  Elongation is a fundament process in amyloid fiber growth, which is normally
characterized by a linear relationship between the fiber elongation rate and
the monomer concentration. However, in high concentration regions, a sub-linear
dependence was often observed, which could be explained by a universal
saturation mechanism. In this paper, we modeled the saturated elongation
process through a Michaelis-Menten like mechanism, which is constituted by two
sub-steps -- unspecific association and dissociation of a monomer with the
fibril end, and subsequent conformational change of the associated monomer to
fit itself to the fibrillar structure. Typical saturation concentrations were
found to be $7-70\mu M$ for A$\beta$40, $\alpha$-synuclein and etc.
Furthermore, by using a novel Hamiltonian formulation, analytical solutions
valid for both weak and strong saturated conditions were constructed and
applied to the fibrillation kinetics of $\alpha$-synuclein and silk fibroin.
