| Phase transition is a very common phenomenon in nature. Besides Martensite phase transitions in Shape Memory Alloys which we are familiar with, there are also phase transition phenomena in biological systems. This thesis will focus on one of them---phase transition in bacterial flagellar filaments. The motility of most bacteria was propelled by the bacterial flagellar filaments. The polymorphism of the filaments is represented as different handedness and pitch macroscopically. A filament is made up only one single protein called flagellin, flagellin are arranged one on top of another into a protofilament and eleven protofilaments are lined side by side forming a filament. Experimental work including the recent crystallization of flagellin, as well as theoretical studies, show how the microscopic packing properties and underlying bistability of flagellin may give rise to different static structures of the filament. In this relatively new research area, both the experiment and theoretical model for the polymorphism of the filament are not enough. The work of this thesis will give a complete picture of phase transition in bacterial flagellar filaments. This thesis starts with an introduction of this new phase transition phenomenon in biological systems; then a detail analysis of the existing experiment will be given, to obtain the transition rule of the filament; based on the experiment analysis results and the Landau phase transition theory, the local free energy in two dimension is constructed, then the stability criteria was given based on the free energy above. Next the non-local free energy which describes the interface property was derived based on Ginzburg-Landau theory, the non-local interface energy can give the profile of both curvature and twist interface between the left handed normal and right handed semicoiled forms. Finally a description of the numerical simulation by using the Finite Element Method is presented; two cases are simulated here, firstly the filament is under the quasi-static loading, then the kinetic evolution is also simulated based on Time-Dependent Ginzburg-Landau Theory. |
