= Effective displayed-library coverage
\b[Effective displayed-library coverage] describes which molecular sequence variants actually occur as usable displayed molecules, rather than merely being allowed by a library design or present in its input material. In <mRNA display>, successful <translation (biology)>, fusion formation and recovery connect input sequences to usable <genotype-phenotype linkages>. Duplicates of one sequence do not replace missing sequences.
An original occupancy model has $S$ equally likely sequence variants, $N$ independent input molecules and usable-display probability $f$ for each input. A particular variant is usable with probability $f/S$ per input. Therefore
$$
\Pr(\text{a specified variant is absent})=(1-f/S)^N,\qquad
\mathbb E[D]=S\left[1-(1-f/S)^N\right],
$$
where $D$ is the number of distinct represented variants. The expected usable molecule count is $Nf$, which is generally larger than the expected distinct count.
For $S=1000$, $N=300{,}000$ and stipulated $f=0.01$, the expected usable count is 3000 but the expected distinct count is only about 950.2. An average 95% represented fraction does not imply 95% probability that every variant is represented. By the <union bound>,
$$
\Pr(\text{at least one variant absent})\leq S(1-f/S)^N
\leq S e^{-Nf/S}.
$$
A sufficient condition for that probability to be at most $\delta$ is $N\geq(S/f)\log(S/\delta)$. With $\delta=0.05$ in this example, this requires at least 990,349 input molecules under the stated model.
\Image[/display-library-mean-versus-complete-coverage.png]
{title=Expected represented fraction and probability of complete coverage answer different library-design questions}
The figure uses a <Poissonized displayed-library occupancy model>: independently Poisson-distributed usable counts per variant with mean $\lambda=Nf/S$. The expected represented fraction is $1-e^{-\lambda}$, while the probability of all $S$ variants being present is $(1-e^{-\lambda})^S$. That is an explicitly different count model, providing a convenient comparison rather than an exact fixed-input formula.
Real variants need not have equal abundance or display probability. In the original RNA-peptide fusion experiment, the selected and unselected encoding-sequence ratios were affected by preferential <translation (biology)> and <reverse transcription>.<RNA-peptide fusions for the in vitro selection of peptides and proteins by Richard W. Roberts and Jack W. Szostak>[Figure 5 legend: input mixtures and biased unselected ratios]{ref} Coverage calculations therefore need assumptions about sequence representation and process losses; nominal library size alone does not validate them.
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