Relaxedfba

plotRelaxedFBA(sol, model, tol)

Print a summary of a relaxedFBA solution, reporting which steady state constraints and which reaction bounds were relaxed to make the model feasible

USAGE:

plotRelaxedFBA (sol, model, tol)

INPUTS:

• sol – relaxedFBA solution structure with fields:

  • .stat - solution status (1 if relaxedFBA solved successfully)

  • .v - reaction rate

  • .r - relaxation on the steady state constraints S*v = b

  • .p - relaxation on the lower bounds of reactions

  • .q - relaxation on the upper bounds of reactions

• model – COBRA model structure with fields:

  • .S - m x n stoichiometric matrix

  • .b - m x 1 right hand side of S*v = b

  • .lb - n x 1 lower flux bounds

  • .ub - n x 1 upper flux bounds

  • .rxns - n x 1 reaction identifiers

  • .mets - m x 1 metabolite identifiers

  • .SIntRxnBool - n x 1 boolean, true for internal reactions

OPTIONAL INPUTS:

tol – tolerance below which a relaxation is treated as zero (default: 100 * LP feasTol)

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 with fields:

  • .S - m x n stoichiometric matrix

  • .b - m x 1 right hand side of S*v = b

  • .c - n x 1 linear objective coefficients

  • .lb - n x 1 lower flux bounds

  • .ub - n x 1 upper flux bounds

  • .csense - m x 1 constraint sense for each row of S (E, L or G)

  • .C - additional inequality constraint matrix (C*v <= d)

  • .d - right hand side of the C*v <= d constraints

  • .dsense - constraint sense for each row of C

• param – structure containing the relaxation options:

  • .excludedReactions - bool vector of size n, reactions excluded from relaxation

  • .excludedReactionLB - n x 1 bool vector, reactions whose lower bound is excluded from relaxation

  • .excludedReactionUB - n x 1 bool vector, reactions whose upper bound is excluded from relaxation

  • .excludedMetabolites - bool vector of size m, metabolites excluded from relaxation

  • .toBeUnblockedReactions - n x 1 vector indicating reactions to be unblocked

  • .nbMaxIteration - stopping criterion, maximum number of iterations

  • .epsilon - stopping criterion tolerance

  • .theta - parameter of the capped-L1 approximation

  • .maxUB - maximum finite upper bound used when relaxing bounds

  • .minLB - minimum finite lower bound used when relaxing bounds

  • .maxRelaxR - maximum relaxation of any bound or equality constraint permitted

  • .printLevel - verbosity of the progress output

  • .gamma0 - zero-norm weight on the reaction rate

  • .gamma1 - one-norm weight on the reaction rate

  • .lambda0 - zero-norm weight on relaxation of steady state constraints

  • .lambda1 - one-norm weight on relaxation of steady state constraints

  • .alpha0 - zero-norm weight on relaxation of reaction bounds

  • .alpha1 - one-norm weight on relaxation of reaction bounds

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 * .lb * .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 * .c - linear objective coefficients * .A - constraint matrix used in place of .S for a generic LP problem * .rxnNames - reaction descriptions * .metNames - metabolite descriptions * .SConsistentRxnBool - boolean, true for stoichiometrically consistent reactions * .SExtRxnBool - boolean, true for external reactions * .evars - extra variable identifiers * .evarlb - lower bounds on extra variables * .evarub - upper bounds on extra variables * .evarc - objective coefficients on extra variables * .ctrs - additional constraint identifiers

• 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

  • .extraVarRelax: * 0 = do not allow to relax bounds on extra variables * 1 = do not allow to relax bounds on extra variables with finite bounds * {2} = allow to relax bounds on all extra variables

  • .extraConstraintRelax: * 0 = do not allow to relax extra constraints * {1} = allow to relax extra constraints

  • .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

  • .toBeUnblockedEvars - nEvars x 1 vector indicating the extra variables to be unblocked * toBeUnblockedEvars(i) = 1 : impose vEvar(i) to be positive * toBeUnblockedEvars(i) = -1 : impose vEvar(i) to be negative * toBeUnblockedEvars(i) = 0 : do not add any constraint (default)

  • .excludedEvars - nEvars x 1 bool vector indicating the extra variables to be excluded from relaxation * excludedEvars(i) = false : allow to relax bounds on extra variable i (default) * excludedEvars(i) = true : do not allow to relax bounds on extra variable i

  • .excludedEvarLB - nEvars x 1 bool vector indicating

  the extra variables with lower bounds to be excluded from relaxation (overridden by excludedEvars)

  • excludedEvarLB(i) = false : allow to relax lower bounds on extra variablen i (default)

  • excludedEvarLB(i) = true : do not allow to relax lower bounds on extra variable i

  • .excludedEvarUB - nEvars x 1 bool vector indicating

  the extra variables with upper bounds to be excluded from relaxation (overridden by excludedEvars)

  • excludedEvarUB(i) = false : allow to relax upper bounds on extra variable i (default)

  • excludedEvarUB(i) = true : do not allow to relax upper bounds on extra variable i

  • .excludedCtrs - nCtrs x 1 bool vector indicating the extra constraints to be excluded from relaxation * excludedCtrs(i) = false : allow to relax steady state constraint on extra constraints i (default) * excludedCtrs(i) = true : do not allow to relax steady state constraint on extra constraints i

  • .lambda - 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

  • .alpha0 - zero-norm weight on relaxation of reaction bounds

  • .alpha1 - one-norm weight on relaxation of reaction bounds

  • .lambda0 - zero-norm weight on relaxation of steady state constraints

  • .lambda1 - one-norm weight on relaxation of steady state constraints

  • .gamma0 - zero-norm weight on the reaction rate

  • .gamma1 - one-norm weight on the reaction rate

  • .maxUB - maximum finite upper bound used when relaxing bounds

  • .minLB - minimum finite lower bound used when relaxing bounds

  • .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

  • vEvar - extra variable value

  • pEvar - relaxation on lower bound of extra

  variables * qEvar - relaxation on upper bound of extra variables * rCtrs - relaxation on steady state of extra constraints

• relaxedModel – model structure that admits a flux balance solution

Fleming RMT, Haraldsdottir HS, Le HM, Vuong PT, Hankemeier T, Thiele I. Cardinality optimisation in constraint-based modelling: Application to human metabolism, 2022 (submitted).

• April 2026: expanded to deal with extra variables and constraints (model.C,

model.D, model.E) - Tania Barata