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Represent a dilute polymer by two beads connected by a spring obeying Hooke's law, with stiffness $H>0$ and bead drag coefficient $\zeta >0$. Neglect bead inertia and hydrodynamic interactions. For component-first velocity gradient $K_{ij}={\partial }_j{u}_i$, the configuration Fokker-Planck probability current isThe factor two comes from relative motion of both beads. At rest its normalized probability density is ${\psi }_{\mathrm{eq}}=(H/(2\pi k_{B}T){)}^{3/2}{e}^{-H{R}^2/(2k_{B}T)}$. The conformation tensor $C=⟨\mathbf{R}{\mathbf{R}}^T⟩$ has dimensions of length squared; $(H/(k_{B}T))C$ is its dimensionless version. The Hookean dumbbell in planar extension is a particular imposed flow in this model.

The drag of a translating polymer depends on conformation and hydrodynamic interactions. A nondraining coil has $\zeta \sim 6\pi \eta R_h$; a long aligned rod has ${\zeta }_{\parallel }\sim 2\pi \eta L/\log (L/a)$ and roughly twice that transverse drag; a freely draining model sums the monomer drag coefficients. Mechanical power is $\zeta U^2$. These models cannot be interchanged without changing the assumed conformation and solvent-mediated coupling.

Lipids are a chemically diverse family of chemical substances, conventionally associated with biological origin and solubility in nonpolar solvents. They include triglycerides, phospholipids, and cholesterol. The name does not specify one molecular formula, one repeating monomer, or one kind of polymer. Some lipids have both hydrophilic and hydrophobic regions rather than being uniformly nonpolar. See IUPAC Gold Book: lipids.

Structure explains the different jobs. A triglyceride has three fatty acid-derived acyl groups attached to glycerol and serves as an energy store. A typical glycerophospholipid instead combines two such chains with a phosphate-containing head, making it an amphiphile suited to a lipid bilayer. Cholesterol has a different structure and also contributes to cell membrane properties. Thus a lipid is a chemical building material, while a lipid bilayer is an organization of many molecules. See Clark, Douglas and Choi, Biology 2e, 3.3: lipid structures and functions.

Connecting segments into chains reduces their independent translational mixing entropy per segment by factors $1/N_A$ and $1/N_B$. Consequently the tolerable positive contact penalty tends to zero as both chains become long. Equal chain lengths have ${\chi }_c=2/N$; favorable unlike interactions can still produce a miscible polymer pair.

For component-first velocity gradient $L_{ij}={\partial }_j{u}_i$, ideal incompressible magnetic field stretching gives the displayed evolution of $M_{ij}=B_i{B}_j$. It has the same upper-convected derivative as a polymer conformation tensor with negligible relaxation. Finite polymer relaxation time introduces a local return toward thermal equilibrium, whereas magnetic diffusion acts through spatial gradients and preserves the solenoidal magnetic-field constraint. The analogy does not identify those dissipative mechanisms.

Molar mass turns a weighed sample into a count of specified particles. For a pure chemical substance with specified chemical formula and isotopic abundances, its molar mass iswhere $m$ is the sample mass and $n$ its amount of substance. The SI unit is $\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}$; the common laboratory unit is $\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$, with $1\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}=10^{-3}\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}$. Equal values of amount of substance correspond to equal counts of specified entities, but heavier entities give greater masses.

Combining this definition with molecular count and amount of substance givesThe Avogadro constant is exact under the modern definition of the mole. The mass $M/N_{\mathrm{A}}$ is the mean mass per specified entity; if isotopes differ within the sample, individual molecules need not all have that mean mass.^[ref]

Specify what $n$ counts: water molecules, hydrogen chemical atoms, and oxygen chemical atoms give different values of amount of substance for the same sample. For a mixture, dividing total mass by total amount of substance of its chosen entities gives a composition-dependent mean. A polymer sample can contain many chain molar masses; its number-average molar mass and weight-average molar mass answer different averaging questions. Relating either to degree of polymerization also requires a unit-mass convention and end-group accounting.

The distribution of molar masses among the chains in a polymer sample. Number-weighted and mass-weighted descriptions assign different relative weights to long chains: for populations $N_i$ of chains with molar mass $M_i$, the number-average molar mass is $M_n={\sum }_i{N}_i{M}_i/{\sum }_i{N}_i$, while the weight-average molar mass is $M_w={\sum }_i{N}_i{M}_i^2/{\sum }_i{N}_i{M}_i$. A single average does not determine the full distribution.

A monomer is a chemical substance whose molecules can undergo polymerization and contribute units to the essential structure of a polymer. A monomer molecule is the individual reactant; its incorporated monomeric unit is the portion identifiable as derived from that reactant. Incorporation can change chemical bonds and release small byproducts, so the unit in the chain need not have the complete molecular formula of the feed molecule. See IUPAC: monomer molecule and IUPAC, A brief guide to polymer terminology: monomeric units.

Three different counts answer three different questions. Monomer molecules describe the feed; monomeric units trace that feed's contributions to the product; a constitutional repeating unit describes the smallest chemical pattern repeating along a regular chain. In Nylon 6,10, two different monomers supply one interior constitutional repeating unit,Its molecular formula is ${\mathrm{C}}_{16}{\mathrm{H}}_{30}{\mathrm{N}}_2{\mathrm{O}}_2$. The diamine feed is ${\mathrm{C}}_6{\mathrm{H}}_{16}{\mathrm{N}}_2$, and the diacid-chloride feed is ${\mathrm{C}}_{10}{\mathrm{H}}_{16}\mathrm{C}{\mathrm{l}}_2{\mathrm{O}}_2$; subtracting two $\mathrm{HCl}$ gives the interior-repeat budget. For a finite linear chain made from $n$ of each reactant, there are $2n-1$ joining connections, hence $2n-1$ formal $\mathrm{HCl}$ eliminations, rather than $2n$. One amine end and one acid-chloride end remain in that ideal unmodified chain. This is a stoichiometry check on the depicted reaction, not a claim that the washed experimental thread retains those ends. The reacting end-group chemistry is documented in Hammond, lecture 5, p. 4; the specified Nylon 6,10 reactant pair is reported in Polymer Day, nylon demonstration.

Ethene shows the reverse counting mismatch. One $\mathrm{C}{\mathrm{H}}_2=\mathrm{C}{\mathrm{H}}_2$ monomer molecule contributes a two-carbon $-\mathrm{C}{\mathrm{H}}_2-\mathrm{C}{\mathrm{H}}_2-$ monomeric unit to ideal linear polyethylene, without a small byproduct in that growth step. But the smallest constitutional repeating unit of its uniform backbone is just $-\mathrm{C}{\mathrm{H}}_2-$: a chain containing $m$ incorporated ethene monomeric units contains $2m$ such interior one-carbon repeats. End groups are additional bookkeeping. The frequently drawn $[-\mathrm{C}{\mathrm{H}}_2-\mathrm{C}{\mathrm{H}}_2-{]}_m$ is useful for tracing the feed, although it is not the smallest constitutional repeat. The minimum-pattern definition is given in IUPAC: constitutional repeating unit; the ethene growth chemistry is described by ^[ref].

A segment of a freely jointed chain or Gaussian chain instead belongs to a statistical model. It can represent many chemical bonds and several constitutional repeating units; its length and the segment count are chosen to describe chain conformation at the intended scale. The three-repeats-per-segment drawing in Figure "From two liquid layers to a Nylon 6,10 thread: monomers, a constitutional repeating unit, and statistical segments" is a deliberately arbitrary coarse-graining example, not a measured Kuhn length. Always state the counting convention before substituting a segment count for a degree of polymerization. This separation lets chemistry track which chemical atoms entered the material while statistical models track how the material bends and extends.

Awarded to Pierre-Gilles de Gennes for extending methods for studying ordering in simple systems to complex matter, especially liquid crystals and polymers.^[ref]

A nucleic acid is a polymer built from linked nucleotides. The two principal kinds are DNA and RNA. Each nucleotide contributes a sugar, a nucleobase, and phosphate to the chain; the nucleobase is only the letter-bearing component, not the entire nucleotide. DNA uses deoxyribose, whereas RNA uses ribose. A chain of six nucleotides is therefore not six free nucleobases placed beside one another. See NHGRI: nucleic acids.

The chemical scaffold and the variable lettering do different jobs. Covalent bonds maintain the sugar–phosphate backbone, while the order of attached nucleobases specifies the nucleic acid sequence. A phosphodiester bond links the $3^{\prime }$ position of one sugar to the $5^{\prime }$ position of the next through phosphate. These position numbers refer to carbon chemical atoms in the sugar; they define a strand's direction even when it bends or forms a circle. See Brown, Genomes, 2nd edition: the human genome.

This division is useful: the backbone supplies a connected physical record, and base pairing allows its letters to recognize partners. DNA replication copies that record, and transcription uses a DNA template to make RNA. RNA can also fold into functional shapes through base pairing within one chain; a nucleic acid sequence specifies chemical order, not a requirement for a straight shape. The beauty is that the same kind of polymer can carry a message and form a structure that acts on messages.

A nucleotide contains a nucleobase, a sugar, and one or more phosphate groups. In DNA and RNA, linked nucleotides form a polymer whose nucleotide sequence carries information. The sugar is deoxyribose in DNA and ribose in RNA. See NHGRI: nucleotide.

Measure the pressure difference across a solvent-permeable barrier that retains the solute, at known temperature. For dilute polymers, the zero-concentration intercept of reduced pressure determines the number-average molar mass. Leakage, aggregation and ionic contributions can change the inferred molecular population.

The common part of the magnetohydrodynamic-viscoelastic analogy is stretching and advection of a line element. With the component-first velocity gradient $L_{ij}={\partial }_j{u}_i$, the ideal tensor equation isIts left side is an upper-convected derivative, the same deformation law that appears for a polymer conformation tensor when its relaxation is neglected.

There are two important physical differences. First, the solenoidal magnetic-field constraint prevents a magnetic field line from having an isolated source or sink inside the fluid. A polymer strand can have molecular endpoints, so there is no corresponding no-endpoint constraint on the strands. Second, a stretched polymer has a preferred relaxed distribution of configurations and a finite polymer relaxation time; it can recoil relative to the solvent through molecular drag and thermal motion. An ideal magnetic field has no analogous local tendency to return to a prescribed coil size. Real polymers also have a finite contour length, a limitation absent from the ideal field-stretching law.

For example, a normalized Hookean dumbbell conformation equation has the dissipative extensionThis is local relaxation toward an isotropic thermal configuration. It remains nonzero for a spatially uniform stretched $C$. By contrast, constant magnetic diffusivity ${\eta }_m$ adds ${\eta }_m{\nabla }^2\mathbf{B}$ to the resistive induction equation. The product rule gives the resistive evolution of the magnetic second-moment tensor:The magnetic correction involves spatial gradients, vanishes for a uniform magnetic field, and does not create field-line endpoints: ${\partial }_t(\nabla \cdot \mathbf{B})={\eta }_m{\nabla }^2(\nabla \cdot \mathbf{B})$. It permits slippage and magnetic reconnection while preserving the no-monopole constraint. After averaging, its gradient-product term generally needs additional information beyond the averaged $M$. One cannot replace it indiscriminately by the local polymer relaxation term, or even by ${\eta }_m{\nabla }^{2}M$ alone.

ToxB greatly reduces recognizable actin cables and actin patches. Untreated biological cells contain concentrated cortical structures, especially in buds, whereas treated biological cells show only weak diffuse staining even at ten times the exposure. The disappearance concerns actin filament organization and detectable F-actin, not necessarily destruction of the actin protein itself. actin monomer sequestration could leave substantial total actin present while eliminating most visible polymer.

Loss of actin cables should impair directed secretory-vesicle delivery and polarized cell growth; loss of actin patches should impair actin-dependent endocytosis. Disruption of the actomyosin machinery can also compromise cytokinesis. The rounded, poorly polarized appearance is consistent with these effects, but the images do not independently measure secretion, division rate or viability. A vehicle control and equal-exposure images are needed for quantitative fluorescence comparison; a longer exposure is useful for finding residual structures but prevents direct comparison of the displayed brightnesses.

ToxB strongly reduces sedimentable wild-type actin, whereas act1-117 shows little additional reduction from its already low baseline. Wild-type actin is mainly in the pellet without drug and mainly in the supernatant with drug. Using the approximate bar heights, the pelleted fraction changes from $740/(740+120)\approx 0.86$ to $280/(280+820)\approx 0.25$. Thus the drug suppresses polymer accumulation but does not eliminate the measured pellet entirely.

For act1-117, the pellet is only about one-third of the recovered signal even without drug, and remains near that fraction with drug: approximately $340/(340+570)\approx 0.37$ and $360/(360+660)\approx 0.35$. The mutant is therefore relatively resistant to the drug's further effect under this assay, but it is not a wild-type-quality polymerizer with an unchanged high polymer mass. These are approximate within-pair fractions; unequal total recoveries and loading limit precise quantitative comparison. sedimentation also requires confirmation that pellet material is filamentous rather than aggregated.

Sequestration of assembly-competent G-actin explains both inhibition of polymer formation and loss of pre-existing cellular structures. In a simple model, free actin monomer $G$ binds drug $D$ to form an assembly-incompetent complex $GD$. At a actin filament end the net addition rate is $v=k_{+}[G]-k_{-}$, so lowering free $[G]$ reduces elongation and can drive net shortening below the relevant actin critical concentration $C_c=k_{-}/k_{+}$. Formation of new nuclei is also strongly reduced when free actin monomer is depleted. Existing actin filaments can consequently disappear through ordinary turnover even though ToxB never binds them.

Binding could instead or additionally alter the actin monomer conformation or nucleotide state needed for productive assembly. The resistant mutant supports an interaction with actin, but resistance could reflect reduced drug affinity, an allosteric change or altered assembly properties. The available endpoints do not uniquely distinguish these possibilities. In particular, they provide no positive evidence for direct actin filament severing or actin filament capping by ToxB. A real example of actin-monomer sequestration and resistant-actin genetics is latrunculin A; this is an analogy, not a chemical identification of ToxB.^[ref]

A pyrene-actin assay can follow assembly continuously. A small labelled fraction mixed with otherwise unlabelled actin gives an increasing signal as the labelled subunits enter actin filaments, allowing comparison of lag, growth and plateau. Drug-only and labelled-monomer controls are essential because ToxB might alter fluorescence directly; comparable label incorporation and a polymer-mass calibration are needed to compare wild-type and mutant actin. An institutional description explains this polymerization-dependent fluorescence readout.^[ref]

Direct single-filament imaging by total internal reflection fluorescence microscopy can separately measure actin filament appearance, elongation and shortening, rather than inferring all three from bulk polymer mass. It requires matched surface and label conditions, since immobilization can perturb assembly. light scattering or turbidity can also follow assembly, but aggregates and changes in actin filament length or bundling complicate mass estimates. electron microscopy can verify actin filament morphology and length at selected times, although sampling and preparation make it less convenient for a continuous kinetic curve. Combining a bulk assay with direct actin filament imaging addresses different ambiguities.

Experiment 2: relative to untreated wild-type actin, actin monomer sequestration predicts a longer apparent actin nucleation lag, a smaller elongation slope, and a lower steady-state polymer mass. At sufficiently strong binding the trace could remain nearly flat rather than displaying clearly separated phases. The wild-type sedimentation data support a lower endpoint, but do not establish the exact kinetic changes or whether the time to plateau is longer or shorter.

Experiment 3: the halo result does not determine the drug-free actin polymerization curve of act1-129. A mutation that mainly raises drug affinity could have a nearly wild-type curve without drug. A mutation that intrinsically destabilizes actin filaments or impairs actin nucleation could instead prolong the lag, reduce growth and lower the plateau. Thus a wild-type-like untreated curve is a useful conditional reference prediction, not a result that follows from cellular hypersensitivity.

Experiment 4: act1-129 is about sixfold more sensitive in the cellular assay. If that reflects a stronger direct effect of ToxB on its actin monomers, predict greater suppression than in experiment 2: a longer or effectively absent actin nucleation phase, slower or undetectable elongation, and a still lower plateau relative to its own drug-free control. It could be almost flat. However, cellular hypersensitivity could also arise from reduced functional reserve or altered interactions with other proteins; it does not guarantee stronger inhibition of purified actin. These alternatives are exactly what the comparison with experiment 3 resolves. No numerical lag, slope or plateau can be calculated from halo sizes alone.

The weak phalloidin signal could include sparse or very short residual actin filaments, together with nonspecific background; it is not a direct image of G-actin. phalloidin preferentially binds filamentous actin, so genuinely specific residual signal would indicate some polymeric actin. Long exposure also amplifies autofluorescence, out-of-focus light and nonspecific staining. Vehicle controls, unstained biological cells, matched acquisition settings and background subtraction are needed before assigning a small residual signal to actin filaments.

For wavelength $\lambda$ and numerical aperture $\mathrm{NA}$, the Rayleigh criterion gives a conventional lateral resolution of roughly $d\approx 0.61\lambda /\mathrm{NA}$, typically a few hundred nanometres under these conditions, far larger than the roughly 7-nm actin filament diameter. It cannot resolve every short actin filament, distinguish all closely spaced actin filaments, or infer polymer mass directly from the displayed intensity. The tenfold exposure change is not itself a calibrated tenfold decrease in F-actin. Fixation, dye accessibility and detector saturation introduce further uncertainty.

Super-resolution fluorescence microscopy could localize remaining labelled structures more precisely, while electron microscopy or cryo-electron tomography could test ultrastructure, subject to preservation and sampling limits. Biochemical separation of polymeric and soluble actin, with the mass-balance and aggregation controls already discussed, supplies an independent population measurement. A live actin reporter can test dynamics but must be checked for perturbation of assembly. Agreement between these approaches would be stronger than concluding complete actin filament absence from weak conventional staining alone.

A rising force branch means that an already exposed polypeptide segment is being stretched while the remaining structure resists further extraction. At a force peak a structural barrier is crossed; protein denaturation or release of additional chain increases the available polymer contour length, so the force falls at almost the same imposed separation. Stretching then loads the next remaining barrier. The sawtooth pattern therefore reports a sequence of mechanically resistant intermediates, not a regular spring oscillation. The cantilever deflection gives force, while its own deformation must be removed from the imposed separation to infer molecular extension.

A membrane protein also requires work to disrupt membrane anchoring, helix packing and contacts with lipids or neighboring molecules. The retinal-containing interior contributes a bulky, specifically packed scaffold; exposing chain must overcome these steric hindrance as well as hydrogen bond and electrostatic interactions. A force peak need not correspond one-to-one to one complete helix, and its height depends on loading rate and pathway, not just an equilibrium folding Gibbs free energy.

Salt changes electrostatic screening and specific ion binding. Screening can reduce either repulsion or attraction, so “more salt always weakens the protein” is false. The primary study underlying these kinds of traces identifies relatively salt-insensitive steric hindrance around the retinal and additional salt-sensitive barriers. Its interpretation associates some added barriers with screening of destabilizing electrostatic repulsion. The supplied traces likewise show a changed set of intermediate peaks, rather than a uniform rescaling of one curve.^[ref]

Exact assignment of every arrow to a residue requires structural and statistical information beyond one representative trace. No such assignment is inferred here. The printed reference to titin in the following paragraph is inconsistent with the stated specimen and the membrane-protein traces; the physical interpretation here concerns bacteriorhodopsin, while the worm-like-chain expression is a generic polymer approximation.