Analysis of the Conradi-Kahle Algorithm for Detecting Binomiality on Biological Models
aa r X i v : . [ q - b i o . M N ] D ec Analysis of the Conradi-Kahle Algorithm forDetecting Binomiality on Biological Models ⋆ Alexandru Iosif and Hamid Rahkooy JRC-COMBINE Aachen, Germany CNRS, Inria and the Universit´e de Lorraine, Nancy, France [email protected]@inria.fr
Abstract.
We analyze the Conradi-Kahle Algorithm for detecting bi-nomiality. We present experiments using two implementations of the al-gorithm in Macaulay2 and Maple on biological models and assess theperformance of the algorithm on these models. We compare the two im-plementations with each other and with Gr¨obner bases computations upto their performance on these biological models.
Keywords: binomial ideals, Gr¨obner bases, biological models.
We study the problem of binomiality of polynomial ideals. Given an ideal with afinite set of generators, we would like to know if there exists a basis for the idealsuch that its elements have at most two monomials. Such an ideal is called a binomial ideal . We use the Conradi-Kahle Algorithm for testing whether an idealis binomial. Our investigations are focused on implementing this algorithm andperforming computations. Binomial ideals offer clear computational advantagesover arbitrary ideals. They appear in various applications, e.g., in biological andchemical models.Binomial ideals have been extensively studied in the literature [6,12,13].Eisenbud and Sturmfels in [6] have shown that Gr¨obner bases [1] can be usedto test binomiality. Recently, biochemical networks whose steady state ideals arebinomial have been studied in the field of
Algebraic Systems Biology [5,7,18].Mill´an and Dickenstein in [17] have defined
MESSI Biological Systems as a gen-eral framework for modifications of type enzyme-substrate or swap with inter-mediates, which includes interesting binomial systems [17].In the context of biochemical reaction networks, Mill´an, Dickenstein, Shiuand Conradi in [18] present a sufficient condition on the stoichiometric matrix for binomiality of the steady state ideal. Conradi and Kahle [4] proved that thiscondition is necessary for homogeneous ideals and proposed an algorithm. The ⋆ This work has been supported by the bilateral project ANR-17-CE40-0036/DFG-391322026 SYMBIONT. We would like to thank the authors of [14] for providing uswith the polynomials that we used in our computations. Alexandru Iosif and Hamid Rahkooy
Conradi-Kahle Algorithm is implemented in Macaulay2 [11]. Iosif, Conradi andKahle in [3] use the fact that the irreducible components of the varieties of bi-nomial ideals admit monomial parametrization in order to reduce the dimensionof detecting total concentrations that lead to multiple steady states.Our contribution in this article is analysing efficiency and effectiveness of theConradi-Kahle Algorithm, using Gr¨obner bases for reduction, applied to somebiological models. We first discuss the complexity of the algorithm and reduceit to the complexity of computing a Gr¨obner basis for a preprocessed input setof polynomials. Then we present our computations in Macaulay2 [9] and Maple[15] and compare the algorithm with simply computing Gr¨obner basis of theinput ideal which shows the strength of the algorithm. The experiments are per-formed on biological models in the BioModels repository , which is a repositoryof mechanistic models of bio-medical systems [2,8]. Our intial motivation was tounderstand the advantages and disadvantages of the method in [18] for testingbinomiality of chemical reaction networks. As the Conradi-Kahle Algorithm fol-lows the idea of the method in [18] with more subtle reduction steps, we ratheruse the Conradi-Kahle Algorithm to check binomiality of ideals coming frombiomodels, although none of our steady state ideals are homogeneous. The Conradi-Kahle Algorithm is based on the sufficient condition by Mill´an,Dickenstein, Shiu and Conradi [18] for binomiality of steady state ideals. Thelatter states that if the kernel of the stoichiometric matrix has a basis with aparticular property then the steady state ideal is binomial. Conradi and Kahleconverted this into a sufficient condition for an arbitrary homogenous ideal I generated by a set F of polynomials of fixed degree. They proved that I isbinomial if and only if the reduced row echelon form of the coefficient matrix of F has at most two non-zero elements in each row. This leads to the Algorithm1 which is incremental on the degrees of the generators.Now we analyze the complexity of Algorithm 1. – Steps and . can be ignored. – Step . Let t denote the number of distinct monomials in F min and m :=max( s, t ). Computing the reduced row echelon form of A can be done inat most m ω steps, where ω is the constant in the complexity of matrixmultiplication. – Step . needs at most st operations which is less or equal than m ω , so weignore this term. – Steps . can be bounded by tm , which itself can be bounded by m ω , henceignored. – Step . This can be done via computing a Gr¨obner basis of h B i . Anotherway to do this, is by means of Gaussian elimination on the correspondingMacaulay matrix of B . nalysing the Conradi-Kahle Algorithm 3 Algorithm 1 (Conradi and Kahle, 2015)
Input:
Homogeneous polynomials f , . . . , f s ∈ K [ X ], where K is a field. Output:
Yes if the ideal h f , . . . , f s i is binomial. No otherwise.1: Let B := ∅ , R := K [ x , . . . , x n ] and F := { f , . . . , f s } .2: while F = ∅ do
3: Let F min be the set of elements of minimal degree in F .4: F := F \ F min .5: Compute the reduced row echelon form A of the coefficient matrix of F min .6: if A has a row with three or more non-zero entries then return No and stop8: end if
9: Let M be the vector of monomials in F min .10: Let B ′ be the set of entries of AM .11: B := B ∪ B ′ .12: R := K [ x , . . . , x n ] / h B i .13: Redefine F as the image of F in R .14: end while return Yes . – Step . is equivalent to reducing F modulo h B i , which can be done viareducing F modulo a Gr¨obner basis of h B i . Another method to do this isvia Gaussian elimination over the Macaulay matrix of F ∪ B .Following Mayr and Meyer’s work on the complexity of computing Gr¨obnerbases [16], computations in steps 11 and 12 of the algorithm can be EXP-SPACE.Conradi and Kahle observe through experiments that these steps can be per-formed via graph enumeration algorithms like breadth first search, which makesit more efficient than Gr¨obner bases in practice [4]. In this article we do not usesuch graph enumeration algorithms in our implementations. This is the subjectof a future work. We consider 20 Biomodels from the BioModels repository [2,8] whose steadystate ideal is generated by polynomials in Q ( k , . . . , k r )[ x , . . . , x n ] where k , . . . , k r are the parameters and x , . . . , x n are the variables corresponding to thespecies. Our polynomials are taken from [14]. We use Algorithm 1 to test binomi-ality of these biomodels. We emphasise that in our computations we do not assignvalues to the parameters k , . . . , k r and we work in Q ( k , . . . , k r )[ x , . . . , x n ]. Wehave implemented Algorithm 1 in Maple [10] and also use a slight variant of theimplementation of the algorithm in the Macaulay2 package Binomials [11,12].We also test binomiality of an ideal given by a set of generating polynomials viacomputing a Gr¨obner basis of the ideal, using Corollary 1.2 in [6]. Our computa-tions are done on a 3.5 GHz Intel Core i7 with 16 GB RAM. In our computationswe used Macaulay2 1.12 and Maple 2019.1. Alexandru Iosif and Hamid RahkooyBiomodel C-K (M2) C-K (Maple) Bin (C-K) GB (M2) GB (Maple) Bin (GB)2 0.1 1 No9 0.04 0.2 Yes 0.5 0.001 Yes28 0.04 0.1 No30 0.5 0.2 No46 0.02 0.2 No 100 80 No85 0.04 0.6 No86 0.08 6 No102 0.04 0.2 No103 0.1 0.9 No108 0.01 0.03 No152 0.3 400 No153 0.4 500 No187 0.02 0.07 No 0.06 0.1 No200 0.05 1 No205 0.6 50 No243 0.04 0.3 No 0.01 0.05 No262 0.05 0.02 Yes 0.01 0.02 Yes264 0.7 0.03 Yes 2 0.04 Yes315 0.02 0.2 No335 0.04 0.8 No 30 90 No
Table 1: CPU times (in seconds) for Algorithm 1 and Gr¨obner bases.Table 1 shows the results of our computations. Biomodel columns in the tableshows the number of the biomodel. The columns C-K (M2) and C-K (Maple)show the CPU timings in seconds of executing Algorithm 1 in Macaulay2 andMaple, respectively. In the column Bin (C-K), Yes means that the algorithmsuccessfully determined that the ideal is binomial, while No means that thealgorithm cannot determine whether the ideal is binomial or not. The columnsGB (M2) and GB (Maple) are the timings of Gr¨obner bases computations ofthe input polynomials in Macaulay2 and Maple, respectively. The Macaulay2and Maple timings are rounded to the first nonzero digit. Bin (GB) column isblank if the Gr¨obner basis computation did not finish after 600 seconds. Yesin the latter column means that Gr¨obner basis computation finished and showsthat the ideal is binomial, while No shows that the Gr¨obner basis computationfinished but it detected that the ideal is not binomial.None of the ideals in the biomodels that we have studied are homogeneous.Therefore, in order to use Algorithm 1 we need to homogenise the ideals. Conse-quently, if the algorithm returns No , we are not able to say whether the ideal isbinomial or not (see [4, Section 4]). As one can see from the column Bin (C-K),the Conradi-Kahle Algorithm is able to test binomiality only for Biomodels 9,262 and 264. If Gr¨obner bases computations finish, then they can test binomial-ity for every ideal. However, as one can see from the related columns, this is notthe case. Actually in most of the cases, Gr¨obner bases computations did not fin-ish within 600 seconds. One can see from the table that whenever Gr¨obner bases nalysing the Conradi-Kahle Algorithm 5 computations give a yes answer to the binomiality question, then the Conradi-Kahle Algorithm also can detect this as well. In the Yes cases, the timings forboth methods in both Macaulay2 and Maple are very close.Algorithm 1 returns the output within at most a few seconds, however, mostof the Gr¨obner bases computations did not finish in 600 seconds. The advan-tage of testing binomiality using Gr¨obner bases computations can be seen inBiomodels 46, 187, 243 and 335, where Gr¨obner bases computations—althoughslower—show that the ideal is not binomial, but the Conradi-Kahle Algorithmcannot detect this in spite of its fast execution. With a few exceptions, we do notobserve significant difference between Macaulay2 and Maple computations, nei-ther for the Conradi-Kahle Algorithm nor for the Gr¨obner bases computations.We would like to emphasise that the Conradi-Kahle Algorithm is complete overhomogeneous ideals. However, in this article we are interested in ideals comingfrom some biological models which are inhomogeneous, and this might affect theperformance of the algorithm. In future we will do experiments on homogeneousideals in order to better understand the performance of the algorithm in thatcase. Acknowledgement.
We would like to thank the anonymous referees for theircomments.
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