relaxedFBA¶
- relaxFBA_cappedL1(model, param)¶
Finds the mimimal set of relaxations on bounds and steady state constraints to make the FBA problem feasible. The zero-norm is appproximated by capped-L1 norm
USAGE:
[solution] = relaxFBA_cappedL1(model, param)
- INPUTS:
model: COBRA model structure param: Structure containing the relaxation options:
excludedReactions - bool vector of size n indicating the reactions to be excluded from relaxation * excludedReactions(i) = false : allow to relax bounds on reaction i * excludedReactions(i) = true : do not allow to relax bounds on reaction i
excludedReactionLB - n x 1 bool vector indicating the reactions with lower bounds to be excluded from relaxation
excludedReactionLB(i) = false : allow to relax lower bounds on reaction i (default)
excludedReactionLB(i) = true : do not allow to relax lower bounds on reaction i
excludedReactionUB - n x 1 bool vector indicating the reactions with upper bounds to be excluded from relaxation
excludedReactionUB(i) = false : allow to relax upper bounds on reaction i (default)
excludedReactionUB(i) = true : do not allow to relax upper bounds on reaction i
excludedMetabolites - bool vector of size m indicating the metabolites to be excluded from relaxation * excludedMetabolites(i) = false : allow to relax steady state constraint on metabolite i * excludedMetabolites(i) = true : do not allow to relax steady state constraint on metabolite i
gamma - weight on zero norm of fluxes
lamda - weight on relaxation on steady state constraint (overridden by excludedMetabolites)
alpha - weight on relaxation on bounds (overridden by excludedReactions)
- OUTPUT:
solution: Structure containing the following fields:
stat - status
1 = Solution found
0 = Infeasible
-1 = Invalid input
r - relaxation on steady state constraints \(S*v = b\)
p - relaxation on lower bound of reactions
q - relaxation on upper bound of reactions
v - reaction rate
\[\begin{split}min ~&~ c^T v + \gamma_1 ||v||_1 + \gamma_0 ||v||_0 + \lambda_1 ||r||_1 + \lambda_0 ||r||_0 \\ ~&~ + \alpha_1 (||p||_1 + ||q||_1) + \alpha_0 (||p||_0 + ||q||_0) \\ s.t. ~&~ S v + r = b \\ ~&~ l - p \leq v \leq u + q \\ ~&~ r \in R^m \\ ~&~ p,q \in R_+^n\end{split}\]m - number of metabolites, n - number of reactions
- relaxedFBA(model, param)¶
Finds the mimimal set of relaxations on bounds and steady state constraints to make the FBA problem feasible. The optional parameters, excludedReactions and excludedMetabolites override all other relaxation options.
\[\begin{split}min ~&~ c^T v + \gamma ||v||_0 + \lambda ||r||_0 + \alpha (||p||_0 + ||q||_0) \\ s.t ~&~ S v + r \leq, =, \geq b \\ ~&~ l - p \leq v \leq u + q \\ ~&~ r \in R^nMets \\ ~&~ p,q \in R_+^nRxns\end{split}\]nMets - number of metabolites, nRxns - number of reactions
USAGE:
[solution] = relaxedFBA(model, param)
- INPUTS:
- model: COBRA model structure with the fields:
.S
.b
.ub
.ub
.mets (required if model.SIntRxnBool absent)
.rxns (required if model.SIntRxnBool absent)
- OPTIONAL INPUTS:
- model: COBRA model structure with the fields
.csense
.C
.d
.dsense
.SIntRxnBool
param: Structure optionally containing the relaxation parameters:
.internalRelax: * 0 = do not allow to relax bounds on internal reactions * 1 = do not allow to relax bounds on internal reactions with finite bounds * {2} = allow to relax bounds on all internal reactions
.exchangeRelax: * 0 = do not allow to relax bounds on exchange reactions * 1 = do not allow to relax bounds on exchange reactions of the type [0,0] * {2} = allow to relax bounds on all exchange reactions
.steadyStateRelax: * 0 = do not allow to relax the steady state constraint S*v = b * {1} = allow to relax the steady state constraint S*v = b
.toBeUnblockedReactions - nRxns x 1 vector indicating the reactions to be unblocked * toBeUnblockedReactions(i) = 1 : impose v(i) to be positive * toBeUnblockedReactions(i) = -1 : impose v(i) to be negative * toBeUnblockedReactions(i) = 0 : do not add any constraint (default)
.excludedReactions - nRxns x 1 bool vector indicating the reactions to be excluded from relaxation * excludedReactions(i) = false : allow to relax bounds on reaction i (default) * excludedReactions(i) = true : do not allow to relax bounds on reaction i
.excludedReactionLB - nRxns x 1 bool vector indicating
the reactions with lower bounds to be excluded from relaxation (overridden by excludedReactions)
excludedReactionLB(i) = false : allow to relax lower bounds on reaction i (default)
excludedReactionLB(i) = true : do not allow to relax lower bounds on reaction i
.excludedReactionUB - nRxns x 1 bool vector indicating
- the reactions with upper bounds to be excluded from relaxation (overridden by excludedReactions)
excludedReactionUB(i) = false : allow to relax upper bounds on reaction i (default)
excludedReactionUB(i) = true : do not allow to relax upper bounds on reaction i
.excludedMetabolites - nMets x 1 bool vector indicating the metabolites to be excluded from relaxation * excludedMetabolites(i) = false : allow to relax steady state constraint on metabolite i (default) * excludedMetabolites(i) = true : do not allow to relax steady state constraint on metabolite i
.lamda - weighting on relaxation of relaxation on steady state constraints S*v = b
.alpha - weighting on relaxation of reaction bounds
.gamma - weighting on zero norm of fluxes
.nbMaxIteration - stopping criteria - number maximal of iteration (Default value = 100)
.epsilon - stopping criteria - (Default value = 1e-6)
- .theta - initial parameter of the approximation (Default value = 0.5)
Theoretically, the greater the value of step parameter, the better the approximation of a step function. However, practically, a greater inital value, will tend to optimise toward a local minima of the approximate cardinality optimisation problem.
.printLevel (Default = 0) Printing the progress of
the algorithm is useful when trying different values of theta to start with the appropriate parameter giving the lowest cardinality solution. * .relaxedPrintLevel (Default = 0) Printing information on relaxed reaction bounds and steady state constraints * .maxRelaxR (Default = 1e4), maximum relaxation of any bound or equality constraint permitted
- OUTPUT:
solution: Structure containing the following fields:
stat - status
1 = Solution found
0 = Infeasible
-1 = Invalid input
r - relaxation on steady state constraints S*v = b
p - relaxation on lower bound of reactions
q - relaxation on upper bound of reactions
v - reaction rate
relaxedModel model structure that admits a flux balance solution
Authors: - Hoai Minh Le, Ronan Fleming .. Please cite: Fleming RMT, Haraldsdottir HS, Le HM, Vuong PT, Hankemeier T, Thiele I. Cardinality optimisation in constraint-based modelling: Application to human metabolism, 2022 (submitted).