Influence of ligand-receptor interactions on force-extension behavior within the freely jointed chain model

We study the influence of receptor-ligand interactions on the force response of single polymer chains theoretically. The extension of the chain is modeled in terms of freely jointed chain or elastic freely jointed chain (EFJC) models. The situation involving noninteracting bonds is solved exactly, while effects of interactions are treated within a mean-field approximation. The form with shorter bonds governs the low force situation, while the form with longer bonds is relevant in the high force regime. We further discuss the accuracy of approximate relations, which were used to describe the response of the EFJC model.

Figure 1

Schematic representation of the models investigated. (a) Freely jointed chain (FJC) and (b) elastic freely jointed chain (EFJC). (c) Binding equilibrium between the empty receptor, the ligand, and the occupied receptor. (d) Polymer with empty and occupied receptors.

Figure 2

Force-extension profiles of the EFJC model with a Kuhn length $a=1.0\mathrm{nm}$ and elasticity constants (a) $k=0.20\mathrm{nN}$, (b) $k=2.0\mathrm{nN}$, and (c) $k=20\mathrm{nN}$. The limiting profiles of the FJC model and of the purely elastic term are shown as well.

Figure 3

Force-extension profile of EFJC model and approximate models 1 and 2 for a chain with Kuhn length $a=1.0\mathrm{nm}$ and elasticity constants (a) $k=0.20\mathrm{nN}$ and (b) $k=2.0\mathrm{nN}$. The relative deviation of models 1 and 2 from the EFJC is shown below as panels (c,d).

Figure 4

Interplay of ligand-receptor interactions and polymer extension for different chemical potentials. The ligand-receptor model assumes no interactions between the occupied sites. The two forms of EFJC model are characterized with the parameters $a_0=1.0\mathrm{nm}$ and $k_0=2.0\mathrm{nN}$ in the empty form, and $a_1=1.2\mathrm{nm}$ and $k_1=1.0\mathrm{nN}$ in the occupied form. (a) Fractional occupation versus the force (b) and force versus the relative extension.

Figure 5

Effect of interactions on ligand-receptor interactions and polymer extension. The ligand-receptor model assumes interactions between the occupied sites as described with a mean-field model, and the two forms obey the FJC model. The parameters are $a_0=1.0\mathrm{nm}$ in the empty form and $a_1=1.2\mathrm{nm}$ in the occupied form. The chemical potential is fixed at $\beta \mu =-5$. (a) Fractional occupation versus the force and (b) force versus the relative extension and different chemical potential. For the interaction parameter $\gamma =-4$, the mean-field model predicts a critical point (full circle). For $\gamma =-7$, the mean-field model predicts an instability loop (dashed line) inducing a first-order phase transition (open circles).The limiting dependencies for the occupied and empty chain are also indicated.