An Assessment of Groebner Basis Methods for a Simplified Biomolecular Folding Model

Groebner Basis methods offer attractive alternatives to analytically compute solutions of systems of polynomial equations in a variety of contexts including biomolecular folding. However, the increase in size of the Groebner Basis as the number of polynomials increases may be challenging and the growth for specific classes of polynomials is generally unclear. Here, we addressed this growth in a simple model for the folding of a biomolecule based on a chain of N beads linked rigidly and which could interact with each other using spring-like forces. We expressed the conditions for equilibria as systems of polynomial equations. We used the Groebner Basis methods to obtain explicit solutions for N = 3, 4, and 5. We noted that the maximum size of the Groebner Basis as a function of the polynomial system size was like a power law for N ≤ 10. The rigorous solutions obtained compare favorably with numerical solutions. These results offer an assessment of the applicability of Groebner Basis methods for this class of problems.