Past exam of the biological physics course of the University of Oxford / 2021 / 6

Magnetic tweezers apply force and controlled turns to a single torsionally constrained DNA tether joining a surface and a magnetic bead. A magnetic-field gradient pulls the bead; rotation of the field rotates the bead and changes imposed turns. Imaging measures bead height, hence DNA extension. At sufficient excess twisting and suitable force, DNA supercoiling creates writhed loops that shorten the tether. The extension-versus-turn curve supplies a calibration relating the removal or addition of supercoiling to vertical motion. Multiple attachments at both ends are needed conceptually to constrain rotation; freely swivelling DNA would relax the signal.

During transcription initiation, RNA polymerase binds a promoter in a closed complex and then opens a local DNA bubble, exposing template for RNA synthesis. In bacterial initiation a sigma factor helps recognize the promoter. Opening changes local helical twist. Because the total linking number is constrained,Unwinding the bubble means $\Delta T{w}_{\mathrm{bubble}}<0$, so the remainder acquires a positive turn increment. A negatively supercoiled tether therefore loses some negative writhe and becomes longer; a positively supercoiled tether gains positive writhe and becomes shorter. Ordinary DNA compaction instead shortens both tethers. Comparing both signs separates local unwinding from compaction rather than assigning every height change to opening.

If $q$ is the number of turns unwound, $s$ the magnitude of the extension change per imposed turn, and $c$ the compaction, the symmetric-slope approximation predicts $\Delta z_{-}=sq-c$ and $\Delta z_{+}=-sq-c$. HenceMeasured unequal slopes can be retained in the two separate calibration equations. Multiplying $q$ by the helical repeat converts turns to an estimated number of unwound base pairs, subject to the mechanical model and the distinction between twist and writhe.

Repeated extension transitions reveal open-complex lifetimes, closed-complex lifetimes and their dependence on mechanical torque. Initial short RNA synthesis can occur while RNA polymerase remains associated with the promoter, including abortive initiation and DNA scrunching. Addition of nucleotides allows progression to promoter escape; an unwound bubble can then persist with elongation, so loss of the bubble is not the definition of escape. Successive cumulative unwinding events can identify renewed access to the same promoter after clearance. These are the measurement principles demonstrated by the paired-sign experiments.^[ref]

The experiment observes mechanics rather than directly imaging individual opened base pairs. Drift, nonspecific binding, tether integrity, force-dependent buckling and limited time resolution can mimic or obscure steps. A control lacking productive promoter recognition and agreement of the opposite-sign responses help identify the origin of a signal.

A point mutation changes an individual nucleotide position, introducing heritable sequence variation. Replication errors and spontaneous chemical damage are possible origins; mutation does not occur because a future change would be useful. Natural selection changes the frequencies of variants through their effects on reproduction, while genetic drift causes random frequency changes, especially in small populations. A sequence difference fixed between lineages is a substitution; many new mutations disappear and never become substitutions. Under a simple neutral diploid model, $2N_e\mu$ new neutral variants per generation times fixation probability $1/(2N_e)$ gives substitution rate $\mu$, motivating a molecular clock for suitable neutral sites.

To infer a phylogenetic tree, compare homologous, preferably orthologous sequences and align corresponding sites. A nucleotide substitution model converts observed sequence differences into estimated evolutionary branch lengths, allowing for multiple changes at one site. For example, in the equal-frequency, equal-rate Jukes-Cantor model, a site changes to any alternative with rate $\mu /3$, where $\mu$ is its total substitution rate. The probability of matching its initial state after elapsed time $t$ is $1/4+(3/4)e^{-4\mu t/3}$. Thus for an evolutionary separation with expected substitutions per site $d$, the observed mismatch fraction is $p=(3/4)(1-e^{-4d/3})$ andFor small $p$, $d\simeq p$; near saturation $p=3/4$, large evolutionary distances become difficult to distinguish. The correction does not rescue fully saturated data. Real analyses may allow unequal nucleotide frequencies, transition biases and variation of rates between sites.

Tree topology and branch lengths can be estimated using maximum likelihood estimation or Bayesian phylogenetic inference, with support assessed from resampling or posterior uncertainty. A gene tree need not equal the species tree, because incomplete lineage sorting, genetic recombination and horizontal gene transfer can produce different histories. Rooting with an appropriate outgroup establishes ancestral direction; pairwise differences alone do not supply a root.

Branch lengths measured in substitutions per site are not yet calendar times. Under a strict molecular clock with rate $\mu$ per site per year, a branch of duration $t$ has length $\mu t$; two contemporaneously sampled descendants separated at time $t$ have total path length $d=2\mu t$, hence $\boxed{t=d/(2\mu )}$. Calibration comes from independently dated constraints such as fossil calibration, geological information or sufficiently informative sampling dates. A dated common ancestor is not automatically identical in age to the earliest observed fossil. A relaxed molecular clock permits branch-specific rates while imposing a statistical model on them.

Without an external temporal calibration or an independently known rate, multiplying all times by a constant and dividing all rates by that constant leaves expected sequence divergence unchanged. Absolute dates are therefore unidentifiable from sequence divergence alone. A useful time-calibrated phylogenetic tree reports uncertainty from sequence sampling, substitution models, rate variation, topology and calibration bounds, rather than displaying an unjustified exact date for each ancestor.

An ion channel supplies a water-accessible pathway through the otherwise strongly insulating, hydrophobic lipid bilayer. Moving a hydrated ion directly into that low-permittivity region is energetically expensive. The channel's polar pore and coordinating groups reduce the barrier while retaining selectivity. Transport is passive: the electrochemical potential difference drives net movement, and the channel does not itself pump an ion uphill. For an ideal species of charge number $z$,The net electric current of a pathway permeable only to that species reverses at its Nernst equilibrium potential. An open channel can carry zero net electric current at that voltage even while individual ions cross both ways.

The selectivity filter balances loss of the hydration shell against favorable coordination within the pore. In a typical potassium channel, backbone carbonyl oxygen atoms create appropriately arranged binding sites for a substantially dehydrated potassium ion. The smaller sodium ion is not automatically preferred simply because it fits geometrically: its hydration is stronger and the filter does not compensate its dehydration in the same way. Geometry, flexibility, coordination chemistry and ion occupancy all matter. Sodium-selective and calcium-selective channels use different arrangements, often involving charged side chains; a single rigid-sieve explanation is inadequate.

Rapid passage coexists with transient binding because several nearby sites have comparable free energies and relatively low barriers between them. Multiple ions occupying a pore influence one another through electrostatic interactions, enabling correlated exchange and displacement rather than irreversible trapping at one deep site. High throughput therefore requires favorable passage kinetics as well as strong discrimination against unwanted species. A more tightly bound species can even block a pathway instead of being its fastest permeant.

Gating and selectivity are distinct. A voltage-sensing domain couples membrane potential to activation-gate conformation; ligand-gated ion channels couple binding to opening; mechanosensitive ion channels couple membrane stress to opening. Ion-channel inactivation introduces additional unavailable states. None of these mechanisms alone identifies which ion the open pore prefers.

For $N$ similar independent channels, a local Ohmic approximation with outward-positive electric current isHere open probability controls availability, single-channel electrical conductance controls current per open pore, and $V-E_{\mathrm{rev}}$ is the electrical driving factor in this approximation. Nonlinear current-voltage relations and mixed-ion permeation may require a fuller model. Single-channel recording resolves discrete openings; reversal shifts when ionic concentrations change test selectivity; gating electric currents test sensor motion; structural evidence identifies the pore and its coordinating groups. Together these distinguish rapid selective permeation from the separate process of opening the gate.

A small swimming member of the bacteria samples too little spatial separation across its own body to compare weak chemical gradients accurately. Temporal sensing in bacterial chemotaxis instead compares recent receptor activity with an adapted memory. Straight runs carry it between locations; tumbles reorient it. When conditions improve, tumbling becomes less frequent and favorable runs last longer. An unfavorable direction is abandoned sooner. Repetition creates a biased random walk without requiring the cell to identify the best direction at each tumble. Rotational diffusion and environmental obstacles eventually destroy the benefit of maintaining a direction indefinitely.

An order-of-magnitude transport estimate compares the time $\ell /v$ to swim a distance $\ell$ with the time ${\ell }^2/D$ for an attractant to diffuse that distance. Their crossover is a Péclet number of order one:This estimates a useful run scale: over much shorter distances, nutrient diffusion rapidly replenishes the local environment compared with the organism's displacement. Runs of order a second also permit comparison along a substantial path before reorientation. The given $D$ and $v$ determine this scale, not an exact biological mean run length or a complete sensory optimization. If a one-coordinate root-mean-square diffusion length $\sqrt{2D\tau }$ is equated with $v\tau$, the resulting prefactor is two; a three-dimensional root-mean-square convention gives six. Such constants cannot be fixed by a scaling argument alone.

For a cilium of length $d$, compare its advective time $d/u$ with the local diffusion time $d^2/D$. Significant stirring relative to diffusion requires ${\mathrm{Pe}}_d=ud/D\gtrsim 1$, soThis is about $33$ times the stated swimming speed. If the tip travelled on a circular path of radius $d$, that speed would correspond to an illustrative frequency $u/(2\pi d)\simeq 160\mathrm{Hz}$. That geometric conversion is not a specified beat pattern. At microscopic scales the flow has small Reynolds number; useful transport depends on the actual nonreciprocal motion and resulting flow, and rapid motion alone does not guarantee increased uptake. The scaling explains why simply stirring with a very short appendage is demanding when molecular diffusion is already fast.