Prions diseases are characterized by a chain reaction in which infectious misfolded proteins force native proteins into a similar pathogenic structure. Recent studies reinforce the hypothesis that the prion paradigm, the templated growth and spreading of misfolded proteins, can explain the progression of a variety of neurodegenerative disorders. However, our current understanding of this prion-like growth and spreading is rather empirical. Here we show that physics-based reaction-diffusion models correctly predict the growth and spreading of amyloid-beta deposits and tau inclusions in Alzheimer's disease, alpha-synuclein inclusions in Parkinson's disease, and TDP-43 inclusions in amyotrophic lateral sclerosis. We demonstrate that, rather than using complete whole brain models, we can represent the brain through a connectivity-weighted Laplacian graph, which we create from 418 brains of the Human Connectome Project. Our brain network model correctly predicts the neuropathological pattern of Alzheimer's disease and captures the key characteristic features of our whole brain models at a fraction of their computational cost. Our results suggest that misfolded proteins in various neurodegenerative disorders grow and spread according to a universal law that follows the basic physical principles of nonlinear reaction and anisotropic diffusion. A better understanding of the spreading of misfolded proteins could open new therapeutic opportunities towards blocking protein misfolding and promoting protein clearance using antibodies or small molecules.