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 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).