 with actual reaction IDs. Each profile point fixes that reaction
and reoptimizes nuisance fluxes. For nonlinear networks, repeat with a different seed
and more starts before interpreting a profile. A small residual is not an
identifiability result.

1. Inspect the evidence. Check failed starts, residual patterns, mass balance,

active bounds, local sensitivity rank, and profile status. Report threshold-crossing
brackets at their actual grid resolution. Refine the grid if they are too coarse.
Each requested profile gives a one-flux interval under the stated error model;
multiple 95% profiles are not a simultaneous 95% region for the whole network.
If a profile finds a better solution than the baseline, rerun the fit; do not publish
the stale intervals. A failed profile point is unknown, not excluded by the data.

1. Deliver a bounded scientific result. Include model and data hashes, package

versions, source/correction provenance, units and reference flux, fitted predictions,
residual diagnostics, profile plots or a table, and the unresolved flux combinations.
Retain the JSON artifact. Separate point estimates supported by the data from arbitrary
optimizer choices along a flat direction. Suggest additional measurements only after
testing that their predicted labeling changes along that direction.

Worked examples

These executable examples use synthetic, tracer-only data. There is no hidden natural-
abundance correction, and the tracer proportions already include unlabeled material.

Recover a pathway split; then remove the informative measurement

The analytical two-route model sends a two-carbon substrate through either a
carbon-preserving or a carbon-swapping route. Uptake is fixed to 100. An 80% carbon-1
labeled feed and a carbon-1 fragment with M+1 = 0.56 determine the preserving route
as 70 and the swapping route as 30.

.venv-mfa/bin/python "$SKILL_DIR/scripts/mfa.py" fit \
  --model "$SKILL_DIR/assets/branch-model.json" \
  --data "$SKILL_DIR/assets/branch-identifiable.json" \
  --profile straight --profile-points 41 --output branch-fit.json

.venv-mfa/bin/python "$SKILL_DIR/scripts/mfa.py" fit \
  --model "$SKILL_DIR/assets/branch-model.json" \
  --data "$SKILL_DIR/assets/branch-unresolved.json" \
  --profile straight --output unresolved-fit.json

The first fit recovers approximately 70/30. Under its declared Gaussian error model,
the analytical 95% interval for straight is about 67.55–72.45; the script reports
grid brackets enclosing the threshold crossings. The second fit has only the whole-
molecule distribution, which is identical for the two routes. Expect local rank zero
and unresolved_within_bounds; its returned split is an arbitrary optimum.

assets/branch-fluxes.json supplies the 70/30 forward-simulation vector.

Reproduce a published cyclic-network calculation

.venv-mfa/bin/python "$SKILL_DIR/scripts/mfa.py" simulate \
  --model "$SKILL_DIR/assets/tca-model.json" \
  --data "$SKILL_DIR/assets/tca-tracer.json" \
  --fluxes "$SKILL_DIR/assets/tca-fluxes.json" --output tca-simulation.json

.venv-mfa/bin/python "$SKILL_DIR/scripts/mfa.py" fit \
  --model "$SKILL_DIR/assets/tca-model.json" \
  --data "$SKILL_DIR/assets/tca-reference-mdv.json" \
  --profile v3 --profile v7 --output tca-fit.json

The first command reproduces the published rounded glutamate MDV
[0.3464, 0.2695, 0.2708, 0.0807, 0.0286, 0.0039].
The second uses synthetic reference measurements to recover the glutamate branch
flux near 50, while recognizing that this labeling does not resolve the
fumarate/oxaloacetate exchange. A constraint-induced upper edge is not evidence of
a measurement-determined exchange interval.

Interpretation boundaries

• A positional isotopomer string runs carbon 1 to carbon N from left to right.

"100000" means carbon-1 labeled glucose. A mass distribution alone cannot specify
that positional mixture. The adapter handles mfapy's reversed integer-bit ordering.

• Natural-abundance correction and tracer-purity correction are different operations.

Inputs must be in the documented tracer-only basis, with tracer impurity represented
consistently in source mixtures. Do not correct the same contribution twice.

• An N-carbon mass distribution has at most N independent components because it sums

to one. The tool removes one bin and uses the reduced covariance. Retain cross-bin
correlations when available. Diagonal SEM fits are explicitly approximate.

• The symmetric flag means equal averaging of identity and **complete carbon-order

reversal**, as in the bundled fumarate/succinate map. It is not arbitrary molecular
symmetry. Other permutations need an explicitly supported model representation.

• Unsupported in this CLI: nonstationary MFA, isotope effects on reaction rates,

unmodeled pools or compartments, MS/MS joint distributions, multi-element isotope
correction, fractional carbon stoichiometry/pseudo-reactions, and organism-scale
performance guarantees. For these, use a validated specialized model/engine and
retain the same input/provenance and identifiability discipline.

Implementation and validation

scripts/mfa.py is the CLI. scripts/_mfa_model.py validates inputs and adapts them to
the mfapy EMU simulator; scripts/_mfa_fit.py handles feasible flux coordinates,
multistart optimization, diagnostic rank, and profile calculations.
The engine is pinned in assets/requirements.txt.

The repository suite at tests/13c-metabolic-flux/ checks the published reference,
analytical split recovery and likelihood profiles, unresolved routes and exchange,
omitted-bin invariance with correlated errors, parallel tracers, absolute-rate
anchoring, repeated-substrate condensation, symmetry, invalid maps, and CLI behavior.
These checks establish the tested numerical behavior, not biological validation of a
user's model or a measured advantage over any particular language model.

Sources

• mfapy source at the tested revision, version 0.6.3; API documentation.
• Matsuda et al. (2021), mfapy: An open-source Python package for 13C-based metabolic flux analysis.
• Antoniewicz, Kelleher, and Stephanopoulos (2007), Elementary metabolite units (EMU): A novel framework for modeling isotopic distributions.
• The TCA network is adapted from mfapy's MIT-licensed example files; the attribution

and full notice are in assets/mfapy-license.txt.