[Paper Review] Groebner bases of reaction networks with intermediate species

AmirHosein Sadeghimanesh, Elisenda Feliu|arXiv (Cornell University)|Mar 21, 2018

Computational Drug Discovery Methods20 references3 citations

TL;DR

This paper presents a method to efficiently compute Gröbner bases for reaction networks with intermediate species by leveraging the core network (without intermediates) and lifting the basis via linear algebra. The key result is that a Gröbner basis of the core network's steady state ideal always lifts to a Gröbner basis of the extended network's ideal under suitable monomial orders, drastically reducing computation time and preserving binomiality under an additional structural condition.

ABSTRACT

In this work we consider the computation of Groebner bases of the steady state ideal of reaction networks equipped with mass-action kinetics. Specifically, we focus on the role of intermediate species and the relation between the extended network (with intermediate species) and the core network (without intermediate species). We show that a Groebner basis of the steady state ideal of the core network always lifts to a Groebner basis of the steady state ideal of the extended network by means of linear algebra, with a suitable choice of monomial order. As illustrated with examples, this contributes to a substantial reduction of the computation time, due mainly to the reduction in the number of variables and polynomials. We further show that if the steady state ideal of the core network is binomial, then so is the case for the extended network, as long as an extra condition is fulfilled. For standard networks, this extra condition can be visually explored from the network structure alone.

Motivation & Objective

• To reduce the computational cost of computing Gröbner bases for steady state ideals in reaction networks with intermediate species.
• To establish a theoretical framework for lifting Gröbner bases from the core network (without intermediates) to the extended network (with intermediates).
• To determine conditions under which the steady state ideal remains binomial after including intermediates.
• To provide a practical method that avoids full computation on large networks by exploiting the linear structure of intermediate species.

Proposed method

• The method uses the fact that intermediate species generate a square, full-rank linear subsystem in the steady state equations.
• It computes a Gröbner basis for the core network (excluding intermediates) using a suitable monomial order where intermediate variables are prioritized.
• The basis is then lifted to the extended network via substitution of intermediate concentrations using the linear subsystem solution.
• The approach relies on linear algebra to express intermediate concentrations in terms of non-intermediate species and parameters.
• A monomial order is chosen such that intermediate variables are lexicographically ordered and greater than non-intermediate variables.
• For binomiality detection, the method combines the core network's Gröbner basis with an algebraic independence condition on the network structure.

Experimental results

Research questions

• RQ1 Can a Gröbner basis of the steady state ideal of a reaction network with intermediates be efficiently computed by first computing it on the core network?
• RQ2 Under what conditions does the binomiality of the steady state ideal of the core network imply binomiality of the extended network?
• RQ3 How can the structure of the reaction network be used to identify suitable monomial orders that accelerate Gröbner basis computation?
• RQ4 Is there a systematic way to lift a Gröbner basis from the core network to the extended network without recomputing from scratch?
• RQ5 Can the presence of enzymes be treated analogously to intermediates in terms of Gröbner basis lifting?

Key findings

• A Gröbner basis of the steady state ideal of the core network can always be lifted to a Gröbner basis of the extended network using linear algebra, provided a suitable monomial order is chosen.
• The lifting process reduces computation time significantly due to fewer variables and polynomials in the core network.
• If the steady state ideal of the core network is binomial and an additional algebraic independence condition is satisfied, then the extended network's steady state ideal is also binomial.
• The algebraic independence condition can be visually checked from the network structure in standard reaction networks.
• The method avoids full Gröbner basis computation on the extended network, making it computationally advantageous.
• The approach does not extend to enzymes in the same way as intermediates, as demonstrated by counterexamples showing binomiality is not preserved under enzyme addition.