A stretch of adenosine-containing residues at an RNA end. Tail length and associated proteins can influence messenger RNA stability and biological translation.

Removal of an RNA cap. It can expose an end to degradation and can also change recovery in an assay that selects capped RNAs.

The segment of a messenger RNA downstream of its translated coding region. It can contain cis-regulatory elements affecting localization, stability and biological translation.

A localized Drosophila melanogaster transcript with an experimentally studied diffusion-and-entrapment mechanism.^[ref]

A transcript encoding a form of actin, with a localization element recognized by ZBP1.

The chloroplast transcript encoding D1 protein. Changes in its ribosome recruitment can help couple synthesis to demand for replacement D1 protein.

A polycistronic messenger RNA contains multiple protein-coding regions. Their translation can be regulated independently or coupled through local RNA structure and ribosome movement.

The messenger-RNA region upstream of the principal coding start, which can contain structures and upstream reading frames affecting initiation.

A maternally supplied Drosophila transcript localized at the anterior of the oocyte and early embryo, providing a spatial source of Bicoid protein.

A Drosophila transcript with distinct maternal and zygotic contributions. Nanos-dependent regulation of the maternal RNA involves response elements in its 3′ untranslated region and shapes the posterior Hunchback protein distribution.

A Drosophila transcript whose anterior translation is repressed by Bicoid, helping establish a posterior Caudal protein distribution despite broad maternal transcript availability.

A multidrug efflux pump exports multiple chemical substrates using a coupled source of free energy. Broad recognition can arise from overlapping binding sites and conformational flexibility, but binding alone does not establish productive transport.

AcrB is a bacterial proton-coupled multidrug transporter that functions as part of the AcrAB–TolC export assembly. Its asymmetric trimer and multiple drug-binding regions are structural models for studying broad specificity.^[ref]

An ATP-binding cassette transporter couples nucleotide-binding-domain activity to transmembrane transport. For an exporter, productive substrate movement must be measured separately from substrate binding and ATP hydrolysis.

A flippase catalyzes directional movement of selected lipids between membrane leaflets. Such activity can establish or maintain lipid asymmetry, subject to specificity and energy coupling.

A scramblase facilitates relatively nonselective bidirectional movement of lipids between membrane leaflets, tending to dissipate asymmetry when active. Its action differs from directional transport used to maintain asymmetry.

A nucleotide-sugar transporter supplies an activated donor across a membrane to a compartment containing glycosyltransferases. Whole-cell donor abundance does not establish donor availability at a lumen-facing enzyme.

Initial selective capture of a transport carrier before close membrane docking and fusion.

A conserved endosomal cargo-sorting complex participating in retrieval pathways. Associated sorting machinery varies between organisms and cargos.

For real $p>0$ and $0<s<p$, the displayed improper integral converges: its endpoint powers are $x^{s-1}$ near zero and $x^{s-p-1}$ near infinity. The change of variables $t=x^p$ givesHere the beta function, the beta-gamma identity, and the Gamma reflection formula evaluate the integral. The same formula holds for complex $s$ in the Mellin fundamental strip $0<\mathrm{Re}s<p$, by absolute convergence and analytic continuation within the strip. For integer $p$, a sector contour of angle $2\pi /p$ gives a direct residue theorem proof, with a consistent branch of $z^{s-1}$.