Online text: fncbook.com/lu/

Online text: fncbook.com/integration/

Cited passages: Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, §5.6.1, equations 5.6.3–5.6.5: integrating the piecewise linear interpolant.

Recursion following the constructors of a finite inductive datatype. Correctness is naturally proved by structural induction.

// Target: computer-science.bigb

A procedure whose result depends only on its explicit argument values and which has no observable side effects. For terminating calls with reusable results, this makes the value condition of the memoization cache contract particularly simple.

// Target: computer-science.bigb

A specification of the meaning of a mathematical term or notation. A structural definition specifies conditions, such as the group axioms; a definition by a universal property characterizes an object by its arrows. Such a characterization does not by itself prove that the object exists. For example, existence of an exponential object is an additional property of a category.

In Receptor sensitivity in bacterial chemotaxis by Victor Sourjik and others, Victor Sourjik and Howard C. Berg used a cyan fluorescent tag on CheZ as FRET donor and a yellow tag on CheY as FRET acceptor. Escherichia coli biological cells adhered to a polylysine-coated coverslip in a continuously perfused chamber. A fluorescence microscope excited them with a $442$-nanometre laser; split emission bands fed two photon-counting detectors. The yellow/cyan emission ratio fell when attractant was added and rose when it was removed.^[ref]

Saturating attractant increased cyan emission by an amount matching selective bleaching of the yellow tag: removing the FRET acceptor supported relief of FRET donor transfer. Strains carrying the tagged pair but lacking other chemotaxis machinery showed no stimulus-dependent emission ratio change. Weak illumination reduced bleaching drift.^[ref]

The inference is a change in the population of nearby tagged CheY/CheZ pairs associated with CheY phosphorylation. It neither measures each pair's separation nor directly observes motor motion. The readout also averages many biological cells, and perfusion takes finite time. The pathway interpretation belongs to the chemotaxis FRET assay; the general measurement lesson is that molecular association can alter FRET while the geometry of an individual associated pair stays the same.

In FRET photobleaching controls, selectively destroy the FRET acceptor's ability to absorb and test whether FRET donor fluorescence increases. Under unchanged excitation and detection, complete loss of transfer, and an intact FRET donor,This is a destructive comparison. Monitor acceptor loss, donor-only samples exposed to the bleaching illumination, and an unbleached reference region or time trace. Incomplete bleaching, donor damage, movement or label replenishment violate the comparison. During ordinary recordings, gradual unequal bleaching can instead produce an apparent change of FRET without a biological change.^[ref]

FRET sensitized emission is FRET acceptor fluorescence caused by transfer from an excited FRET donor. Excite predominantly the FRET donor and collect both emission bands. Reduced FRET donor emission and increased FRET acceptor emission are useful together, but an acceptor-to-donor emission ratio is not automatically the FRET efficiency: brightness, label abundance and channel collection also enter.^[ref]

FRET efficiency is the probability that an excitation of the FRET donor ends in transfer to the FRET acceptor. Write $k_D=k_{\mathrm{r}}+k_{\mathrm{nr}}$ for the total decay rate without a FRET acceptor, including radiative fluorescence and other decay, and $k_T$ for the transfer rate. For constant competing rates,The integral counts transfer events among all excitations in a population with exponential decay. Substitution gives the separation law above. This short rate argument assumes one FRET donor, one FRET acceptor, weak coupling, an available FRET acceptor, and fixed geometry during the decay; it is not a model for arbitrary interacting labels.^[ref]