Moma

MOMA(modelWT, modelDel, osenseStr, verbFlag, minNorm)

Performs a quadratic version of the MOMA (minimization of metabolic adjustment) approach

USAGE:

[solutionDel, solutionWT, totalFluxDiff, solStatus] = MOMA (modelWT, modelDel, osenseStr, verbFlag, minNorm)

INPUTS:

• modelWT – Wild type model with the fields:

  • .S - m x n stoichiometric matrix

  • .rxns - n x 1 cell array of reaction identifiers

  • .osenseStr - objective sense stored on the model, ‘max’ or ‘min’

• modelDel – Deletion strain model with the fields:

  • .S - m x n stoichiometric matrix

  • .rxns - n x 1 cell array of reaction identifiers

  • .c - n x 1 linear objective coefficient vector

OPTIONAL INPUTS:

• osenseStr – Maximize (‘max’) / minimize (‘min’) (Default = ‘max’)

• verbFlag – Verbose output (Default = false)

• minNorm – Determines the approach to solving the first optimisation problem See minNorm option of optimizeCbModel {(1),(0), ‘one’, ‘zero’, > 0 , n x 1 vector}, where [m,n]=size(S); i.e. 1e-6 is the default, which minimises the Euclidean Norm of the

  subject to atainment of the optimal FBA objective as an additional constraint.

• minNorm – Work with minimum 1-norm flux distribution for the FBA problem (Default = false)

OUTPUTS:

• solutionDel – Deletion solution structure

• solutionWT – Wild-type solution structure

• totalFluxDiff – Value of the linear MOMA objective, i.e. \(\sum (v_{wt}-v_{del})^2\)

• solStatus – Solution status

First solve:

\[\begin{split}max ~&~ c_{wt}^T v_{wt0} \\ ~&~ lb_{wt} \leq v_{wt0} \leq ub_{wt} \\ ~&~ S_{wt}v_{wt0} = 0 \\\end{split}\]

Then solve:

\[\begin{split}min ~&~ \sum (v_{wt} - v_{del})^2 \\ ~&~ S_{wt}v_{wt} = 0 \\ ~&~ S_{del}v_{del} = 0 \\ ~&~ lb_{wt} \leq v_{wt} \leq ub_{wt} \\ ~&~ lb_{del} \leq v_{del} \leq ub_{del} \\ ~&~ c_{wt}^T v_{wt} = f_{wt} \\\end{split}\]

Here \(f_{wt}\) is the optimal wild type objective value found by FBA in the first problem. Note that the FBA solution \(v_{wt0}\) is not used in the second problem. This formulation avoids any problems with alternative optima

First solve

\[\begin{split}max ~&~ c_{wt}^T v_{wt0} \\ ~&~ lb_{wt} \leq v_{wt0} \leq ub_{wt} \\ ~&~ S_{wt}v_{wt0} = 0 \\\end{split}\]

Then solve a regularised problem, the default of which is shown here

\[\begin{split}min ~&~ ||v_{wt}|| \\ ~&~ S_{wt}v_{wt} = b_{wt} \\ ~&~ c_{wt}^T v_{wt} = f_{wt} \\ ~&~ lb_{wt} \leq v_{wt} \leq ub_{wt} \\\end{split}\]

Here \(f_{wt}\) is the objective value obtained in the 1st optimization.

Finally solve:

\[\begin{split}min ~&~ \sum (v_{wt} - v_{del})^2 \\ ~&~ S_{del}v_{del} = 0 \\ ~&~ lb_{del} \leq v_{del} \leq ub_{del}\end{split}\]

NOTE:

1) These formulation allows for selecting for more appropriate
optimal wild type FBA solutions as the starting point as opposed to
picking an arbitrary starting point (original MOMA implementation).

2) The reaction sets in the two models do not have to be equal as long as
there is at least one reaction in common

linearMOMA(modelWT, modelDel, osenseStr, minFluxFlag, verbFlag)

Performs a linear version of the MOMA (minimization of metabolic adjustment) approach

USAGE:

[solutionDel, solutionWT, totalFluxDiff, solStatus] = linearMOMA (modelWT, modelDel, osenseStr, minFluxFlag, verbFlab)

INPUTS:

• modelWT – Wild type model with the fields:

  • .S - m x n stoichiometric matrix

  • .rxns - n x 1 cell array of reaction identifiers

• modelDel – Deletion strain model with the fields:

  • .S - m x n stoichiometric matrix

  • .rxns - n x 1 cell array of reaction identifiers

  • .c - n x 1 linear objective coefficient vector

OPTIONAL INPUTS:

• osenseStr – Maximize (‘max’) / minimize (‘min’) (Default = ‘max’)

• minFluxFlag – Minimize the absolute value of fluxes in the optimal MOMA solution (Default = false)

• verbFlag – Verbose output (Default = false)

OUTPUTS:

• solutionDel – Deletion solution structure

• solutionWT – Wild-type solution structure

• totalFluxDiff – Value of the linear MOMA objective, i.e. \(\sum |v_{wt}-v_{del}|\)

• solStatus – Solution status - solves the problem: (f_wt is the optimal wild type objective value found by FBA)

\[\begin{split}min ~&~ \sum |v_{wt} - v_{del}| \\ ~&~ S_{wt}v_{wt} = 0 \\ ~&~ lb_{wt} \leq v_{wt} \leq ub_{wt} \\ ~&~ c_{wt}^T v_{wt} = f_{wt} \\ ~&~ S_{del}v_{del} = 0 \\ ~&~ lb_{del} \leq v_{del} \leq ub_{del}\end{split}\]

Note

1) This formulation allows for selecting the most appropriate optimal wild type FBA solution as the starting point as opposed to picking an arbitrary starting point (original MOMA implementation).

2) The reaction sets in the two models do not have to be equal as long as there is at least one reaction in common