Efficiently and Effectively Recognizing Toricity of Steady State Varieties

Abstract

We consider the problem of testing whether the points in a complex or real variety with non-zero coordinates form a multiplicative group or, more generally, a coset of a multiplicative group. For the coset case, we study the notion of shifted toric varieties which generalizes the notion of toric varieties. This requires a geometric view on the varieties rather than an algebraic view on the ideals. We present algorithms and computations on 129 models from the BioModels repository testing for group and coset structures over both the complex numbers and the real numbers. Our methods over the complex numbers are based on Gröbner basis techniques and binomiality tests. Over the real numbers we use first-order characterizations and employ real quantifier elimination. In combination with suitable prime decompositions and restrictions to subspaces it turns out that almost all models show coset structure. Beyond our practical computations, we give upper bounds on the asymptotic worst-case complexity of the corresponding problems by proposing single exponential algorithms that test complex or real varieties for toricity

A Graph Theoretical Approach for Testing Binomiality of Reversible Chemical Reaction Networks

“…For ten biomodels the graph theory (and the linear algebra) approach is at least 500 times faster than the computations via Gröbner basis and quantifier elimination. An interesting observation is the quite high number (sixty nine) of the biomodels that are not considered in [16] for the unclear numeric values of their rate constants, whereas our graph theoretical (and linear algebra) approaches on almost all of those cases terminated in less than a second.…”

“…The performance of the implementations is tested on the biomodels from the BioModels repository [5] and the results are compared to the experiments done on the same biomodels using Gröbner basis and quantifier elimination in [16]. Note that in order to be able to test unconditional binomiality of biomodels, we have assumed reversibility with free reverse rate constants, while in [16] binomiality with preassigned values of rate constants are tested. The result of the experiments can be found in the Git repository [22].…”

“…Comparing Algorithm 1 with the Gröbner ba-sis and quantifier elimination computations in [16] shows that for the great majority of the cases, the graph theoretical approach is much faster. More importantly, there are twenty biomodels that Gröbner basis and/or quantifier elimination methods in [16] run out of time (for a six hour limit of computations), while those cases were handled in less than three seconds via both graph theory and linear algebra approaches.…”

“…The latter has been implemented in Maple and Macaulay II in [19,18]. A geometric view towards toricity has been considered by Grigoriev et al in [16], introducing shifted toricity and presenting algorithms using quantifier elimination [8,15,30] and Gröbner bases [3,4] A first order logic test for toricity has been given in [26]. In [25] unconditional binomiality has been introduced and for reversible reactions a polynomial time algorithm, based on a linear algebra approach, has been presented.…”

“…We use a modification of species-reaction graphs and present an algorithm equivalent to the linear algebra approach presented in [25]. We have implemented the graph theoretical algorithm as well the linear algebra algorithm in Maple [20] and compared them with each other and with the algorithms based on Gröbner basis and quantifier elimination presented in [16] via experiments on biological models from the BioModels repository [5]. Our experiments showed that the graph theoretical and linear algebra approaches are not only drastically faster, but they can also handle many cases that were not possible to compute using Gröbner basis and quantifier elimination.…”