Competitive ligand binding kinetics to linear polymers

Different types of ligands compete in binding to polymers with different consequences for the physical and chemical properties of the resulting complex. Here, we derive a general kinetic model for the competitive binding kinetics of different types of ligands to a linear polymer, using the McGhee and von Hippel detailed binding-site counting procedure. The derived model allows the description of the competitive binding process in terms of the size of the ligand, binding, and release rates, and cooperativity parameters. We illustrate the implications of the general theory showing the equations for the competitive binding of two ligands. The size of the ligand, given by the number of monomers occluded, is shown to have a great impact on competitive binding. Ligands requiring a large available gap for binding are strongly inhibited by smaller ligands. Ligand size then has a leading role compared to binding affinity or cooperativity. For ligands that can bind in different modes (i.e., different number of monomers), this implies that they are more effective in covering or passivating the polymer in lower modes, if the different modes have similar binding energies.

Figure 1

(a) Notation for monomer states and conditional probabilities. (b) Comparison of two configurations; cooperativity increases the probability of the second one by an amount ${\omega }_{\alpha \beta }$ (cooperativity parameter for neighboring ligands of types $\alpha$ and $\beta$). (c) Comparison of effective equilibrium constant for the binding of an isolated-, singly-, and doubly contiguous ligand, respectively.

Figure 2

Polymer coverage of modes 1 and 2, and total coverage, as a function of time for noncooperative system. Starting with a naked polymer, ligands binding to $m_1=30$ and $m_2=60$ sites. (a) Binding rates $k_{b1}=k_{b2}=0.8{\mathrm{s}}^{-1}$, and release rates $k_{r1}=k_{r2}=0.06{\mathrm{s}}^{-1}$. (b) Binding rates $k_{b1}=k_{b2}=0.8{\mathrm{s}}^{-1}$, and release rates $k_{r1}=0.06{\mathrm{s}}^{-1}$ and $k_{r2}=0.0006{\mathrm{s}}^{-1}$. (c) Binding rates $k_{b1}=0.8{\mathrm{s}}^{-1}$ and $k_{b2}=80{\mathrm{s}}^{-1}$, and release rates $k_{r1}=k_{r2}=0.06{\mathrm{s}}^{-1}$.

Figure 3

Comparison of equilibrium coverage of modes 1 and 2, and total coverage, as a function of mode 2 size. Model parameters: $m_1=30$, $k_{r1}=k_{r2}=0.06{\mathrm{s}}^{-1}$; noncooperative: ${\omega }_{11}=1,$ $k_{b1}=k_{b2}=1{\mathrm{s}}^{-1}$ (black lines); noncooperative enhancement of mode 1: ${\omega }_{11}=1,$ $k_{b1}=2{\mathrm{s}}^{-1}$, $k_{b2}=1{\mathrm{s}}^{-1}$ (green lines); and cooperative mode 1: ${\omega }_{11}=2,$ $k_{b1}=k_{b2}=1{\mathrm{s}}^{-1}$ (red lines).