Elastic Bond Network Model for Protein Unfolding Mechanics

Recent advances in single molecule mechanics have made it possible to investigate the mechanical anisotropy of protein stability in great detail. A quantitative prediction of protein unfolding forces at experimental time scales has so far been difficult. Here, we present an elastically bonded network model to describe the mechanical unfolding forces of green fluorescent protein in eight different pulling directions. The combination of an elastic network and irreversible bond fracture kinetics offers a new concept to understand the determinants of mechanical protein stability.

Figure 1

(a) Applying force to a protein structure (here GFP) via two amino acids $i$ and $j$, using cysteine engineering combined with single molecule force spectroscopy. (b) GFP modeled as an elastic bond network. (Inset) Network connections are springs with identical spring constant that may irreversibly break.

Figure 2

Illustration of the force-action matrix ${\alpha }_{lm}{|}_{ij}$ when force is applied to the GFP network via nodes $(132,212)$, $(3,132)$, and $(6,221)$. The width of the network connections codes for the values of ${\alpha }_{lm}{|}_{ij}$, light (red) colors denote extension, dark (blue) colors denote compression. Cutoff radius $R_C=6.75\AA$.

Figure 3

Comparison of experimental fracture force distributions (open circles) for GFP with predicted fracture force distributions from the elastic bond network model (solid lines). Parameters used: cutoff radius $R_C=6.75\AA$, transition state position $\Delta x_1=0.28\mathrm{nm}$, zero-force dissociation rate $k_0=1\times {10}^{-3}{\mathrm{s}}^{-1}$. Calculated $g_N(F_{ij})$ have been normalized to match observed number of events. Experimental conditions were as in 14. Broken lines: two-state analysis as described in 14. Apparent transition state positions $\Delta x_{\mathrm{app}}$ are given in the figure.

Figure 4

(a) Scheme of a serial and a parallel combination of $N$ identical bonds. The amount of the applied force transduced to individual bonds (i.e., ${\alpha }_{lm}{|}_{ij}$) is $\alpha =1$ in the serial configuration, while $\alpha =1/N$ in the parallel configuration. (b) Fracture probability densities $g_N(F)$, calculated with Eq. (2) for different number $N$ of bonds loaded.