Past exam of the biological physics course of the University of Oxford / 2019 / 6

The hydrophobic core of a lipid bilayer makes transfer of a hydrated ion into bare lipid energetically costly. An ion channel supplies a water-accessible route across that barrier. Its narrow selectivity filter presents coordinating chemical groups that compensate for loss or rearrangement of the ion's hydration shell. Selectivity depends on the free-energy balance of hydration, coordination, electrostatics and geometry, rather than a sieve that passes all smaller ions.

A potassium-channel filter presents backbone carbonyl oxygens in a geometry favourable for potassium; smaller sodium does not automatically gain the same energetic compensation. Sodium and calcium channels use different filter architectures: charged side chains and their coordination environments favour particular ions, with ion occupancy and competing species also affecting permeation. In canonical vertebrate voltage-gated sodium channels, the DEKA selectivity ring illustrates chemical discrimination distinct from potassium-channel carbonyl coordination. The precise mechanism therefore depends on the channel family.

Rapid flux requires low successive free-energy barriers, not permanent binding of the preferred ion. Several occupied sites, nearby water and ion–ion interactions can facilitate passage without requiring the ion to cross the lipid. For scale, a $100\mathrm{pS}$ open channel driven by $100\mathrm{mV}$ carries $10\mathrm{pA}$, approximately $6\times 10^7$ monovalent ions per second. A nanometre-scale route with favourable coordination can support such rapid, thermally driven motion. An ion channel is not an ATP-driven pump: net flux follows the electrochemical potential difference,which vanishes at the species' Nernst equilibrium potential. ATP-consuming pumps maintain the gradients that make sustained channel currents possible.

Permeation selectivity and opening probability are separate properties. Voltage-sensor movement, ligand binding or mechanical tension can regulate access to the pore, while inactivation can terminate conduction during a maintained stimulus. Patch-clamp unitary currents establish discrete conducting states; reversal potentials under changed ionic compositions quantify selectivity. Structure determination locates the filter and gates, and mutations or specific blockers test their functional roles. Agreement between these measurements is stronger evidence than structure or current shape alone.

A charged biological molecule attracts counterions and repels ions of the same sign. Thermal diffusion opposes complete localization, producing a mobile ionic atmosphere whose charge partly cancels the molecule's electric field at larger distances. Solvent polarization also reduces the field through its permittivity, but that dielectric response is distinct from the mobile-ion screening cloud.

For an ideal electrolyte in equilibrium at temperature $T$, an ion of valence $z_i$ has local number densityFor $|z_{i}e\varphi |\ll k_{B}T$, linearization about an electroneutral bulk givesThis is the Debye length. Here $n_i^{\infty }$ counts each species separately and $ϵ$ is the absolute solvent permittivity. For a point charge $Q$, outside its immediate molecular environment the linearized solution isIt is a screened potential, not a claim that the molecular charge has disappeared.

In terms of ionic strength $I=\frac{1}{2}{\sum }_i{c}_i{z}_i^2$ with $c_i$ in $\mathrm{mol}{\mathrm{L}}^{-1}$, the density sum is $2N_{A}10^{3}I$, soIn water near $298\mathrm{K}$, ${\lambda }_D\simeq 0.304\mathrm{nm}/\sqrt{I/(1\mathrm{mol}{\mathrm{L}}^{-1})}$: about $9.6\mathrm{nm}$ at $1\mathrm{mM}$ and $0.79\mathrm{nm}$ at $0.15\mathrm{M}$. Increasing ion density or valence shortens the range; temperature and solvent permittivity also matter. The displayed temperature dependence holds at fixed permittivity, which itself changes with temperature in water.

The derivation assumes dilute mobile ions, a continuum uniform dielectric and weak potentials. Near highly charged DNA or protein surfaces, nonlinear Poisson-Boltzmann equation behaviour, counterion condensation, finite ion sizes, specific binding and correlations can matter. At physiological salt the predicted length approaches molecular scales, so the formula is a useful far-field scale with limited microscopic accuracy, rather than a complete account of the first hydration layer.

Fluorescence localization precision can be much finer than the image width: a fluorescent molecule produces a diffraction-broadened point spread function, but the centre of that distribution can be estimated much more precisely than its width. Fit the measured camera counts to a calibrated point-spread model with position, brightness and background parameters. For an ideal Gaussian image of standard deviation $s$ and $N$ independent detected photons, the centroid precision scales asPixelation and background worsen this limit. A useful approximate variance for square pixels of width $a$ and background rms noise $b$ photons per pixel isThis estimate requires a correctly specified isolated image and appropriate noise model. Drift, optical aberration, dye motion, linkage offset, bleaching and motion blur introduce further errors; localization precision alone does not guarantee absolute positional accuracy.^[ref]

In the Myosin V measurements, total internal reflection fluorescence microscopy reduced background and a two-dimensional Gaussian fit localized a single dye attached to a light chain with roughly $1.5\mathrm{nm}$ precision. The motor's centre advances about $37\mathrm{nm}$ per cycle. A hand-over-hand mechanism instead moves the rear head about $74\mathrm{nm}$ past the stationary front head, exchanging their roles. A label displaced by $x$ from the midpoint therefore gives alternating displacements $37+2x$ and $37-2x$ nm; near a head this tends towards alternating $74$ and $0$ nm. An inchworm model would give approximately uniform $37$ nm label displacements. Resolved unequal alternating steps therefore distinguish the mechanisms, with interpretation depending on the label geometry.^[ref]

For sub-diffraction imaging, arrange that only a sparse, spatially separable subset of fluorophores emits in each frame, localize those emitters, then combine many frames into a map. Photoactivation or reversible stochastic switching provides the sparse subsets. Distinct nearby molecules can then be localized at different times, circumventing overlap of their simultaneously emitted diffraction patterns. The achievable structural resolution depends on photon-limited precision, label density and linkage, repeated blinking, drift correction and specimen motion; arbitrarily many fits to the same dye do not supply missing labels elsewhere. This is the principle behind PALM and STORM, with their different switching strategies.^[ref]