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A signal anchor combines targeting or initiation of membrane passage with a retained membrane-spanning segment. Its orientation determines which flanking part enters the ER lumen; a single internal signal can yield either orientation.^[ref]

In the figure's stipulated internal-anchor example, the N-terminus remains in the cytosol and the C-terminus enters the ER lumen. Reversing that orientation would reverse which end is protected from an externally added protease. Merely counting one membrane-spanning segment cannot choose between the two.

A signal peptide is a short targeting segment of a newly made polypeptide chain. A familiar ER signal sequence lies near the N-terminus and is removed during entry; Günter Blobel and Bernhard Dobberstein's 1975 experiments linked a larger immunoglobulin light chain precursor to a processed product.^[ref]

Cleavage is a separate question from recognition. Not every targeting signal is a cleaved signal peptide: an internal signal anchor can remain part of a transmembrane protein. Conversely, loss of a segment does not by itself prove membrane crossing. Compare the independent size and accessibility readouts in the microsomal protein import experiment.

A serine protease uses a serine side chain as a catalytic nucleophile in peptide-bond hydrolysis. In the chymotrypsin family, a catalytic arrangement involving serine, histidine and aspartate promotes formation and hydrolysis of a covalent acyl-enzyme intermediate. The catalytic mechanism and substrate-recognition pocket are distinct: related serine proteases can prefer different cleavage sites. ^[ref]

Hoare logic proves properties of programs through Hoare triples $\{P\}C\{Q\}$. A precondition $P$ constrains initial states and a postcondition $Q$ constrains returned states. The basic rules establish partial correctness; termination requires a further argument.^[ref]

For a program expression $E$ whose evaluation is defined and has no side effects, the assignment rule isThe substitution replaces the free occurrences of $x$ in the desired postcondition. For example, obtaining $x\ge 1$ after x := x+1 requires $x+1\ge 1$ beforehand, so $x\ge 0$ suffices. Integer overflow must be excluded or handled by the stated arithmetic model.

Sequential composition joins a first postcondition to the next precondition. For a while loop, proving $\{I\wedge b\}C\{I\}$ establishesHere $I$ is the loop invariant. For while x > 0 do x := x-1, starting with an exact nonnegative integer, $x\ge 0$ is preserved and the false exit guard yields $x=0$. The decreasing nonnegative integer $x$ is also a loop variant, establishing total correctness for this example.^[ref]

An integer arithmetic result outside the range of its chosen representation. A programming language may specify wrapping, trapping or another behavior; a proof over mathematical integers must account for that difference.

A description of the information needed to determine a program's next permitted step. In the summation loop verification, this includes the current variable values and the position in the instructions.

Section of Precalculus 2e by Jay Abramson. The citation panel credits Jay Abramson as whole-book author; the preface gives no individual chapter allocation. OpenStax publishes and hosts the book. The parent record establishes whole-book publication on 21 December 2021; no separate section revision date is asserted.

Online text: openstax.org/books/precalculus-2e/pages/11-2-arithmetic-sequences

Cited passages: Arithmetic Sequence definition; Recursive Formula for an Arithmetic Sequence; Explicit Formula for an Arithmetic Sequence.

Online text: openstax.org/books/precalculus-2e/pages/10-4-rotation-of-axes

Section of Precalculus 2e by Jay Abramson. The citation panel names Jay Abramson as whole-book author and gives publication on 21 December 2021; the preface supplies no individual chapter allocation. No separately credited section author or revision date is established.

Cited passages: Identifying Conics without Rotating Axes; Using the Discriminant to Identify a Conic.

Online text: openstax.org/books/precalculus-2e/pages/10-1-the-ellipse

Section of Precalculus 2e by Jay Abramson. The citation panel names Jay Abramson as whole-book author and gives publication on 21 December 2021; the preface supplies no individual chapter allocation. No separately credited section author or revision date is established.

Cited passages: Writing Equations of Ellipses in Standard Form, Figure 2; Ellipse definition and Deriving the Equation of an Ellipse Centered at the Origin, Figure 5; Citation panel authors/title/publication date.

The radius of gyration of a dilute linear polystyrene coil in a good solvent depends on its molar mass as $R_g=K{M}^{\nu }$, with a material- and solvent-dependent amplitude. A manufacturer-supplied light-scattering calibration for polystyrene in benzene or toluene near room temperature is the displayed relation, quoted over $M\simeq 2\times 10^5$ to $5\times 10^7\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$. The exponent is close to the Flory approximation $3/5$. Values outside the quoted mass range are extrapolations, not extra measured data. Using $c^{\ast }=M/[N_A(4\pi /3)R_g^3]$ then gives the polymer overlap concentration scaling $c^{\ast }\propto M^{1-3\nu }$. ^[ref]

A nascent chain is a polypeptide chain still being synthesized by a ribosome. Its exposed targeting information can direct entry into the endoplasmic reticulum before biological translation finishes.

A coil-stretch transition is a sharp change from a coiled to a substantially extended polymer conformation under sufficiently strong extensional flow. The driving extensional strain rate must compete with the chain's relaxation rate, and the available residence time matters. A long polymer relaxation time lets a weak but sustained flow produce a large effect. The Hookean dumbbell in planar extension below isolates this competition; its unbounded growth signals a model limitation, rather than a prediction of infinitely long molecules.

A characteristic time for a chosen polymer deformation to relax after forcing stops. Its value depends on the measured quantity and the chain model; one polymer can have multiple relaxation modes.

Polymer adsorption binds a polymer near an attractive interface through contacts of some of its monomers. The binding energy competes with the loss of conformational entropy. A chain need not lie flat: noncontacting loops and tails can extend into the solvent. In a weak-adsorption blob model, the layer thickness exceeds the Kuhn length and local chain statistics remain approximately those of the free chain below that thickness.

A tensile blob is the scale below which thermal conformations of a stretched polymer are weakly disturbed by tension $f$. Balancing stretching work $f{\xi }_t$ with $k_{B}T$ gives ${\xi }_t\sim k_{B}T/f$. For free-chain size law $R\sim b{g}^{\nu }$, each blob contains $g_t\sim ({\xi }_t/b{)}^{1/\nu }$ monomers. A stretched string of blobs has extension $z\sim (n/g_t){\xi }_t$ and stretching free energy of order $k_{B}Tn/g_t$. This description requires blobs larger than individual statistical segments and smaller than the unperturbed coil.

Reptation is the snake-like motion of a long polymer through the constraints imposed by neighboring chains or a fixed porous matrix. A tube model restricts large transverse motion while allowing curvilinear motion along the chain contour. For a flexible chain with additive segment friction, contour diffusion scales as $N^{-1}$ and the time to renew a contour of length proportional to $N$ scales as $N^3$. With coil size squared proportional to $N$, this gives a centre-of-mass diffusion coefficient proportional to $N^{-2}$. Applying this tube picture to weak-field DNA gel electrophoresis gives an inverse-length mobility over its appropriate regime; field-induced orientation and trapping can invalidate that simple law.

The end-to-end vector joins the two ends of a polymer chain. In an isotropic unforced polymer solution, its vector expected value is zero; a useful measure of chain size is its root-mean-square length. For an ideal chain with $N$ independent isotropic steps of length $b$, the cross terms in the square of the step sum have zero expected value, so $R_{\mathrm{rms}}^2=N{b}^2$. For a long Gaussian chain, its radius of gyration obeys $R_g^2=R_{\mathrm{rms}}^2/6$. If $N$ counts chemical repeat units instead of independent Kuhn length segments, the statistical length used in $R_{\mathrm{rms}}^2=N{b}^2$ must be calibrated to that choice of $N$; it is not automatically the physical bond length or the Kuhn length itself.

A polymer melt is a dense fluid of polymer chains with little or no solvent. For a long flexible chain in a homogeneous melt, other chains screen its long-range excluded volume interactions: distant segments of the chosen chain displace statistically similar segments belonging to its neighbours. On scales above local stiffness, equilibrium chain dimensions are consequently approximately those of an ideal chain, even though monomers still cannot physically overlap. Entanglements can strongly affect dynamics without changing this leading large-scale equilibrium size law. Crystallinity, strong orientational order or unusual topological constraints require separate treatment.

A polymer entanglement is a constraint on a chain's motion caused by the inability of neighbouring chain contours to pass through one another. It is not a permanent chemical cross-link. In a dense polymer melt, these constraints can create long relaxation times and a transient rubbery plateau. Chains eventually release constraints through mechanisms such as reptation and motion of their neighbours. Entanglement therefore strongly affects dynamics even when the equilibrium size of a long chain remains approximately that of an ideal chain because excluded-volume interactions are screened.