  We present an analytically solvable model for self-assembly of a molecular
complex on a filament. The process is driven by a seed molecule that undergoes
facilitated diffusion, which is a search strategy that combines diffusion in
three-dimensions and one-dimension. Our study is motivated by single molecule
level observations revealing the dynamics of transcription factors that bind to
the DNA at early stages of transcription. We calculate the probability that a
complex made up of a given number of molecules is completely formed, as well as
the distribution of completion times, upon the binding of a seed molecule at a
target site on the filament (without explicitly modeling the three-dimensional
diffusion that precedes binding). We compare two different mechanisms of
assembly where molecules bind in sequential and random order. Our results
indicate that while the probability of completion is greater for random
binding, the completion time scales exponentially with the size of the complex;
in contrast, it scales as a power-law or slower for sequential binding,
asymptotically. Furthermore, we provide model predictions for the dissociation
and residence times of the seed molecule, which are observables accessible in
single molecule tracking experiments.
