Zeros and factors of polynomials with positive coefficients and protein- ligand binding
Many physiological processes involving the interaction of a ligand with a macromolecule, such as oxygen with hemoglobin, can be described by a binding polynomial of degree equal to the number of binding sites in the molecule. The coefficients are positive numbers which are determined from the chemical reactions which fill from one to n of the binding sites. Certain characteristics of this binding process can be described in terms of structural subunits or linkages between binding sites which are represented by a factorization of the binding polynomial into polynomials with non-negative coefficients (positive polynomials). A p-irreducible polynomial is defined to be a positive polynomial which cannot be factored into a product of positive polynomials. The regions in the complex plane in which the zeros of p-irreducible polynomials of degrees 3 and 4 must be located are determined. A consequence of this is that a positive quartic with real zeros does not vanish in \(| \arg z| <\tan^{-1}(\sqrt{2})\). Factorization into p-irreducible polynomials is not necessarily unique for \(n>3\) and criteria for multiple factorizations are developed. The region is determined in which a pair of non-real zeros of a positive polynomial whose remaining zeros are real must lie and this result gives the zero-free sector for all such polynomials of degree \(k+2\). The proof requires deriving a set of inequalities involving the elementary symmetric functions \(\sigma_ i\) of k positive numbers. It is also conjectured that \(\sigma^ 3_{i+1}+\sigma^ 2_ i \sigma_{i+3}+\sigma_{i-1} \sigma^ 2_{i+2}>\sigma_{i+1}(2\sigma_ i\sigma_{i+2}+\sigma_{i- 1}\sigma_{i+3})\) and \(\sigma^ 2_ i(\sigma^ 2_ i-\sigma_{i- 1}\sigma_{i+2})>\sigma^ 2_{i-1}(\sigma^ 2_{i+1}-\sigma_ i\sigma_{i+2})\).
scientific article; zbMATH DE number 3918505