Nucleated polymerisation processes are involved in many growth phenomena in nature, including the formation of cytoskeletal filaments and the assembly of sickle hemoglobin and amyloid fibrils. Closed form rate equations have, however, been challenging to derive for these growth phenomena in cases where secondary nucleation processes are active, a difficulty exemplified by the highly non-linear nature of the equation systems that describe monomer dependent secondary nucleation pathways. We explore here the use of fixed point analysis to provide self-consistent solutions to such growth problems. We present iterative solutions and discuss their convergence behaviour. We establish a range of closed form results for linear growth processes, including the scaling behaviours of the maximum growth rate and of the reaction end-point. We further show that a self-consistent approach applied to the master equation of filamentous growth allows the determination of the evolution of the shape of the length distribution including the mean, the standard deviation, and the mode. Our results highlight the power of fixed-point approaches in finding closed form self-consistent solutions to growth problems characterised by the highly non-linear master equations.
Biopolymers Crystallization