# relaxedFBA¶

relaxFBA_cappedL1(model, param)[source]

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)[source]

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
• model.S
• model.SIntRxnBool

Optional inputs

• model – COBRA model structure

• model.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
• 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)
• .theta0 - 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 theta0 to start with the appropriate parameter giving the lowest cardinality solution.

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