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RNA polymerase first recognizes and binds a gene promoter to form a closed complex with the local DNA duplex still paired. Local strand separation then produces a transcription bubble in an open promoter complex, making the template available for RNA synthesis. Recognition, binding, and opening are distinct events; observing binding alone does not demonstrate strand opening.

In magnetic tweezers, a DNA tether joins a fixed surface to a magnetic bead, with attachments that prevent free swivelling. The magnets pull upward and can impose turns; the bead height reports tether extension. For a torsionally constrained tether, linking number is partitioned into twist and writhe, $Lk=Tw+Wr$. If opening the transcription bubble removes helical twist while $Lk$ is fixed, compensating twist or writhe appears elsewhere. In a supercoiled regime, changes in writhe change the extension, making an opening event visible as a bead-height step.

The sign provides an important control. On a positively supercoiled tether, local unwinding requires more positive compensating writhe and generally shortens the tether. On a negatively supercoiled tether, the same unwinding removes some negative writhe and generally lengthens it. A bending or compaction contribution can have the same sign in both cases, whereas the torsional contribution reverses. Comparing signs, together with an extension-versus-turns response measured for the tether, distinguishes local unwinding from mere binding-induced bending. These claims assume the tether remains torsionally constrained and lies in the calibrated supercoiled regime, rather than an arbitrary stretched-DNA regime.

The given facilitated target search time contains a rising term $L\lambda /D_1$ and a falling term $V/(D_3\lambda )$. Longer sliding explores more neighbouring DNA without another bulk encounter, reducing the cost of three-dimensional relocation; but it also spends longer revisiting local contour. Roughly $L/\lambda$ local scans, each taking time of order ${\lambda }^2/D_1$, give the first term. The second represents the bulk encounter cost for an effective capturable contour range of order $\lambda$, with a Smoluchowski association rate scaling as $D_3\lambda$. Numerical geometrical factors are already absorbed in the stipulated scaling formula.

Differentiating at fixed positive $L,V,D_1,D_3$ givesThe second derivative $2V/(D_3{\lambda }^3)>0$ proves the minimum is unique. The two contributions are equal there. The formal curve diverges at both zero and infinite $\lambda$; real sliding lengths are bounded, for example by the available contour, so an inaccessible optimum must be replaced by the nearest physical boundary. In dimensionless form, $\tau /{\tau }_{\min }=(u+u^{-1})/2$ with $u=\lambda /{\lambda }_{\ast }$. The right panel of the original illustration shows its minimum and the two contributing terms. The supplied expression does not include a factor of 6000; it should not silently be identified with the preceding earliest-encounter estimate.

A skeletal-muscle fibre contains parallel myofibrils assembled from repeating sarcomeres between Z discs. Thin actin filaments extend from the Z discs; bipolar thick myosin II filaments occupy the central region. Overlap between them allows many myosin heads to interact with actin in parallel.

A rise in cytosolic calcium exposes actin binding sites through the troponin–tropomyosin regulatory system. ATP binding releases a strongly attached myosin head; hydrolysis primes its lever arm, attachment and product release permit a force-generating stroke, and another ATP allows detachment. The cycle converts chemical free energy into mechanical work rather than consuming actin or shortening the filaments themselves.

Relative filament sliding draws the Z discs together. The I band and H zone shorten as overlap increases, whereas the thick-filament A-band length is approximately unchanged. Calcium removal restores inhibition and permits relaxation. The many-head ensemble sustains force even though individual cross-bridges repeatedly attach and detach. This sliding-filament description concerns myosin II in muscle; the processive two-head walking of Myosin V in a is a related but distinct motor arrangement.

In spatially gated myosin attachment, for a freshly available head encountering an isolated site, take $P=0$ before the attachment window. Converting time derivatives to position derivatives with $dx/dt=v$ gives $v{P}^{\prime }=k_{on}(1-P)$ in the binding window, $P^{\prime }=0$ in the protected bound region, and $v{P}^{\prime }=-k_{off}P$ after detachment becomes allowed. Continuity givesFor $v<k_{on}\Delta x$, the plateau exceeds $1-e^{-1}$ and approaches one for much slower motion. For $v>k_{on}\Delta x$, the binding probability is lower, becoming $k_{on}\Delta x/v$ at very high speed. The spatial detachment tail has length $v/k_{off}$, so faster sliding makes a longer low-probability tail. The following is an original calculation of those profiles; the force panel is derived in c.

Reset condition. Geometric separation $D\gg x_0$ does not alone guarantee detachment before the next site: independently available encounters also need $(D-x_0)k_{off}/v\gg 1$. Without that extra timescale separation the head may still be attached to an earlier site.

For completeness, the periodic steady-state correction can be written explicitly. Put $a=k_{on}/v$, $b=k_{off}/v$, $A_0=1-e^{-a\Delta x}$ andHere $Q_0$ is the probability of entering the new window still attached to an earlier site. In the window the new-site probability is $P(x)=1-e^{-ax}-Q_{0}a(e^{-bx}-e^{-ax})/(a-b)$; after it, use its value $p_b=P(\Delta x)$ in the plateau and tail above. For $a=b$, replace $a(e^{-bx}-e^{-ax})/(a-b)$ by $ax{e}^{-ax}$. This follows from the old-site probability $Q(x)=Q_0{e}^{-bx}$ and new attachment rate $a[1-P-Q]$. The isolated-site result is recovered as $Q_0\to 0$.

For the work of a spatially gated myosin head, the mechanical force exerted by an attached head in the increasing-$x$ direction is $f(x)=-dU/dx=\kappa (x_0-x)$. It assists motion before $x_0$ and resists it afterwards. For one site encounter the expected work isIn the narrow binding-window limit, retain the capture probability $p_b=1-e^{-k_{on}\Delta x/v}$ but neglect the attachment displacement compared with $x_0$. The positive work is $p_b\kappa x_0^2/2$. After $x_0$, use $s=x-x_0$ to obtain the negative contribution $-p_b\kappa {\int }_0^{\infty }s{e}^{-k_{off}s/v}ds=-p_b\kappa (v/k_{off}{)}^2$. HenceAn independent energy calculation gives the same factor: binding near zero loads energy $\kappa x_0^2/2$, while an exponentially distributed post-threshold attachment time has $⟨t^2⟩=2/k_{off}^2$, leaving mean elastic energy $\kappa v^2/k_{off}^2$ to be lost upon detachment. The factor of two in that second moment is essential.

One site passes every distance $D$, so average work per distance, or power divided by $v$, givesThe force tends to $\kappa x_0^2/(2D)$ as $v\downarrow 0$ and changes sign at $v=k_{off}{x}_0/\sqrt{2}$ in this idealized model. Above that speed forced sliding is resisted on average. The formula uses the reset condition in b, as well as $\Delta x\ll x_0$; if reset fails, replace $p_b$ by the periodic capture probability to leading order in $\Delta x/x_0$. For arbitrary window width, the isolated-site work is exactly ${\int }_0^{\Delta x}(k_{on}/v)e^{-k_{on}x/v}[\kappa (x_0-x{)}^2/2-\kappa v^2/k_{off}^2]dx$. The loaded spring energy must be supplied by the binding/power-stroke chemistry; this model does not specify an ATP efficiency. Real force-dependent transition rates and energy budgets limit extrapolation of the toy force-velocity law.

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.

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.

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.

The term $V/(D_3\lambda )$ is the bulk-diffusion search cost: increasing the effective length of DNA searched after each encounter enlarges the useful capture region and reduces this contribution. The term $L\lambda /D_1$ is the accumulated sliding cost. Approximately $L/\lambda$ DNA landing-and-sliding episodes, each taking order ${\lambda }^2/D_1$, give $L\lambda /D_1$. These are scaling estimates; geometric constants and correlated revisits are omitted by the supplied model.

Thus the facilitated target search is slow as $\lambda \to 0$, because too little sequence is examined per encounter, and slow at very large $\lambda$, because sliding and repeated local exploration take too long. A finite intermediate minimum balances 1D scanning against 3D relocation. The right panel of the figure in solution shows the two contributions and their sum. The model requires a useful range of sliding lengths below the full chromosome length, finite positive diffusion coefficients and effective relocation between episodes; it is not a claim that arbitrarily large $\lambda$ is physically available.