Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S12

The protein primary structure is the ordered protein sequence along a polypeptide, from its N-terminus to its C-terminus. In cells, information in a DNA coding sequence is copied into messenger RNA by transcription. During translation, the ribosome reads successive three-base codons; transfer RNA molecules match these by their anticodons and supply the corresponding amino acids. The genetic code is degenerate: different codons can encode the same amino acid. A start establishes the reading frame and a stop terminates translation. Peptide bonds link the amino acids into the backbone, leaving chemically different side chains. Thus the sequence specifies the order of side-chain interactions, rather than a literal coordinate list for the folded molecule.

Protein secondary structure describes local backbone arrangements. An alpha helix is stabilized by a regular pattern of backbone hydrogen bonds, typically between the carbonyl of residue $i$ and the amide of residue $i+4$. Beta sheets assemble extended strands through inter-strand hydrogen bonding, with parallel or antiparallel strand orientations. Turns and loops connect these elements; not every region belongs to a helix or sheet. The peptide bond is approximately planar, while backbone torsion angles provide conformational freedom subject to steric restrictions.

Protein tertiary structure is the complete three-dimensional organization of one chain, often arranged into protein domains. Protein quaternary structure describes the organization of multiple chains in an assembly. Soluble globular proteins often bury nonpolar side chains and expose polar ones; membrane proteins can instead expose hydrophobic surfaces to the surrounding lipid. Other stabilizing interactions include hydrogen bonds, electrostatic interactions, Van der Waals forces and sometimes covalent disulfide bonds. These competing contributions define active sites, binding pockets and mechanically useful structures; folded proteins are dynamic ensembles, not perfectly rigid objects.

The connection from sequence to structure is thermodynamic and kinetic. Folding usually decreases the chain's conformational entropy, but favorable interactions and changes in solvent entropy, including the hydrophobic effect, can lower the total Gibbs free energy. Under suitable conditions the native ensemble is favored over unfolded alternatives. A change in sequence can therefore change the balance of accessible conformations and alter folding or function. Reversible refolding of a denatured protein in an appropriate environment demonstrates that the sequence can contain the information needed for its native structure.^[ref]

The environment is part of the specification: temperature, pH, ionic conditions, ligands and post-translational modifications matter. Molecular chaperones can prevent aggregation and help chains escape kinetic traps without providing a residue-by-residue structural template. Some sequences are intrinsically disordered proteins or disordered regions, and some require partners for a stable fold. Thus primary structure strongly constrains structure, but does not guarantee one unique rigid shape independent of conditions.

Let the interface be $y=0$, with glass at $y<0$ and water at $y>0$, and choose $x$ parallel to the incident beam's tangential component. Define $k=\omega /c$ as the vacuum wavenumber, with glass and water refractive indices $n_g$ and $n_w$. Continuity of phase along the interface fixes $k_x=k{n}_g\sin \theta$. In water, the dispersion relation is $k_x^2+k_y^2=k^2{n}_w^2$, so above the critical angle,Choose the positive imaginary part because a field growing exponentially into the water would be unphysical. The evanescent field in total internal reflection is thereforeIt propagates in phase parallel to the interface but decays normally. Here $E_0$ is the transmitted field amplitude at the interface; relating it to the incident amplitude requires the appropriate polarization-dependent Fresnel equations. If the symbol $k$ instead denotes the incident glass wavenumber, replace every vacuum $k$ in the formula by $k/n_g$; equivalently $k_x=k\sin \theta$ and $\kappa =k\sqrt{{\sin }^2\theta -(n_w/n_g{)}^2}$. This states the wavenumber convention left implicit in the source.

The excitation rate in the weak, unsaturated fluorescence regime is proportional to the time-averaged squared electric field amplitude. Thus the illumination profile relevant to excitation isThe amplitude decay length $1/\kappa$ is twice the intensity decay length. This profile is often called the evanescent excitation intensity; it does not imply a propagating normal energy flux through an ideal lossless totally reflecting interface.

Total internal reflection fluorescence microscopy restricts excitation to a shallow layer adjacent to the glass, typically on the scale of hundreds of nanometres for visible wavelengths and suitable angles. Fluorophores near the basal cell membrane are excited while much of the cytoplasm and bulk solution remain dark. The resulting reduction in out-of-focus fluorescence and autofluorescence improves the signal-to-background ratio for individual molecules, allows camera-based parallel observation, and reduces unnecessary excitation of deeper regions. Its limitation is the same selectivity: it does not image arbitrary molecules deep inside a cell. This near-interface selectivity was directly demonstrated in cell–substrate contact imaging.^[ref]

Under the simplest interpretation, each spot contains two equally bright, independently bleaching green fluorescent protein emitters. Losing one halves the fluorescence and losing the other removes it. A stable protein dimer carrying one active label per subunit is consistent with this result. The conclusion is two active emitters in the observed spot; it does not by itself rule out unresolved coincident proteins, incomplete fluorescent maturation or blinking. Permanent membrane binding makes loss by dissociation less plausible, but lateral motion out of the observation region would still need experimental control.^[ref]

Let their independent photobleaching times be $T_A,T_B\sim \mathrm{Exp}(k_b)$. Then $t_1=\min (T_A,T_B)$ has rate $2k_b$, while the memoryless remaining lifetime $t_2-t_1$ has rate $k_b$ and mean $1/k_b$, matching the stated single-emitter calibration. The two-emitter photobleaching completion time is the maximum, not one single exponential and not a sum of two equal-rate exponential variables. Its cumulative distribution function isDifferentiating gives its probability density function:with zero density for $t<0$. It integrates to one, and its mean is $1/(2k_b)+1/k_b=3/(2k_b)$. This uses two active fluorophores at the start of observation, constant illumination and independent irreversible bleaching with the calibrated rate.

Cells interconvert several forms of free energy. Chemical concentration differences and the chemical potentials of metabolites can drive reactions, including ATP hydrolysis. The usable free-energy difference is that between reactants and products under the actual cellular conditions; breaking a chemical bond alone does not release energy without the rest of the reaction. Oxidation-reduction reactions transfer electrons down an electrochemical potential difference and can drive ion pumping in an electron transport chain. Light absorption can create excited states and charge separation in photosynthesis, eventually supporting chemical synthesis.

An electrochemical gradient stores free energy in both an ion concentration difference and a membrane potential. Chemiosmosis couples downhill ion flow to ATP synthesis by chemiosmosis; ATP-powered pumps can run the conversion the other way. Molecular motors and other mechanochemical enzymes convert chemical free energy into mechanical work or elastic strain, while ion gradients can also power transport and rotary motion. These couplings conserve total energy but dissipate some free energy as heat. Equilibrium thermal fluctuations alone cannot sustain directional work without a maintained nonequilibrium resource.

Use number concentrations and write the electrochemical potential of a sodium ion as $\tilde{\mu }={\mu }^{\circ }+k_{\mathrm{B}}T\ln c+e\psi$. Define the inward sodium-motive force as available work per positive charge for transport from outside to inside. With $V_m={\psi }_i-{\psi }_o$,Its negative is the signed inward free-energy cost per charge; stating this direction removes a common sign ambiguity.

Long-open sodium channels establish electrochemical equilibrium, $k_{\mathrm{B}}T\ln (c_{i0}/c_o)+e{V}_{m0}=0$. This does not generally require equal concentrations: impermeant counterions can support a nonzero equilibrium membrane potential. Let $n_0=c_{i0}V$. For a large fixed-concentration external reservoir, after pumping $n$ ions out, $c_i(n)=(n_0-n)/V$ and $V_m(n)=V_{m0}-ne/C$. The sodium export cost from a capacitive vesicle for exporting another ion isThe initial-equilibrium relation was used in the last step. The first term is the incremental electrical cost and the second is the change in chemical cost from their initial balanced values.

If the intended initial reference is uncharged with $V_{m0}=0$ and $c_{i0}=c_o$, then $n_0=c_{o}V$ and the requested form in the listed variables isThat equal-concentration reference is an additional assumption. For example, $c_{i0}=2c_o$ and $V_{m0}=-(k_{\mathrm{B}}T/e)\ln 2$ also give zero initial sodium flux, but a different ion inventory. Thus the open-channel condition alone cannot determine $n_0$ from $c_{o}V$. If concentrations are molar rather than number densities, use the corresponding Avogadro constant conversion and $RT$ for molar free energies. Treating the capacitor charge discretely gives a charging-energy increment $(n+1/2)e^2/C$ rather than the differential $n{e}^2/C$; the distinction is negligible for the thousands of ions used next.

Use the initial equal-concentration, zero-potential reference, or equivalently quote component changes relative to the initial electrochemical equilibrium. At $n=n_0/2=5000$, with $C=10^{-16}\mathrm{F}$,At room temperature $T=300\mathrm{K}$, the chemical component isThe electrical cost dominates by about $447$. The associated change in membrane potential is $-ne/C=-8.01\mathrm{V}$. Such a large voltage is a consequence of the ideal fixed-capacitance, perfectly impermeable model; an actual lipid membrane would not generally remain intact and nonconducting over this entire extrapolation.

The electrical component is a straight line through zero, $n{e}^2/C$. The chemical component, $k_{\mathrm{B}}T\ln [n_0/(n_0-n)]$, starts at zero, increases convexly, and diverges as $n\to n_0^{-}$ because the remaining internal concentration vanishes. Their sum lies above both: nearly electrical over most of the stated range, but eventually diverging with the chemical term. For a nonzero initial equilibrium potential, the separate absolute electrical and chemical components have opposite constant offsets; the total curve and the quoted changes remain the same. The final few ions require a discrete-particle treatment rather than a literal continuum divergence.