  The twin-arginine translocation (Tat) system transports folded proteins of
various sizes across both bacterial and plant thylakoid membranes. The
membrane-associated TatA protein is an essential component of the Tat
translocon, and a broad distribution of different sized TatA-clusters is
observed in bacterial membranes. We assume that the size dynamics of TatA
clusters are affected by substrate binding, unbinding, and translocation to
associated TatBC clusters, where clusters with bound translocation substrates
favour growth and those without associated substrates favour shrinkage. With a
stochastic model of substrate binding and cluster dynamics, we numerically
determine the TatA cluster size distribution. We include a proportion of
targeted but non-translocatable (NT) substrates, with the simplifying
hypothesis that the substrate translocatability does not directly affect
cluster dynamical rate constants or substrate binding or unbinding rates. This
amounts to a translocation model without specific quality control.
Nevertheless, NT substrates will remain associated with TatA clusters until
unbound and so will affect cluster sizes and translocation rates. We find that
the number of larger TatA clusters depends on the NT fraction $f$. The
translocation rate can be optimized by tuning the rate of spontaneous substrate
unbinding, $\Gamma_U$. We present an analytically solvable three-state model of
substrate translocation without cluster size dynamics that follows our computed
translocation rates, and that is consistent with {\em in vitro}
Tat-translocation data in the presence of NT substrates.
