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Dermatan sulfate is a family of sulfated glycosaminoglycans containing repeating iduronic acid and N-acetylgalactosamine units. The sugar linkage pattern and sulfate substitutions help define the material; chain length and substitution vary, so the name does not identify one small molecule with a unique molecular mass.^[ref]

Its uronic-acid stereochemistry can be created after a chain has been assembled. Chondroitin-glucuronate 5-epimerase converts D-glucuronate residues in a chondroitin chain into L-iduronate residues in a dermatan chain. This enzyme is distinct from heparosan-N-sulfate-glucuronate 5-epimerase, which acts in a different polysaccharide context.^[ref]

An ideal segment of 20 complete two-sugar repeats contains 20 uronic-acid residues and 20 amino-sugar residues, or 40 sugar residues altogether. Epimerizing an eligible uronic-acid residue changes its configuration without increasing that total. Adding a sulfate group changes substitution without adding another sugar residue.

These distinctions guide an experiment: measuring chain extension, uronic-acid configuration and sulfate content asks three separate questions. A total carbohydrate measurement alone cannot determine which operation occurred. To compare two preparations meaningfully, record the property actually measured rather than assuming every dermatan-sulfate chain has identical composition.

In protein chemistry, a heterodimer is an assembly of two different protein subunits, conventionally written $AB$. A homodimer has two copies of the same subunit type, $A_2$. Different types can have similar protein folds and nearly equal molecular masses; “different” refers to molecular identity, not simply size or shape. An $A_2{B}_2$ protein complex contains two types but four chains, so it is not itself a heterodimer.

The alpha/beta tubulin heterodimer is the repeating building block of a microtubule. The alpha and beta protein subunits have similar folds but distinct sequences and nucleotide environments. Their structural relationship was resolved by electron crystallography of zinc-induced tubulin sheets: the 1998 model reached $3.7$ angstrom resolution. This is a concrete example of a two-chain protein quaternary structure, rather than the protein tertiary structure of a single chain.^[ref]

Stoichiometry gives a useful upper bound. If an idealized mixture contains $3\mu \mathrm{mol}$ of $A$ and $2\mu \mathrm{mol}$ of $B$, and only $AB$ can assemble, at most $2\mu \mathrm{mol}$ of heterodimer can form; at least $1\mu \mathrm{mol}$ of $A$ remains outside $AB$. The actual amount also depends on binding equilibrium and experimental conditions: chain counting alone does not guarantee complete association.

In molecular flow, the gas mean free path is large compared with the channel or vessel dimensions, so molecules encounter walls more frequently than one another. The Knudsen number $\mathrm{Kn}=\lambda /L$ characterizes this regime.

For a small aperture of area $A$ between equilibrium gas reservoirs, the vacuum conductance is $C=A\overline{v}/4$, where $\overline{v}$ is the mean molecular speed. At fixed temperature this scales as the inverse square root of molecular mass. Continuum viscous-flow formulas need not apply.^[ref]

The molecular mass is the mass of a molecule; relative molecular mass compares it with a reference mass.

A nuclear pore complex is an assembly of proteins spanning the nuclear envelope and regulating passage between the cell nucleus and cytosol. It is a selective gateway, not simply an unregulated hole: messenger RNA export and import of nuclear proteins require appropriate transport machinery. See Alberts et al.: nuclear pore complexes.

The nuclear pore complex occupies a junction of both nuclear envelope membranes. Its protein scaffold surrounds an aqueous passage connecting cytosol to the cell nucleus, rather than the intermembrane space. Diffusion permits small soluble cargo to pass; selective receptors transport other cargo. Molecular mass alone is not a universal cutoff.^[ref]

The 375 meV pair should shift most fractionally, towards zero bias, because C–H stretching has strong hydrogen displacement. A reduced-mass estimate givesso the pair would move approximately from $\pm 375$ to $\pm 275$ mV. A frozen-carbon estimate gives $1/\sqrt{2}$, or about 265 mV; the actual normal mode includes coupled motion. Whole-molecule modes change much less: the molecular mass rises only from approximately 41 to 44, giving a translational estimate $\sqrt{41/44}\simeq 0.965$.

Use $u$ as a spectroscopic wavenumber, in the same units throughout. Write the mass absorption coefficient of the absorbing species asHere $S$ is the spectral line strength, the integrated mass absorption coefficient; it is not the line-centre coefficient. If ${\kappa }_{\nu }$ has units ${\mathrm{m}}^2\mathrm{k}{\mathrm{g}}^{-1}$ and $\nu$ is in $\mathrm{c}{\mathrm{m}}^{-1}$, then $S$ has units ${\mathrm{m}}^2\mathrm{k}{\mathrm{g}}^{-1}\mathrm{c}{\mathrm{m}}^{-1}$. For absorber mass density ${\rho }_a$, the volume coefficient in Beer-Lambert law is ${\rho }_a{\kappa }_{\nu }$.

Lifetime broadening. Finite lifetimes of the upper and lower states destroy optical coherence, even for an isolated molecule. Exponential coherence decay gives a Lorentzian line shape. For population decay rates ${\Gamma }_u,{\Gamma }_l$ and no extra dephasing, its wavenumber half-width is $({\Gamma }_u+{\Gamma }_l)/(4\pi c)$, with $c$ expressed in the length unit reciprocal to $\nu$. This sets a natural linewidth and can matter at extremely low collision rates; it is generally much smaller than atmospheric collisional or thermal widths.

Thermal Doppler broadening. A Maxwell-Boltzmann distribution of molecular line-of-sight velocities givesThis Gaussian distribution has a $1/e$ half-width ${\Delta }_D$, determined by absorber molecular mass $m_a$ and temperature. It is especially relevant in the rarefied upper atmosphere, where collisions are infrequent.

Pressure broadening. Collisions perturb transition frequencies and phases. In the impact regime, exponential phase decorrelation givesHere ${\Delta }_L$ is the half-width at half maximum, proportional to the coherence-destroying collision rate. It depends on perturber density, molecular collision properties, and temperature; it is often parameterized by ${\Delta }_L\propto p(T_{\ast }/T{)}^{n_T}$ for a specified broadener. It dominates much of the dense lower atmosphere. The Lorentzian line shape has algebraic $\delta {\nu }^{-2}$ wings, whereas the Doppler profile has rapidly decreasing Gaussian wings. Where both mechanisms matter, their convolution is a Voigt profile. A pure pressure-broadened profile should not be extrapolated to arbitrarily low pressure while discarding the finite Doppler core.

Take a weak line in the radiative sense: the integrated slant optical depth is small wherever absorption materially contributes, not merely a small greenhouse-gas mass mixing ratio. Normalize every local spectral line profile so that $\int {\kappa }_{\nu }(p)d\nu =S$, with the line strength independent of pressure and, as stipulated, temperature. Keep a finite Doppler or natural core, or a Voigt profile, so that a pressure-only zero-width core at the top does not invalidate a uniform optical-depth expansion. The line must also be narrow enough to use $B_{{\nu }_0}$ across it.

To first order,Thus the detailed normalized line shape drops out, even if its width varies with pressure. Substituting in the deficit in c givesThe neglected term is second order when the relevant squared optical depths are integrable and uniformly small. A very narrow saturated line need not be weak simply because $q\ll 1$. More generally the displayed first-order integrated limit follows from the finite integrated opacity even if an idealized line centre makes a pointwise expansion nonuniform; a finite physical core is the straightforward sufficient assumption.

For an explicitly perturbative temperature estimate, evaluate the bracket at $T_{\mathrm{bare}}$:At fixed $p_s,g,q,S$ and radiative conditions, increasing $\kappa =R_b/c_{p,b}$ makes every level $0<r<1$ cooler relative to the surface, enlarging the bracket and the warming. Increasing $c_{p,b}$ at fixed $R_b$ has the opposite effect. The limit $\kappa \to 0$ is isothermal and gives zero line-induced deficit. A change of background molecular mass alone need not change this leading result: for ideal gases with the same active degrees of freedom, $R_b$ and $c_{p,b}$ scale together and their ratio is unchanged. Comparing only the height-coordinate lapse rate $g/c_{p,b}$ would miss that the absorber column is naturally a pressure-coordinate quantity.

A phosphorothioate diastereomer is a stereochemical form of a phosphorothioate substitution that differs in configuration without changing connectivity. In a nucleic-acid phosphate joining different nucleotide environments, replacing a non-bridging oxygen with sulfur can make phosphorus a stereogenic center. The resulting $R_P$ and $S_P$ forms have the same elemental composition but a different spatial arrangement at that center.

They are diastereomers in the usual nucleic-acid context because the rest of the RNA already contains fixed stereochemistry, including its ribose units. Changing one phosphorus configuration while preserving those other centers does not produce the mirror image of the entire RNA. It is therefore misleading to call such a pair enantiomers merely because one phosphorus label changes from R to S. Assignment of the labels uses the Cahn-Ingold-Prelog priority rules, not the order in which a nucleotide sequence is written.

Two independently stereogenic substituted phosphates give four combinations: $(R_P,R_P)$, $(R_P,S_P)$, $(S_P,R_P)$ and $(S_P,S_P)$. Their molecular masses are the same. A mass measurement alone cannot determine either the configuration at one site or the proportions in an unresolved mixture.

A simple activity model shows why mixture composition can matter. Suppose equal amounts of two purified forms separately give product fractions $0.8$ and $0.2$ in a stipulated assay. If molecules act independently and the assay response is linear, a mixture containing fraction $f$ of the first form gives $p=0.8f+0.2(1-f)$. At $f=1/2$, $p=0.5$. These invented numbers illustrate averaging; competition for limiting components can invalidate that model.

The U6 metal rescue experiment supplies an actual chemical distinction. Sulfur substitutions with opposite phosphorus configurations at the same U6 position showed differing metal-rescue behavior, including differences between branching and exon ligation.^[ref] Position, configuration and measured reaction therefore belong in the description of a phosphorothioate reagent.

A phosphorothioate substitution replaces an oxygen of a phosphate group with sulfur. In a nucleic acid backbone, the usual experimental substitution replaces a non-bridging oxygen, while retaining the oxygen bridges connecting neighboring nucleotides. It changes backbone chemistry without deleting a nucleotide or changing its nucleobase sequence. The altered group is a phosphate analogue rather than a disulfide bond.

The two non-bridging positions matter. For a phosphate joining different nucleotide environments, replacing one oxygen or the other can give different configurations at phosphorus. Such phosphorothioate diastereomers can interact differently with an RNA-folding environment or a metal ion. The 2013 U6 metal rescue experiment tested both $R_P$ and $S_P$ substitutions at selected sites, rather than treating every sulfur replacement as equivalent.^[ref]

An original counting example shows the extra information required. At three independently stereogenic, substituted phosphates, two possible configurations at each site give $2^3=8$ configuration combinations. All eight can have the same nucleotide sequence and elemental formula. Measuring molecular mass alone therefore cannot tell which configuration combination was supplied. If every oxygen-to-sulfur replacement is made once at a selected site, each adds approximately $32-16=16$ unified atomic mass units; three add about $48$, regardless of those configuration choices. These rounded atomic masses are illustrative, not exact isotope masses.

Replacing oxygen with sulfur also changes metal–ligand interactions. A loss of RNA activity in magnesium followed by preferential rescue with a different metal can test whether the original oxygen helped position a catalytic ion. It can also reflect altered folding or assembly, so the comparison needs controls. The substitution is a targeted chemical perturbation whose interpretation depends on position, configuration and the reaction being measured.

A polymeric osmotic laxative; 3350 denotes its approximate average molecular mass. Retention of water in the intestinal lumen accounts for its local action.^[ref]

A svedberg, symbol $\mathrm{S}$, is a non-SI unit of sedimentation coefficient, with $1\mathrm{S}=10^{-13}\mathrm{s}$. A sedimentation coefficient is a settling velocity divided by the applied acceleration, so its physical dimension is time. It is a transport property under specified conditions, rather than the time required for a particle to cross a sample.^[ref]

For example, let a particle have sedimentation coefficient $s=20\mathrm{S}=2\times 10^{-12}\mathrm{s}$ in a centrifugal acceleration $a=5\times 10^5\mathrm{m}{\mathrm{s}}^{-2}$. Its local settling speed is $v=sa=10^{-6}\mathrm{m}{\mathrm{s}}^{-1}$, or one micrometre per second. The acceleration in a rotating sample varies with radius, so this local calculation is not a whole-tube travel time.

Two joined particles do not generally have the sum of their separate sedimentation coefficients: joining changes both the net driving force and the resistance to motion. A label such as 16S ribosomal RNA therefore describes sedimentation behavior, not a molecular mass of16 units or a count of16 components.