Department of Plant Sciences, 234 Herzl st., Weizmann Institute of Science, Rehovot 76100, Israel

Department of Bioengineering, 9500 Gilman Drive, University of California San Diego, La Jolla, CA 92093-0412, USA

Wyss Institute for Biologically Inspired Engineering and Department of Genetics, 77 Avenue Louis Pasteur, Harvard Medical School, Boston, MA 02115, USA

Abstract

Background

Constraint-based modeling is increasingly employed for metabolic network analysis. Its underlying assumption is that natural metabolic phenotypes can be predicted by adding physicochemical constraints to remove unrealistic metabolic flux solutions. The loopless-COBRA approach provides an additional constraint that eliminates thermodynamically infeasible internal cycles (or loops) from the space of solutions. This allows the prediction of flux solutions that are more consistent with experimental data. However, it is not clear if this approach over-constrains the models by removing non-loop solutions as well.

Results

Here we apply Gordan’s theorem from linear algebra to prove for the first time that the constraints added in loopless-COBRA do not over-constrain the problem beyond the elimination of the loops themselves.

Conclusions

The loopless-COBRA constraints can be reliably applied. Furthermore, this proof may be adapted to evaluate the theoretical soundness for other methods in constraint-based modeling.

Background

Constraint-based modeling has become a successful framework for the analysis of large and complex stoichiometric biochemical networks

Loop law constraints on metabolic networks

**Loop law constraints on metabolic networks.****(a)** Metabolic network reconstructions frequently have sets of reactions that cycle all metabolites internally. The fluxes of these reactions are therefore unconstrained. **(b)** Metabolic network solutions are found within a convex space, which is enclosed by known constraints on metabolite inputs, outputs, and known fluxes. Loops result in unconstrained dimensions in the solution space (blue). By implementing loopless-COBRA constraints, all loop-containing solutions are removed, leaving only solutions that do not contain loops (orange).

A new approach, called loopless-COBRA, was recently presented

The loopless-COBRA method was shown to work in various scenarios, and the paper that presented the approach provides an explanation for why this method works. However, there is no mathematical proof for its formulation as an optimization problem. Specifically, it does not demonstrate that the additional MILP constraints do not over-constrain the problem and eliminate some non-loop containing solutions. Since constraint-based methods attempt to only eliminate impossible

Here, we address this issue by presenting a mathematical proof for the completeness and soundness of the loopless-COBRA method, thereby adding fundamental support and rigorous proof for the constraints presented by Schellenberger et al.

Results and discussion

Formal definition of loop-law constraints

In loopless-COBRA, the constraints added to the linear problem are:

where _{
i
} are the flux variables and _{
int
} is the null-space matrix of _{
int
}(the stoichiometric matrix of internal reactions). The third constraint (
_{
i
} can have any value if _{
i
} = 0. We can rewrite all these constraints succinctly as:

Formal definition of loops

In order to prove that this constraint eliminates loops (and only loops), we must first find a mathematical formulation for a loop, using the same notation as above. We thus define a loop as a nonzero vector
_{
int
} ·

According to this definition, a flux distribution (
_{
i
} is either zero or has the same sign as _{
i
}) and is itself a loop (i.e. _{
int
} ·

We have now finished laying the groundwork for our mathematical proof that loopless-COBRA is sound and complete. In order to do that, we are left only to show that Equation 1 is satisfiable if and only if Equation 2 is unsatisfiable (in other words, there are no loops).

Gordan’s theorem

We start our proof by quoting Gordan’s theorem: For all

We will show that statement (

As a guidance for the following sections, one can see that statement (_{
int
}. The only difference is that

At first glance, statement (_{
int
})

Illustrative example for Corollary 2

**Illustrative example for Corollary 2.** This example shows a small network with 3 internal reactions (_{2−4}). The flux directions were chosen according to the direction of the arrows. The matrix **(a)** A flux distribution is shown where all 3 internal reactions are active and form a loop. Therefore there is a solution (_{2} = _{3} = _{4} = 1) for the mass balance equation ^{⊤}**(b)** A loopless flux distribution, in this case _{4}is not active. There is no solution for ^{⊤}

Corollary 1

For all
^{
n
} exactly one of the following two statements is true:

Proof

First, define a new matrix
_{
i
} = 0 and where columns corresponding to _{
i
} = −1 are multiplied by −1. Statement (1_{
i
} = 0, and negating values corresponding to _{
i
} = −1 (as previously done for

Likewise, statement (1_{
i
} = −1 are negated in
_{
i
} = 0 have no other constraints in (1

Therefore, Corollary 1 is directly derived from Gordan’s theorem. □

Since constraint-based models usually use a vector of real values (

Corollary 2

For all

Proof

Defining _{
i
} ≡ sign(_{
i
}), we get this directly from Corollary 1. Note that −sign(_{
i
}) can be used in (2b), since the existence of

This adjustment now allows us to apply Corollary 2 to constraint-based problems and show that it eliminates loops (see example in Figure

Corollary 3

Adding the following constraint:

is equivalent to eliminating all loops in a flux distribution

Proof

Using Corollary 2, all that must be shown is that statement (2a) is equivalent to having a loop. This is apparent since

Note that the trivial case

Applying Corollary 3 in loopless-COBRA

The added constraints in loopless-COBRA

We claim here that both formulations are equivalent. The fundamental theorem of linear algebra states that the nullspace, null(^{⊤}). Therefore, we can say that null(^{⊤}), so we can rewrite the constraint above as:

which is obviously equivalent to the constraint in Corollary 3.

Conclusions

Our results prove that the constraints proposed by Schellenberger, et al.
_{
r
}
_{
f
}

In conclusion, this proof provides theoretical credibility for the loopless-COBRA constraint. However, as with any algorithmic MILP implementation, care must still be taken with respect to numerical limitations and the convergence of the optimization algorithm.

Lastly, we believe this proof may be extended to similar methods addressing loop elimination. We also hope that similar proofs will appear for other methods, since more rigorous mathematical treatments are needed in many published algorithms in computational biology to prove or disprove their correctness.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

EN conceived of the study, participated in its design and coordination, and helped to draft the manuscript. NL participated in the design of the study and drafted the manuscript. RM participated in its design, coordination, and drafting the manuscript. All authors read and approved the final manuscript.

Acknowledgements

We thank Or Sheffet, Bernhard Ø. Palsson and Uri Barenholz for mathematical assistance and helpful discussions. EN is grateful to the Azrieli Foundation for the award of an Azrieli Fellowship.