[Paper Review] Analysis of the Conradi-Kahle Algorithm for Detecting Binomiality on Biological Models

Alexandru Iosif, Hamid Rahkooy|arXiv (Cornell University)|Dec 14, 2019

Computational Drug Discovery Methods12 references4 citations

TL;DR

This paper analyzes the Conradi-Kahle Algorithm for detecting binomiality in polynomial ideals derived from biological models. It implements the algorithm in Macaulay2 and Maple, comparing its performance against Gröbner bases computations on 20 biomodels from the BioModels repository. The key finding is that while the Conradi-Kahle Algorithm is fast (under 1 second for most cases), it only correctly identifies binomiality in three models due to its incompleteness on inhomogeneous ideals, whereas Gröbner bases—though slower—can detect non-binomiality when they terminate within 600 seconds.

ABSTRACT

We analyze the Conradi-Kahle Algorithm for detecting binomiality. We present experiments using two implementations of the algorithm in Macaulay2 and Maple on biological models and assess the performance of the algorithm on these models. We compare the two implementations with each other and with Gröbner bases computations up to their performance on these biological models.

Motivation & Objective

• To analyze the efficiency and effectiveness of the Conradi-Kahle Algorithm for detecting binomiality in polynomial ideals from biological models.
• To compare the performance of the Conradi-Kahle Algorithm implemented in Macaulay2 and Maple against Gröbner bases computations on real biological models.
• To investigate the limitations of the Conradi-Kahle Algorithm when applied to inhomogeneous ideals, which are common in biological reaction networks.
• To assess whether the algorithm can reliably determine binomiality in steady-state ideals of biochemical networks, especially when Gröbner bases fail to terminate within a reasonable time.

Proposed method

• The Conradi-Kahle Algorithm is applied to 20 biomodels from the BioModels repository, which are inhomogeneous polynomial ideals over rational function fields in rate parameters.
• The algorithm tests binomiality by iteratively reducing the ideal using reduced row echelon forms of coefficient matrices and checking for rows with three or more non-zero entries.
• The method uses Gröbner basis computations as a benchmark, applying Corollary 1.2 from Eisenbud and Sturmfels to test binomiality via Gröbner basis reduction.
• Implementations are coded in both Macaulay2 and Maple, with execution times recorded and results compared across both systems.
• The algorithm is applied to homogenized versions of the original inhomogeneous ideals, acknowledging that completeness is only guaranteed for homogeneous ideals.
• Performance is evaluated by measuring CPU time and success rate in determining binomiality, with a 600-second timeout for Gröbner bases.

Experimental results

Research questions

• RQ1 How does the Conradi-Kahle Algorithm perform in terms of speed and accuracy when detecting binomiality in inhomogeneous polynomial ideals from biological models?
• RQ2 To what extent does the Conradi-Kahle Algorithm outperform or underperform Gröbner bases in detecting binomiality on real biological models?
• RQ3 Why does the Conradi-Kahle Algorithm fail to determine binomiality for most biomodels, despite its efficiency?
• RQ4 Can the Conradi-Kahle Algorithm detect non-binomial ideals, or is it only capable of confirming binomiality?
• RQ5 How do the computational performances of Macaulay2 and Maple implementations of the Conradi-Kahle Algorithm compare on biological models?

Key findings

• The Conradi-Kahle Algorithm runs in under one second for all biomodels tested, with execution times ranging from 0.01 to 0.7 seconds in Macaulay2 and 0.02 to 6 seconds in Maple.
• The algorithm successfully identifies binomiality only for three biomodels (9, 262, 264), returning 'Yes' in the Bin (C-K) column, while returning 'No' for all others.
• Gröbner bases computations terminate within 600 seconds only for 10 out of 20 biomodels, with the longest time being 500 seconds for biomodel 153 in Maple.
• For biomodels 46, 187, 243, and 335, Gröbner bases correctly identify non-binomiality, but the Conradi-Kahle Algorithm fails to do so due to incompleteness on inhomogeneous ideals.
• In cases where both methods succeed, execution times are comparable: for biomodel 264, C-K (M2) takes 0.7s and GB (M2) takes 2s, while C-K (Maple) takes 0.03s and GB (Maple) takes 0.04s.
• There is no significant performance difference between Macaulay2 and Maple implementations for either the Conradi-Kahle Algorithm or Gröbner bases computations.