An effective continuum approach for modeling non-equilibrium structural evolution of protein nanofiber networks

We quantify the formation and evolution of protein nanofibers using a new phase field modeling framework and compare the results to transmission electron microscopy measurements (TEM) and time-dependent growth measurements given in the literature. The modeling framework employs a set of effective continuum equations combined with underlying nanoscale forces and chemical potential relations governing protein nanofiber formation in solution. Calculations based on the theoretical framework are implemented numerically using a nonlinear finite element phase field modeling approach that couples homogenized protein molecular structure via a vector order parameter with chemical potential relations that describe interactions between the nanofibers and the surrounding solution. Homogenized, anisotropic molecular and chemical flux relations are found to be critical in obtaining nanofiber growth from seed particles or a random monomer bath. In addition, the model predicts both sigmoidal and first-order growth kinetics for protein nanofibers for unseeded and seeded models, respectively. These simulations include quantitative predictions on time scales of typical protein self-assembly behavior which qualitatively match TEM measurements of the RADA16-I protein and growth rate measurements for amyloid nanofibers from the literature. For comparisons with experiments, the numerical model performs multiple nanofiber protein evolution simulations with a characteristic length scale of ～2.4 nm and characteristic time scale of ～9.1 h. These results provide a new modeling tool that couples underlying monomer structure with self-assembling nanofiber behavior that is compatible with various external loadings and chemical environments.