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

Hamiltonian Dynamics of Saturated Elongation in Amyloid Fiber Formation