def degrees_of_freedom(model):
"""
Return the degrees of freedom, i.e. number of "free variables".
Parameters
----------
model : cobra.Model
The metabolic model under investigation.
Notes:
------
This specifically refers to the dimensionality of the right nullspace
of the S matrix, as dim(Null(S)) corresponds directly to the number of
free variables in the system [1]_. The forumla used calculates this using
the rank-nullity theorem [2]_.
References:
-----------
.. [1] Fukuda, K. & Terlaky, T. Criss-cross methods: A fresh view on
pivot algorithms. Mathematical Programming 79, 369-395 (1997).
.. [2] Alama, J. The Rank+Nullity Theorem. Formalized Mathematics 15,
(2007).
"""
s_matrix, _, _ = con_helpers.stoichiometry_matrix(
model.metabolites, model.reactions
)
return s_matrix.shape[1] - matrix_rank(model)
def degrees_of_freedom(model):
"""
Return the degrees of freedom, i.e., number of "free variables".
Parameters
----------
model : cobra.Model
The metabolic model under investigation.
Notes
-----
This specifically refers to the dimensionality of the (right) null space
of the stoichiometric matrix, as dim(Null(S)) corresponds directly to the
number of free variables in the system [1]_. The formula used calculates
this using the rank-nullity theorem [2]_.
References
----------
.. [1] Fukuda, K. & Terlaky, T. Criss-cross methods: A fresh view on
pivot algorithms. Mathematical Programming 79, 369-395 (1997).
.. [2] Alama, J. The Rank+Nullity Theorem. Formalized Mathematics 15,
(2007).
"""
s_matrix, _, _ = con_helpers.stoichiometry_matrix(
model.metabolites, model.reactions
)
return s_matrix.shape[1] - matrix_rank(model)
def absolute_extreme_coefficient_ratio(model):
"""Return the absolute max and absolute non-zero min coefficients."""
# S-Matrix with absolute values:
s_matrix, _, _ = con_helpers.stoichiometry_matrix(model.metabolites,
model.reactions)
abs_matrix = np.absolute(s_matrix)
absolute_max_coef = np.amax(abs_matrix)
absolute_non_zero_min_coef = abs_matrix[abs_matrix > 0].min()
return (absolute_max_coef, absolute_non_zero_min_coef)
def matrix_rank(model):
"""
Return the rank of the model's stoichiometric matrix.
Parameters
----------
model : cobra.Model
The metabolic model under investigation.
"""
s_matrix, _, _ = con_helpers.stoichiometry_matrix(model.metabolites,
model.reactions)
return con_helpers.rank(s_matrix)
def absolute_extreme_coefficient_ratio(model):
"""
Return the maximum and minimum absolute, non-zero coefficients.
Parameters
----------
model : cobra.Model
The metabolic model under investigation.
"""
s_matrix, _, _ = con_helpers.stoichiometry_matrix(model.metabolites,
model.reactions)
abs_matrix = np.abs(s_matrix)
return abs_matrix.max(), abs_matrix[abs_matrix > 0].min()
def matrix_rank(model):
"""
Return the rank of the model's stoichiometric matrix.
Parameters
----------
model : cobra.Model
The metabolic model under investigation.
"""
s_matrix, _, _ = con_helpers.stoichiometry_matrix(
model.metabolites, model.reactions
)
return con_helpers.rank(s_matrix)
def number_steady_state_flux_solutions(model):
"""
Return the amount of steady-state flux solutions of this model.
Parameters
----------
model : cobra.Model
The metabolic model under investigation.
"""
s_matrix, _, _ = con_helpers.stoichiometry_matrix(
model.metabolites, model.reactions
)
n_matrix = con_helpers.nullspace(s_matrix)
return n_matrix.shape[1]
def number_independent_conservation_relations(model):
"""
Return the amount of conserved metabolic pools.
Parameters
----------
model : cobra.Model
The metabolic model under investigation.
"""
s_matrix, _, _ = con_helpers.stoichiometry_matrix(
model.metabolites, model.reactions
)
ln_matrix = con_helpers.nullspace(s_matrix.T)
return ln_matrix.shape[1]
def absolute_extreme_coefficient_ratio(model):
"""
Return the maximum and minimum absolute, non-zero coefficients.
Parameters
----------
model : cobra.Model
The metabolic model under investigation.
"""
s_matrix, _, _ = con_helpers.stoichiometry_matrix(
model.metabolites, model.reactions
)
abs_matrix = np.abs(s_matrix)
return abs_matrix.max(), abs_matrix[abs_matrix > 0].min()
def number_independent_conservation_relations(model):
"""
Return the number of conserved metabolite pools.
This number is given by the left null space of the stoichiometric matrix.
Parameters
----------
model : cobra.Model
The metabolic model under investigation.
"""
s_matrix, _, _ = con_helpers.stoichiometry_matrix(model.metabolites,
model.reactions)
left_ns = con_helpers.nullspace(s_matrix.T)
return left_ns.shape[1] if len(left_ns) > 1 else 0
def number_independent_conservation_relations(model):
"""
Return the number of conserved metabolite pools.
This number is given by the left null space of the stoichiometric matrix.
Parameters
----------
model : cobra.Model
The metabolic model under investigation.
"""
s_matrix, _, _ = con_helpers.stoichiometry_matrix(
model.metabolites, model.reactions
)
ln_matrix = con_helpers.nullspace(s_matrix.T)
return ln_matrix.shape[1]
def find_inconsistent_min_stoichiometry(model, atol=1e-13):
"""
Detect inconsistent minimal net stoichiometries.
Parameters
----------
model : cobra.Model
The metabolic model under investigation.
atol : float, optional
Values below the absolute tolerance are treated as zero. Expected to be
very small but larger than zero.
Notes
-----
See [1]_ section 3.3 for a complete description of the algorithm.
References
----------
.. [1] Gevorgyan, A., M. G Poolman, and D. A Fell.
"Detection of Stoichiometric Inconsistencies in Biomolecular
Models."
Bioinformatics 24, no. 19 (2008): 2245.
"""
if check_stoichiometric_consistency(model):
return set()
Model, Constraint, Variable, Objective = con_helpers.get_interface(model)
unconserved_mets = find_unconserved_metabolites(model)
LOGGER.info("model has %d unconserved metabolites", len(unconserved_mets))
internal_rxns = con_helpers.get_internals(model)
internal_mets = set(met for rxn in internal_rxns
for met in rxn.metabolites)
get_id = attrgetter("id")
reactions = sorted(internal_rxns, key=get_id)
metabolites = sorted(internal_mets, key=get_id)
stoich, met_index, rxn_index = con_helpers.stoichiometry_matrix(
metabolites, reactions)
left_ns = con_helpers.nullspace(stoich.T)
# deal with numerical instabilities
left_ns[np.abs(left_ns) < atol] = 0.0
LOGGER.info("nullspace has dimension %d", left_ns.shape[1])
inc_minimal = set()
(problem,
indicators) = con_helpers.create_milp_problem(left_ns, metabolites, Model,
Variable, Constraint,
Objective)
LOGGER.debug(str(problem))
cuts = list()
for met in unconserved_mets:
row = met_index[met]
if (left_ns[row] == 0.0).all():
LOGGER.debug("%s: singleton minimal unconservable set", met.id)
# singleton set!
inc_minimal.add((met, ))
continue
# expect a positive mass for the unconserved metabolite
problem.variables[met.id].lb = 1e-3
status = problem.optimize()
while status == "optimal":
LOGGER.debug("%s: status %s", met.id, status)
LOGGER.debug("sum of all primal values: %f",
sum(problem.primal_values.values()))
LOGGER.debug("sum of binary indicators: %f",
sum(var.primal for var in indicators))
solution = [
model.metabolites.get_by_id(var.name[2:]) for var in indicators
if var.primal > 0.2
]
LOGGER.debug("%s: set size %d", met.id, len(solution))
inc_minimal.add(tuple(solution))
if len(solution) == 1:
break
cuts.append(
con_helpers.add_cut(problem, indicators,
len(solution) - 1, Constraint))
status = problem.optimize()
LOGGER.debug("%s: last status %s", met.id, status)
# reset
problem.variables[met.id].lb = 0.0
problem.remove(cuts)
cuts.clear()
return inc_minimal
def matrix_rank(model):
"""Return the rank of the model's stoichiometric matrix."""
s_matrix, _, _ = con_helpers.stoichiometry_matrix(model.metabolites,
model.reactions)
return con_helpers.rank(s_matrix)
def number_steady_state_flux_solutions(model):
"""Return the amount of steady-state flux solutions of this model."""
s_matrix, _, _ = con_helpers.stoichiometry_matrix(model.metabolites,
model.reactions)
n_matrix = con_helpers.nullspace_basis(s_matrix)
return n_matrix.shape[1]
def number_independent_conservation_relations(model):
"""Return the amount of conserved metabolic pools."""
s_matrix, _, _ = con_helpers.stoichiometry_matrix(model.metabolites,
model.reactions)
ln_matrix = con_helpers.nullspace_basis(s_matrix.T)
return ln_matrix.shape[1]