Decoupled molecules with binding polynomials of bidegree (n, 2)

Abstract

We present a result on the number of decoupled molecules for systems binding two different types of ligands. In the case of n and 2 binding sites respectively, we show that there are [Formula: see text] decoupled molecules to a generic binding polynomial. For molecules with more binding sites for the second ligand, we provide computational results.

Keywords: Binding polynomials; Ligand binding; Numerical algebraic geometry.

Fig. 1

A molecule with 4 binding sites (e.g. hemoglobin)

Fig. 2

A molecule with (4,4) sites

Fig. 3

A decoupled molecule with (n,2) sites

Fig. 4

Coefficients of the binding polynomial of bidegree (n,2)

Fig. 5

A converging and a diverging solution path

Fig. 6

Perturbation of normalized binding polynomials

Fig. 7

Two complex conjugate solutions for (3, 2)

Fig. 8

Mixed volumes of the Newton polytopes of (1), numbers in red resp. blue were verified symbolically using Gröbner bases resp. numerically using homotopy continuation (color figure online)