Anomalous diffusion dynamics of associating artificial proteins in hydrogels

Associative polymer gels have attracted a great deal of interest as responsive materials and biomaterials; while a great deal is known about their mechanical properties, knowledge about self-diffusion in these materials is still limited. Using coiled-coil proteins as a model associative polymer system where the number of stickers per polymer and molar mass of chains between stickers are exactly defined, we investigate self-diffusion in associative polymer hydrogels using forced Rayleigh scattering on the length scales ranging from 0.3 to 50 $\mu $m. Although the presence of associative groups reduces the rate of diffusion, ``superdiffusive'' scaling is observed for the first time up to a length scale of 10 $\mu $m. Fickian diffusion is recovered at larger length scales. The anomalous diffusion strongly depends on the temperature and the hydrogel concentration. We propose a simple two state model to capture the interplay between the diffusion of the proteins and the association of the coiled-coil segments. The model is able to capture both the anomalous regime and the Fickian regime, and provides estimation of the apparent diffusivities and the dissociation rates of the coiled-coil domains.