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The equivalent circuit has membrane capacitance in parallel with electrical conductance–battery branches: voltage-dependent sodium and potassium conductances in series with their reversal potentials, a chloride electrical conductance in series with $E_{Cl}$, and other leak conductances. For externally supplied electric current,The model figure in solution includes this circuit and an illustrative passive response. Although chloride electric current is small when $V\simeq E_{Cl}$, a perturbation produces the restoring electric current $\delta I_{Cl}=g_{Cl}\delta V$. A large chloride electrical conductance therefore shunts perturbations, lowers input resistance and shortens the passive time constant, $R_{\mathrm{in}}\simeq 1/\sum g_j$ and ${\tau }_m\simeq C_m/\sum g_j$.

Reducing that electrical conductance increases the voltage response to a given electric current or extracellular-potassium disturbance and weakens stabilization near rest, making repeated excitation easier. There need not be an instantaneous shift of resting voltage if $E_{Cl}$ equals the old rest exactly. In skeletal muscle, loss of ClC-1-mediated stabilization is associated with myotonic hyperexcitability.^[ref]

Later cessation of contraction can be explained by sustained depolarization causing voltage-gated sodium channel ion-channel inactivation and failure to recover: fewer available channels support progressively smaller or less frequent spikes until depolarization block occurs. Accumulated extracellular potassium in restricted spaces and incomplete recovery between spikes can promote that depolarization. This requires the nonlinear voltage-gating dynamics; a passive chloride resistor and capacitor alone cannot generate or terminate an action-potential train. The stated membrane permeability change does not uniquely determine the time to block or prove one particular ionic accumulation history.

Including both end faces of the perfect cylinder, with $r=1\mu \mathrm{m}$ and $\ell =100\mu \mathrm{m}$,The membrane capacitance is $C=0.01\mathcal{A}=6.346\times 10^{-12}\mathrm{F}$. The required net inward electric charge is $Q=C\Delta V=5.394\times 10^{-13}\mathrm{C}$, or $Q/e\simeq 3.37\times 10^6$ monovalent ions. If sodium supplies that net electric charge, Faraday constant $F=N_{A}e$ givesThe final concentration is about $15.0178\mathrm{mM}$, only $0.119\%$ above the starting value. Neglecting the end faces changes the estimate by $1\%$, giving about $17.6\mu \mathrm{M}$. Over $2\mathrm{ms}$ the capacitive electric charge corresponds to mean net electric current $Q/\Delta t\simeq 0.270\mathrm{nA}$; the duration does not otherwise enter the total electric charge.

This is a net capacitive estimate, not a measurement of total sodium influx. Simultaneous outward potassium electric current makes actual sodium entry larger, and most bulk ions remain near their original concentrations during one spike. Long-term pumps restore gradients after repeated firing; they are not required to replace the entire intracellular sodium pool during the two-millisecond voltage rise.

selective permeability sets the driving-force term. Potassium and sodium channels provide narrow coordinating pathways whose free energy balance of hydration and chemical binding favours the appropriate ion. voltage-gated potassium channel carbonyl coordination differs from the charged filter environment in canonical sodium channels, so selective permeation is not a pore-size ranking. Ion substitution shifts the reversal potential, and selective blockers or filter mutations alter the corresponding electric current. An open channel conducts down the electrochemical gradient; ATP-driven pumps maintain the gradients rather than pushing each ion through the pore.

Voltage-dependent activation sets channel availability. Charged residues in the S4-containing voltage-sensing domains move in the membrane electric field and couple to pore opening. A classical voltage-gated potassium channel is a tetramer; a voltage-gated sodium channel pore-forming polypeptide has four homologous domains. ion-channel gating current measurements and sensor-residue mutations connect electric charge motion to opening, while structures separate the sensor from the permeation pathway. The historical $n^4$ and $m^3$ variables describe kinetics phenomenologically and need not map one-to-one onto individual resolved structural movements.^[ref][ref]

Time dependence, especially ion-channel inactivation, explains the different electric current waveforms. Sodium activation is rapid, but sustained depolarization reduces the availability variable $h$, so the sodium electric current decays despite maintained driving voltage. Fast sodium ion-channel inactivation involves intracellular machinery including the interdomain III–IV region. Potassium activation is slower, supplying delayed outward electric current; the classical delayed rectifier is relatively sustained on this time scale. Some voltage-gated potassium channel variants do show rapid N-terminal pore-blocking ion-channel inactivation or slower filter-associated ion-channel inactivation, which should not be assumed identical to sodium's mechanism. voltage clamp steps separate activation from decay; recovery protocols, single-channel recordings and targeted perturbations distinguish gating, ion-channel inactivation and ion conduction. Removal of the Shaker N-terminal blocker and restoration by a free peptide are particularly direct evidence for one voltage-gated potassium channel mechanism.^[ref][ref]

Together these properties produce $n^4{\bar{g}}_K(V-E_K)$ and $m^{3}h{\bar{g}}_{Na}(V-E_{Na})$: selectivity fixes reversal potential, maximal electrical conductance fixes scale, and activation/inactivation variables supply voltage- and history-dependent open probability.

Represent one permissive voltage-gated potassium channel gate by an available closed state and an activated state, with voltage-dependent transition rates ${\alpha }_n(V)$ and ${\beta }_n(V)$. Gain minus loss givesDuring a voltage clamp step the imposed voltage and hence these rates are constant. Integration with $n(0)=n_0$ givesThe fourth power in the electrical conductance describes four permissive activation variables in the Hodgkin-Huxley model phenomenology. If all must be permissive and are statistically independent, their joint probability is $n^4$. Even a simple exponential rise in $n$ then gives a delayed-looking macroscopic potassium electric current. The second panel of the model figure in solution illustrates this. A continuously changing voltage makes the rates time dependent and requires solving the differential equation along that voltage trajectory; one fixed exponential is not the general action-potential solution.

On releasing the tensile load, the extended chain first retracts towards less aligned conformations because they have much greater configurational multiplicity. Viscous friction and internal relaxation determine the time course; stored free energy is dissipated as heat. An unconstrained ideal chain does not shrink to a point: its mean vector is zero but its root mean square end-to-end length is $b\sqrt{N}\simeq 2.37\mathrm{nm}$ for the supplied ideal-chain parameters.

A real polypeptide also has sequence-dependent interactions. If the original native structure is the stable state in the prevailing solvent and the molecule remains intact and capable of folding, it can subsequently refold, gaining favourable intramolecular and solvent free energy. The folded state is not selected by chain entropy alone, and formation of native contacts need not be instantaneous. Kinetic traps, proline isomerization, protein aggregation, misfolding, surface attachments or chemical damage may prevent complete recovery or produce hysteresis. Thus reversible refolding is a physically motivated possibility, not guaranteed simply by removal of force. This distinction connects the recoil of a stretched polymer to the thermodynamics discussed in Question 6(c).

If the ideal-chain segment count and segment length remain fixed, the entropic spring constant is proportional to absolute temperature. ThereforeThis is an increase by about $11.7\%$. The factor $k_{B}T$ multiplies the free energy penalty for the loss of chain configurations; a larger force is required to impose the same normalized weak extension at higher temperature.

Not degrading does not guarantee that a real protein's folded state, effective spring constant or solvent interactions remain unchanged. Thermal unfolding and changes in persistence length can occur without covalent degradation. Those changes modify the experimental curve and may invalidate a fixed-$N,b$ comparison. The boxed value is the explicit ideal-FJC prediction under fixed segment geometry, not an unconditional thermal-stability prediction for the protein.

The coefficient in the linear force law isStretching suppresses the many disordered orientations available to the chain. The Gaussian chain free energy cost $\Delta \mathcal{F}\simeq 3k_{B}T{R}^2/(2N{b}^2)$ has gradient equal to this restoring force. This is entropic elasticity; individual covalent bonds need not stretch to produce the weak-force spring.

At moderate force, even an ideal FJC becomes nonlinear as its orientations align, so failure of a linear fit alone need not imply nonideal chemistry. But failure of the full FJC curve can reflect backbone spring constant and correlated segment angles, nonuniform effective links, remaining folded contacts, force-induced conformational changes, energetic bond compliance, or tether/calibration effects. An extensible FJC or an appropriate semiflexible model includes some of these effects; their parameters require independent justification. The stated inextensible $14\mathrm{nm}$ polymer contour length already excludes a literal $15\mathrm{nm}$ extension.

The first panel of the original model figure below shows the full orientational response and weak-force tangents without tracing the experimental data.

Let a segment of fixed length $b$ make angle $\theta$ with an applied force $\mathbf{f}=f\hat{\mathbf{z}}$. Its energy contribution is $-fb\cos \theta$. Independent freely jointed orientations give the single-segment partition function and chain partition functionThe mean end-to-end projection is the force derivative of $k_{B}T\ln Z_N$:For $|u|\ll 1$, the Langevin function is $u/3+O(u^3)$. Isotropy therefore gives the vector responseThe covariance at zero force provides an independent route: $⟨R_i{R}_j{⟩}_0=N{b}^2{\delta }_{ij}/3$, and the Boltzmann weight's first-order term yields $⟨R_i⟩=f_j⟨R_i{R}_j{⟩}_0/(k_{B}T)$. This is a relation for the mean vector or its projection. The zero-force mean vector vanishes even though the chain has finite root mean square size $b\sqrt{N}$.

The controlled weak-force condition is $fb\ll k_{B}T$, not merely that a short part of a measured curve looks straight. With the supplied $b$, $k_{B}T/b\simeq 10.3\mathrm{pN}$ at $25^{\circ }\mathrm{C}$. A raw force near $30\mathrm{pN}$, as in the figure's labelled low-extension portion, would give $u\simeq 2.9$ and is not asymptotically weak for those literal segment parameters. A force offset or different effective model might change that interpretation, but no such correction should be invented from the plot.

The AFM pulls a tethered molecule through its cantilever. Deflection, multiplied by a calibrated cantilever spring constant, measures force; support motion must be corrected for that deflection and any linker compliance to obtain molecular extension. Initially folded structure can resist extension, then contacts break and more polypeptide polymer contour length becomes available. The readable trace rises with increasing extension, with fluctuations and a strongly rising large-extension part. It should not be interpreted as a direct image of individual amino acids or as establishing a unique microscopic unfolding pathway.

An ideal inextensible freely jointed chain with the stated segment count and length hasIts mean extension is strictly below $14\mathrm{nm}$ at every finite force. Thus the quoted $15\mathrm{nm}$ maximum does not establish ideal FJC behaviour: if both numbers are taken literally, it exceeds the model's polymer contour length by about $7\%$. Finite segment stretch, inaccurate effective segment parameters, additional linkers or an extension-zero/calibration error can account for such a difference. If the numerical descriptions are approximate, the discrepancy may be within their uncertainty; even exact agreement with a polymer contour length would not prove freely jointed statistics. Low-force fluctuations and the full force–extension relation are separate tests.

The source potential rises from zero to $U_0$ over the half-turn from $s$ to $s+\pi$, then falls over the next half-turn. Its slopes are $\pm U_0/\pi$. Torque is minus the angular derivative of potential; in the detached state $U_B$ is angle-independent and exerts no torque. A uniform set of motor phases therefore givesHere $U_0$ is an energy and $T_0$ a torque because the angle is in radians. The average over a full turn divides by $2\pi$, rather than by the bound-state occupation alone.

Differentiating the moving integration limits and using periodicity givesThe optimal phase of a two-state rotary motor follows on $0<s<\pi$: $P_A$ decreases, so the stationary point is a maximum and satisfiesThe crossing displaced by $\pi$ is the corresponding minimum. For slow positive rotation, $s_{\max }\simeq (\omega /k_0)\ln 2$ lies just after the detachment switch. The potential then rises mainly while the motor is detached and falls mainly while it is bound. For fast rotation, $s_{\max }\simeq \pi /2-{\pi }^2{k}_0/(8\omega )$, with only a small phase-dependent occupation available to produce torque. The potential should still be shifted by this optimum, rather than aligned arbitrarily with a rate switch.

The half-turn symmetry gives ${\int }_s^{s+2\pi }{P}_{A}d\theta =\pi$. At the optimum, direct integration of the two exponential pieces gives ${\int }_s^{s+\pi }{P}_{A}d\theta =s_{\max }$. Thus a useful exact check isFor fast rotation, ${\overline{T}}_{\max }\simeq T_0\pi k_0/(8\omega )$ instead tends to zero. The original plots below show the calculated probabilities and optimized potential shapes. The imposed switching rates constitute driven kinetics: their strictly one-way windows are not equilibrium detailed balance at a fixed angle. Sustained work requires the motor's chemical/ion-gradient energy supply, not extraction of work from a passive single thermal bath.

For the driven two-state rotary motor, use positive angular velocity $\omega >0$ and extend all angular functions with period $2\pi$. In the first half-turn the source figure has $k_{AB}=k_0$, $k_{BA}=0$, so the stationary equation is $\omega P_A^{\prime }=-k_0{P}_A$. In the second half-turn the rates are interchanged, giving $\omega P_A^{\prime }=k_0(1-P_A)$. HenceContinuity at $\pi$ gives $1-C=\alpha e^{-\beta \pi }$, while periodicity gives $\alpha =1-C{e}^{-\beta \pi }$. Solving the two linear conditions gives $C=\alpha$ andIn particular $P_A(0)=P_A(2\pi )=\alpha$ and $P_A(\pi )=1-\alpha$, so the two pieces fill in the apparent endpoint gap in the printed piecewise formula. The probability decreases exponentially during the detachment half-cycle and rises exponentially during the attachment half-cycle; $P_B$ is its complement. Both remain between zero and one. They are state probabilities conditional on rotor angle, not separate angular densities normalized individually to one.

For $\omega \ll k_0$, attachment and detachment almost finish within their allotted half-turns: $P_A$ is nearly zero through most of $0<\theta <\pi$ and nearly one through most of $\pi <\theta <2\pi$, with angular boundary layers of width $\omega /k_0$. For $\omega \gg k_0$, it stays near one half with a small nearly triangular modulation. The upper panels of the original figure in part (b) sketch both limits. At exactly zero speed the advective stationary problem is singular; these statements concern the positive-speed limit.

A point mutation changes a nucleotide at one genomic position. Replication errors and chemical damage can generate substitutions; repair and other proofreading processes influence the retained mutation rate. A coding change may be synonymous or alter an amino acid, while a noncoding change can affect regulation. Its evolutionary effect depends on context, so mutation, functional consequence and eventual fixation are distinct events.

Mutation supplies new variants, while genetic drift, selection and population history determine their persistence. Under a simple neutral model, increasing population size raises the number of new variants but lowers each one's fixation probability; the neutral substitution rate is approximately the mutation rate. Selective constraint, generation time and lineage-specific processes can therefore complicate a universal molecular clock. Molecular evolution is not directed towards anticipated needs.

To infer a phylogeny, compare homologous positions in appropriately chosen orthologous sequences and estimate the tree and branch lengths under a substitution model. Shared derived characters support common ancestry, but reversals and parallel substitutions mean raw mismatches underestimate evolutionary distance. In a simple equal-rates four-state illustration, a mismatch fraction $p$ gives corrected pairwise distance $d=-(3/4)\log (1-4p/3)$. The formula's strong divergence as $p\to 3/4$ reflects saturation and loss of reliable historical information, not arbitrarily precise evidence for an ancient split.

The molecular-clock calibration and rate-time identifiability issue is fundamental: branch lengths first measure expected substitutions per site. A clock relates them to time by $d_i=r_i{t}_i$; for two equal-rate lineages separated for time $t$, pairwise distance is $d=2rt$, so $t=d/(2r)$. Absolute times require external information, such as independently dated fossils or geological constraints, or informative sampling dates when the data contain a temporal signal. Sequence differences alone generally identify only rate-times-time, not both quantities separately.

Time-calibrated inference should propagate uncertainty in topology, substitution rates, ancestral-state reconstruction and calibration bounds. Relaxed clocks permit different lineage rates. Fossil minima are not automatically exact split dates; orthology mistakes, incomplete lineage sorting, recombination, horizontal transfer and selection can cause discordance between one gene tree and species history. Multiple independent loci and well-justified calibrations make a stronger historical argument than a single uncalibrated sequence comparison. No unique numerical tree or date can be constructed here without actual sequence observations and calibration data.

Usable biological free energy has several interconvertible forms. At approximately fixed temperature and pressure, chemical reactions are described by chemical potentials, ${\mu }_i={\mu }_i^{\circ }+RT\log a_i$, and ${\Delta }_{r}G={\sum }_i{\nu }_i{\mu }_i$. ATP hydrolysis, for example, has ${\Delta }_{r}G={\Delta }_r{G}^{\circ \prime }+RT\log (a_{\mathrm{ADP}}{a}_{\mathrm{Pi}}/a_{\mathrm{ATP}})$ under the specified biochemical standard-state convention. The activities and proton/ion convention matter. Breaking a bond alone costs energy; the net free-energy change includes the products, solvent and all bond changes, not a store released simply by snapping one bond.

Redox reactions provide free energy when electrons move between donors and acceptors with a favourable potential difference: $\Delta G=-nF\Delta E$. Electron-transfer chains can couple that drop to ion transport, building an electrochemical gradient. For transfer from an outer to an inner reservoir,Both concentration difference and electrical potential contribute. ATP synthase couples downhill ion flow to nucleotide free energy; driven pumps can perform the reverse conversion and maintain gradients used by transporters and excitability.

Mechanical work appears as $Fdx$ or torque times angle, and can be stored elastically, for example $\kappa x^2/2$. Molecular motors convert chemical free energy into movement or force; mechanically driven reactions can conversely reshape chemical transitions. Electrical energy can also be stored in membrane capacitance, while bulk ionic concentration gradients store additional chemical free energy. Light supplies photon energy $h\nu$, which photosynthetic machinery can convert into redox and chemical potential differences.

Coupling is essential: an overall favourable reaction can drive an unfavourable one only when their processes are linked, rather than merely present in the same cell. At fixed conditions the extracted work is bounded by the consumed free-energy decrease. Dissipation produces heat and entropy, so energy conservation does not mean all energy remains available for work. Thermal fluctuations can drive random motion, but a persistent directional cycle requires a nonequilibrium free-energy source; an equilibrium asymmetric structure alone is not a free motor.

A single fluorophore creates a diffraction-broadened point-spread function, but its centre can be fitted much more precisely than that function's width. With Gaussian width $s$ and $N_{\gamma }$ detected independent photons, the shot-noise limit is ${\sigma }_{\mathrm{loc}}\simeq s/\sqrt{N_{\gamma }}$. Pixelation, background photons, dye orientation, motion blur, drift and label-to-target geometry impose additional uncertainty and systematic error. Localization precision is therefore not identical to either optical resolution or position accuracy.^[ref]

In myosin-V tracking, localizing a label on one head or its lever arm distinguishes the paths of individual motor parts from the average motion of the whole motor. A trailing head that becomes the leading head travels roughly twice the stalk advance: about $74\mathrm{nm}$ versus a roughly $37\mathrm{nm}$ motor stride. A near-head label can therefore alternate a large movement with a pause while the partner head advances. Other lever-arm label positions give unequal alternating steps whose sum is approximately twice the stride. Those patterns support hand-over-hand walking rather than a simple inchworm model in which the same head remains in front. The label geometry is needed when interpreting individual step sizes.^[ref]

Super-resolution localization microscopy extends the same principle to a map containing many molecules by separating otherwise overlapping emitters in time. Sparse activation, blinking or transient binding yields individually fitted positions, which are accumulated. Dense sampling, registration and labelling specificity determine the final structural resolution; merely taking more photons from an unresolved simultaneous pair does not make it a single isolated emitter.

In PAINT, fluorescent probes bind transiently so that separated binding events can be localized and accumulated. DNA-PAINT uses reversible complementary-strand recognition at docking sites on a DNA nanostructure. The bound probe supplies a localized spot while freely diffusing probes contribute background; repeated visits build up a positional map, with replenishment rather than permanent reliance on one fluorophore. This is an imaging principle, not a requirement for a particular target sequence or preparation protocol.^[ref]

A conceptual single-molecule measurement follows a labelled bound molecule along an independently mapped DNA contour. Project fitted positions onto contour distance and estimate the mean-squared displacement at several lag times. For unbiased one-dimensional diffusion,Independent localization errors of variance ${\sigma }_{\mathrm{loc}}^2$ add an intercept $2{\sigma }_{\mathrm{loc}}^2$. Finite exposure adds motion blur, and drift, binding transitions and the finite DNA contour must be distinguished from free sliding. A slope fit with those limitations can estimate $D_1$; simply fitting all observed displacements to a noiseless line can bias it.

Retain the printed coefficient. The original PDF really prints $D_1=5\times 10^{10}{\mathrm{m}}^2{\mathrm{s}}^{-1}$, not a negative exponent. In sliding diffusivity with per-step dissociation, for a basepair step $a\simeq 0.34\mathrm{nm}$, an unbiased nearest-neighbour walk has step interval $a^2/(2D_1)$. If the dissociation probability per step is $P$, the mean number of steps is approximately $1/P$, givingFor the printed values, ${\tau }_{\mathrm{bound}}\simeq 1.16\times 10^{-24}\mathrm{s}$ and ${\ell }_{\mathrm{rms}}\simeq 0.34\mu \mathrm{m}$. The latter is a random-walk displacement, not the contour distance travelled in a million steps. The step interval is about $1.16\times 10^{-30}\mathrm{s}$, even shorter than the light-travel time across a basepair by many orders of magnitude. This is an unphysical extrapolation for a protein, not a meaningful experimentally measurable diffusivity.

Consequently no fluorescence camera could temporally resolve trajectories under the stated formal values, regardless of the spatial excursion's length. An actual corrected exponent cannot be inferred uniquely from the paper. If a physically plausible $D_1$ were supplied instead, resolving sliding would require exposures substantially shorter than ${\tau }_{\mathrm{bound}}$ and diffusion displacements larger than localization error, with enough detected photons. The dependence on $D_1$ is explicit above, so no guessed source correction is needed.

Write $A=V/D_3$ and $B=L/D_1$. The given mean-search scaling is $\tau =A/\lambda +B\lambda$. Three-dimensional relocation sees an effective capture length of order $\lambda$, giving a bulk encounter time $V/(D_3\lambda )$. A sliding excursion explores length of order $\lambda$ in time of order ${\lambda }^2/D_1$; sampling order $L/\lambda$ contour segments gives a total sliding cost of order $L\lambda /D_1$. Prefactors, correlations and repeated visits are coarse-grained in this scaling model.

Small sliding lengths make relocation inefficient; very long sliding wastes time revisiting a local DNA region rather than relocating to another. The optimum balances both contributions:Since $d^2\tau /d{\lambda }^2=2V/(D_3{\lambda }^3)>0$, it is the unique minimum for unrestricted positive $\lambda$. At the optimum, bulk and sliding times are equal. With $s=\lambda /{\lambda }_{\ast }$,

The model is meaningful only over an accessible range between molecular steps and the available chromosome length. An optimum outside that range is replaced by a boundary choice; the unphysical divergences at arbitrarily small or large $\lambda$ describe limits of the scaling expression. A common multiplicative searcher-number factor would not change the optimum, but an exact many-searcher first-arrival mean is not generally obtained by dividing an arbitrary single-searcher mean by their number without a waiting-time assumption.

Model the target as a perfectly absorbing sphere of reaction radius $r_0$, with a fixed target and a dilute far-field polymerase density $c_{\infty }$. Steady radial diffusion satisfies ${\nabla }^{2}c=0$, $c(r_0)=0$, $c(\infty )=c_{\infty }$, henceTherefore, for a rate law multiplying number densities,For the earliest diffusive target encounter among independent searchers, with $N_p$ independent well-mixed searchers in volume $V$, the approximate first-arrival rate is $N_p{k}_a/V$, giving ${\tau }_3\simeq V/(N_p{k}_a)$. The target's effective reaction radius includes encounter geometry and is not uniquely determined by its contour length.

A numerical estimate needs temperature, cell volume and a capture-geometry convention, none of which is fixed numerically in the source. State representative estimates $T=300\mathrm{K}$ and $V=1\mu {\mathrm{m}}^3$. The Stokes–Einstein estimate for a sphere of radius $15\mathrm{nm}$ in the stipulated viscosity isUsing the target contour scale $40\times 0.34\mathrm{nm}=13.6\mathrm{nm}$ as an order-of-magnitude $r_0$ gives $k_a\simeq 2.50\times 10^{-19}{\mathrm{m}}^3{\mathrm{s}}^{-1}$ andThis is an idealized $10^{-2}$-second-scale estimate, not a unique geometric prediction. Finite cell shape, crowded motion, accessibility and orientational restrictions can reduce the ideal capture rate.

If nonspecific binding merely sequesters searchers and their free fraction is $0.2$, the three-dimensional capture hazard is reduced fivefold: $\tau \simeq {\tau }_3/0.2\simeq 0.067\mathrm{s}$ for those estimates. If bound molecules instead search productively by sliding, a free-time fraction alone cannot determine the total arrival time or its sign of change. That is the distinct facilitated-search mechanism treated in c.

At initiation, RNA polymerase recognizes a promoter and forms a closed bound complex; bacterial promoter recognition involves its specificity factor. Local strand opening exposes the template in an open complex, after which initial RNA synthesis can precede promoter escape into elongation. Binding, opening and productive escape are distinct transitions.

For DNA unwinding measured by magnetic tweezers, magnetic tweezers report the mechanics of a torsionally constrained DNA tether through the position of an attached magnetic bead. Axial magnetic force extends the tether, while imposed rotation controls its linking constraint. Video localization measures extension changes. The important physical relation isOpening a small helical region decreases twist, so the rest of the tether acquires compensating positive writhe. On a negatively supercoiled tether this removes some writhe magnitude and raises its extension; on a positively supercoiled tether it increases writhe magnitude and lowers extension. A change with opposite signs under the two torsional conditions therefore distinguishes unwinding from simple compaction, which would shorten both tethers. Calibrating extension against imposed turns converts a mechanical step into a change in helical winding. This is a conceptual readout of local opening, rather than a direct optical image of separated bases.^[ref]

A two-state closed–open scheme at fixed voltage relaxes monotonically and cannot by itself produce a sodium-current transient during a maintained step with constant driving force. A minimal sodium scheme therefore includes an unavailable state: $C\rightleftarrows O\rightleftarrows I$, with voltage-dependent activation, inactivation and recovery; closed-state recovery pathways may be added when needed. Opening is initially rapid, occupancy then accumulates in $I$, and recovery on repolarization restores an activatable population.

For the illustrated potassium current, a closed–open model gives sustained opening, but several activation steps can account for its initial delay. A conventional illustrative coarse-grained description iswith each gate following $\dot{q}={\alpha }_q(V)(1-q)-{\beta }_q(V)q$. Fast $m$ activation, subsequent decrease of $h$, and slower increase of $n$ produce the two qualitative records. The powers are phenomenological gate assumptions, not values uniquely inferred from these traces or an exact structural count for every cardiac channel.

A 1-Hz cardiac rhythm has about a second between successive activations, whereas a 100-Hz neuronal rhythm has only $10\mathrm{ms}$; the available resting interval is shorter still after subtracting the action potential. Recovery or deactivation with a timescale comparable to that interval can then leave residual inactivation or potassium activation, giving use-dependent changes and refractoriness. Millisecond or faster relaxation usually completes much more readily during a long diastolic interval. The particular two-state A and B in d would both nearly equilibrate over a full $10\mathrm{ms}$ at their stipulated constant rates; their values alone do not demonstrate high-frequency failure. Real comparison requires voltage-dependent recovery rates and the different cardiac and neuronal waveforms. A frozen-rate model cannot quantitatively predict either firing pattern from frequency alone.

At fixed voltage let $a$ and $b$ be opening and closing rates. The open probability obeysThis illustrates equal open probability with unequal ion-channel relaxation times: both channels have $p_{\infty }=1/2$, but channel A has ${\tau }_A=1\mathrm{ms}$ and channel B has ${\tau }_B=0.1\mathrm{ms}$. Starting closed, $p_A(t)=\frac{1}{2}(1-e^{-t/(1\mathrm{ms})})$ and $p_B(t)=\frac{1}{2}(1-e^{-t/(0.1\mathrm{ms})})$. If channel number, unitary conductance and driving force agree, both have the same steady macroscopic current; B approaches it ten times faster.

An individual channel remains either open or closed. Its mean closed and open dwell times are $1/a$ and $1/b$, respectively: both are $2\mathrm{ms}$ in A and $0.2\mathrm{ms}$ in B. Thus identical equilibrium open probability does not imply identical kinetics or noise correlation time. Ensemble probability relaxation depends on the sum of rates, while the mean dwell in a given state depends on its exit rate.