Modeling Protein-Based Hydrogels under Force

Hydrogels made from structured polyprotein domains combine the properties of cross-linked polymers with the unfolding phase transition. The use of protein hydrogels as an ensemble approach to study the physics of domain unfolding is limited by the lack of scaling tools and by the complexity of the system. Here we propose a model to describe the biomechanical response of protein hydrogels based on the unfolding and extension of protein domains under force. Our model considers the contributions of the network dynamics of the molecules inside the gels, which have random cross-linking points and random topology. This model reproduces reported macroscopic viscoelastic effects and constitutes an important step toward using rheometry on protein hydrogels to scale down to the average mechanical response of protein molecules.

Figure 1

Mechanical response of polyproteins. (a) Projection of the free energy landscape on the pulling coordinate at various forces for a polyprotein $L$ with 8 repeats, showing an accordionlike shape. Dotted lines follow local energy minima. (b) Single molecule unfolding traces generated with the landscape from (a). Inset: Schematics of an eight-domain polyprotein under force.

Figure 2

Protein hydrogels under force. (a) Molecules are depicted as straight lines with 8 spheres along their axis, representing folded domains. A global constant force is applied along the vertical $z$ axis. (b) Illustration of the unfolding dynamics and orientation change during gel stretching. (c) Snapshots of the same gel at three different time points. Scale bars are $20\times 20\times 20\mathrm{nm}$.

Figure 3

Scaling behavior of protein hydrogel extension at constant force. Hydrogels composed of varying number of proteins (from $N=1$ to $N=72$) at a concentration of $c=9.0\times {10}^{-4}\text{molecules}/{\mathrm{nm}}^3$ are subjected to an applied force of 50 pN for 5 sec. Five normalized traces from separately polymerized hydrogels are simulated for each hydrogel size (gray traces) and averaged (black traces). The residuals, $R$, between the average trace and the individual traces are shown below each graph. $N=50$ is the minimum number of molecules needed to produce a deterministic behavior.

Figure 4

Force dependent behavior of protein-based hydrogels. (a) Average of 15 different hydrogels traces of $N=50$ molecules at a concentration of $c=9.0\times {10}^{-4}\text{molecules}/{\mathrm{nm}}^3$, subjected to a constant force of 30, 40, 50, and 60 pN. (b) Comparison between rate constants of single molecule unfolding (circles) and rate constants from gels, determined by two-term exponential fits from (a), plotted as a function of the applied force. By correlating simulated and measured rates of protein hydrogels, one can determine the underlying average single molecule kinetics.