= Processivity
\b[Processivity] describes how many successive catalytic steps an <enzyme> performs on one <substrate (biochemistry)> during a binding encounter before separating from it. A <DNA polymerase> can add successive <nucleotides>; an <exoribonuclease> can remove them. The average run length and its distribution describe a different property from the number of steps performed per second.
An original probability model makes that distinction precise. After one completed step, suppose the enzyme continues to another step with a constant probability $p$, independently at each decision, and otherwise releases the substrate. If $N$ counts all completed steps including that first one, then
$$
\Pr(N=n)=(1-p)p^{n-1},\qquad n=1,2,\ldots,
\qquad \mathbb{E}[N]=\frac{1}{1-p},\quad 0\leq p<1.
$$
This is a <geometric distribution>. With $p=0.9$, the mean is 10 steps. Making each step take twice as long leaves that mean unchanged in the stipulated model. It changes speed, while changing the continuation probability changes <processivity>.
A structural experiment on <Xrn1> connected substrate recognition to repeated cleavage. Mutations near its terminal-nucleotide recognition site changed the degradation pattern of labeled <RNA>, including accumulation of intermediates. The study combined structural information with denaturing-gel readouts rather than inferring enzyme behavior from the <protein> structure alone.<Coupled 5′ nucleotide recognition and processivity in Xrn1-mediated mRNA decay by Martin Jinek and others>[Figure3 and associated Results, His41 and phosphate-pocket mutations]{ref}
The probability model assumes a constant continuation probability and an indefinitely long substrate. A finite chain, a structured region or changing contacts can truncate or alter the run-length distribution. Likewise, a long final product can result from several binding encounters unless the assay distinguishes continued action from release followed by rebinding.
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