We put forward a modified Zipper model inspired by the statics and dynamics of the spontaneous reconstitution of rodlike tobacco mosaic virus particles in solutions containing the coat protein and the single-stranded RNA of the virus. An important ingredient of our model is an allosteric switch associated with the binding of the first protein unit to the origin-of-assembly domain of the viral RNA. The subsequent addition and conformational switching of coat proteins to the growing capsid we believe is catalyzed by the presence of the helical arrangement of bound proteins to the RNA. The model explains why the formation of complete viruses is favored over incomplete ones, even though the process is quasi-one-dimensional in character. We numerically solve the relevant kinetic equations and show that time evolution is different for the assembly and disassembly of the virus, the former exhibiting a time lag even if all forward rate constants are equal. We find the late-stage assembly kinetics in the presence of excess protein to be governed by a single-exponential relaxation, which agrees with available experimental data on TMV reconstruction.