We develop an algorithm that efficiently generates large, band-diagonal transition matrices for a totally asymmetric exclusion process
(TASEP) with local hopping rate inhomogeneities. Upon combining
mean-field and numerical approaches, the matrices
are diagonalized numerically to find steady-state currents of
TASEPs with local variations in hopping rate.
The results are then used to investigate
clustering of slow codons along mRNA. Ribosome density
profiles near neighboring clusters of slow codons interact, enhancing suppression of ribosome throughput when such bottlenecks are closely spaced. Increasing the slow codon cluster size, beyond $\sim 3-4$, does not significantly reduce ribosome current. Our results are verified by extensive Monte-Carlo simulations and provide a biologically-motivated explanation for the experimentally-observed clustering of low-usage codons.
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