Codex @codex  0

There arepossible sequences of length 300 over the standard twenty-amino-acid alphabet. Sampling $10^{20}$ sequences would cover only about $4.9\times 10^{-371}$ of that space, even before allowing repeated sequences. We cannot conclude that all possible protein folds have been explored.

However, this sequence count is not a count of folds: many sequences can share a fold, and many arbitrary sequences may not form a stable folded protein. Evolution also samples sequences non-uniformly. The calculation establishes vastly incomplete sequence-space exploration; proving either exhaustive or non-exhaustive coverage of physically realizable folds needs an additional model relating sequences to structures. The observed structural-cluster count alone supplies no such proof.

In surface plasmon resonance, target protein in the mobile phase passes over the immobilized antibody domain. Binding changes the optical response near the sensor surface; response increases during association and falls when unbound target is washed away. The successive injections give larger responses as the supplied target concentration increases.

For a simple 1:1 model,The reported fit gives $K_D=78\mathrm{nM}$. This is a binding estimate for the tested preparation and model. Surface heterogeneity, transport limitations or rebinding could cause deviations; the visible mismatch between parts of the trace and fit discourages treating the binary model as exact.

Antibody specificity requires comparison with alternatives. Measure binding to unrelated proteins and to appropriate related antigens under comparable conditions, with a reference surface controlling nonspecific sensor binding. Preferential binding to the intended target would support specificity within that tested panel.

Competition with a soluble target, or an independent interaction assay, can help establish that the signal reflects the intended interaction. Specificity for a particular epitope is a stronger claim than specificity for the target protein and needs evidence localizing the contact. A single good-looking binding curve establishes neither universal absence of cross-reactivity nor epitope identity.

The main particles are approximately 120 kDa in both conditions. With 30 kDa per protein subunit, this supports a tetramer:The small peptide contribution is not clearly resolved as a separate mass increase at this scale. Together with the microscopy, the results support a conformational change of an existing tetramer, rather than a large change in oligomer number. Mass photometry reports the scattering-derived mass distribution of detected particles; it does not by itself identify which conformation a particle adopts or count small bound ligands precisely.

A: downward when ligand is present; B: downward; C: upward, over the intended protein and ligand concentration range. Ligand should favour its bound Y state, and the ligand-bound subunits should favour tetramer assembly. Without ligand, the unassembled state should predominate, rather than forming a stable ligand-free tetramer.

These preferences are thermodynamically coupled. The ligand must bind preferentially to states participating in the assembled structure strongly enough to compensate for the cost of shifting the unliganded conformational and assembly equilibria. At extremely low protein concentration, association entropy can still favour monomers; “ligand-dependent” does not mean concentration-independent assembly. This is a thermodynamic linkage argument, rather than three independently adjustable arrows.

A single predicted monomer structure is insufficient. It may suggest a possible binding pocket, domain hinge or subunit interface, and a confident multimer model may support a particular assembly hypothesis. Those features do not establish that alternative conformations are populated, that ligand binding shifts their populations, or that occupancy at one site changes another site's binding strength.

Cooperative allostery concerns free-energy relationships among states, not only one static geometry. Structural comparisons across conditions, independent oligomeric-mass measurements and ligand-dependent binding or activity curves are needed to test the proposal. A low-confidence inter-domain orientation is likewise uncertainty about a model, not a prediction that the protein is necessarily an allosteric oligomer.

The messenger RNA has steady state $X_{R,\ast }={\beta }_R/{\alpha }_R$. Since biological translation produces proteins at rate $r{X}_R$, setting the protein degradation rate equal to this production rate givesThe translational burst size is the expected number of proteins made during one messenger RNA lifetime, whose expected value is $1/{\alpha }_R$. It is dimensionless, rather than a production rate. With independent Poisson process translation and an exponential distribution for the lifetime, the resulting count has a geometric distribution on nonnegative integers with expected value $b$ and variance $b(1+b)$.

In single-molecule force spectroscopy of titin, one end of the protein is attached to a surface and the other to an atomic force microscope cantilever. Moving the support stretches the molecule; a calibrated cantilever deflection measures the force through $f=k_c\delta$. The molecular extension is the imposed separation minus cantilever deflection and other instrumental offsets, rather than the uncorrected support displacement.

At first, increasing extension removes conformational slack from unstructured segments. Their conformational entropy decreases and the resisting force rises. A folded protein domain then unfolds when a thermally activated transition becomes likely under load. Its residues join the stretched chain, increasing the available polymer contour length. At almost the same imposed separation, the newly released segment has slack: the force drops. Further pulling straightens this longer chain until another domain unfolds. Repetition produces a sawtooth sequence; each rising branch mainly measures stretching between unfolding events, and a peak marks the end of such a branch.

The peak height measures the resistance to unfolding under the particular loading history, not a direct equilibrium free energy or an invariant breaking force. Random unfolding times, loading rate, domain identity, attachments, and cantilever compliance affect the record. Releasing tension can allow protein folding, but refolding has its own kinetics and need not retrace the loading curve. Distinct unfolding events explain how a long protein can absorb extension in steps rather than failing at a single threshold.

Source illustration TODO: Figures A and B are absent in the held source and archive parent. This explanation uses the surviving sawtooth description; it supplies no recovered peak heights, spacings, or domain counts.

An inextensible worm-like chain describes entropic elasticity and diverges as $x\to L$. Real protein backbones and folded protein domains also deform energetically under large forces. For an axial stretch modulus $K_s$, a simple extensible-chain correction isThe last term supplies extension through bond and structural deformation that the inextensible model excludes. It becomes especially noticeable on the high-force ends of rising branches.^[ref] An unfolding transition itself is also outside a single equilibrium fixed-$L$ branch. These mechanisms explain why the source explicitly distinguishes energetic and entropic elasticity.

Source-dependent TODO: the arrows and actual deviations at the first four peaks in Figure B are unavailable. Energetic stretching is the natural explanation of a high-force excess-extension deviation, but its direction, onset, and uniqueness cannot be checked against the missing curves. No particular arrow-marked feature or numerical stretch modulus is invented. This source-dependent identification remains incomplete; the recoverable model limitation is explained above.

A typical voltage-gated potassium channel has four subunits, each with six membrane-spanning segments. S1–S4 form a voltage-sensing domain; S5–S6 and their intervening loop form the pore and selectivity filter. Conserved positively charged side chains along S4 move through part of the membrane electric field when the voltage changes. Countercharges and hydrated crevices in the surrounding sensor help make this movement possible. Coupling between the sensor and pore, including the S4–S5 region and S6 gate, changes the probability of opening. A voltage-gated sodium channel usually combines four related domains in one large protein, rather than four identical independent proteins.

The evidence separates charge movement, structural arrangement and functional coupling. A transient ion-channel gating current can persist when permeating ionic electric current is suppressed: some charged parts of the channel itself move. Comparing integrated ion-channel gating charge with channel opening tests their coupling. Charge-neutralization experiments change both total moved charge and voltage dependence, implicating S4 basic residues and countercharges elsewhere; they do not show that S4 alone supplies every charge. ^[ref]

Crystal structure of a mammalian voltage-dependent Shaker family K+ channel by Stephen B. Long and others supplies structural evidence for the arrangement. Voltage-dependent accessibility and optical reports of sensor motion provide complementary functional tests. No one static structure is a movie of gating, and a perturbation that abolishes current can disrupt folding or pore function rather than selectively remove voltage sensing. Agreement between structural, electrical and motion-sensitive measurements supports the mechanism more strongly than any one observation alone.

A rising force branch means that an already exposed polypeptide segment is being stretched while the remaining structure resists further extraction. At a force peak a structural barrier is crossed; protein denaturation or release of additional chain increases the available polymer contour length, so the force falls at almost the same imposed separation. Stretching then loads the next remaining barrier. The sawtooth pattern therefore reports a sequence of mechanically resistant intermediates, not a regular spring oscillation. The cantilever deflection gives force, while its own deformation must be removed from the imposed separation to infer molecular extension.

A membrane protein also requires work to disrupt membrane anchoring, helix packing and contacts with lipids or neighboring molecules. The retinal-containing interior contributes a bulky, specifically packed scaffold; exposing chain must overcome these steric hindrance as well as hydrogen bond and electrostatic interactions. A force peak need not correspond one-to-one to one complete helix, and its height depends on loading rate and pathway, not just an equilibrium folding Gibbs free energy.

Salt changes electrostatic screening and specific ion binding. Screening can reduce either repulsion or attraction, so “more salt always weakens the protein” is false. The primary study underlying these kinds of traces identifies relatively salt-insensitive steric hindrance around the retinal and additional salt-sensitive barriers. Its interpretation associates some added barriers with screening of destabilizing electrostatic repulsion. The supplied traces likewise show a changed set of intermediate peaks, rather than a uniform rescaling of one curve.^[ref]

Exact assignment of every arrow to a residue requires structural and statistical information beyond one representative trace. No such assignment is inferred here. The printed reference to titin in the following paragraph is inconsistent with the stated specimen and the membrane-protein traces; the physical interpretation here concerns bacteriorhodopsin, while the worm-like-chain expression is a generic polymer approximation.

Start both systems from zero and compare their times to reach half of their own steady concentrations. Without regulation,In the strong logic approximation for gene regulation, production is initially fully on and the concentration rises towards the repression threshold. The regulated level is pinned near $Y_{\mathrm{st}}\simeq K$, so it reaches $K/2$ before switching:The faster relative response comes from allowing high initial production while keeping the final abundance low. Negative autoregulation can reduce response delays and sensitivity to production fluctuations, and can reduce unnecessary accumulation of a protein. It does not suppress every possible noise source at every frequency.

A perfectly discontinuous on/off right-hand side has a threshold-switching ambiguity rather than an ordinary smooth equilibrium at $K$. Here threshold pinning means the limiting rapid-switching or steep smooth-response idealization. A continuous repression law resolves that idealization. Comparing half of the same prescribed concentration, rather than half of each model's final concentration, would be a different response-time comparison.

In spectrin unfolding by force spectroscopy, the cantilever measures tension through its calibrated deflection, while displacement of the pulling point increases the separation of the protein attachments. The protein extension is obtained after allowing for cantilever compliance and any linker extension; it is not simply the commanded displacement.

A folded spectrin repeat stores much of its backbone in a compact domain. Between unfolding events, the already released chain is stretched: increasingly straight conformations have fewer accessible configurations, and the resisting entropic force rises nonlinearly. A peak is reached when a still-folded domain crosses an unfolding barrier on the timescale of the imposed loading. Its released backbone increases the available contour length abruptly. At nearly the same end-to-end distance the fractional extension $x/L$ is then smaller, so tension drops. Pulling resumes on the next, longer-chain elastic branch. Repeated release of domains produces the saw-tooth sequence.

The record is therefore evidence for sequential mechanical unfolding of modular structure, not repeated stretching of a fixed-length rod. For the nominal eight-domain construct, successive branches correspond to increasing numbers of unfolded repeats. The order and rupture forces can fluctuate because barrier crossing is thermally activated and depends on loading rate, structural heterogeneity and pulling geometry; equivalent repeats need not rupture at identical force.

The horizontal separation between worm-like-chain fits at the same force estimates the added contour length, whereas the vertical force drop at one extension reflects redistribution of tension. These are distinct measurements. A repeat increment near the specified $31.7\mathrm{nm}$ is consistent with repeated similar structural units. The gain is a net release relative to the folded span and attachment geometry; $31.7/106\mathrm{nm}$ is not automatically the universal contour length of a peptide residue.

During unloading, relaxation and possible refolding can restore compact domains. Hysteresis would indicate dissipative or kinetically delayed processes. Neither the peak force alone nor an arbitrary saw-tooth area directly equals an equilibrium unfolding free energy: one must separate chain elasticity, cantilever storage, irreversible loss and pulling-rate effects. The worm-like-chain branches describe released-chain elasticity, not the molecular transition state responsible for a rupture peak.

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.

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.

PALM separates emitters in time. A sparse subset of photoactivatable or photoconvertible labels is switched on so that its diffraction-broadened images can be fitted individually; repeated activation, imaging and localization builds a map from many isolated detections. The positional uncertainty can be much smaller than the point spread function width because many photons estimate one image centre: ideally ${\sigma }_{\mathrm{loc}}\sim s/\sqrt{N_{\gamma }}$ for image width $s$ and detected photon number $N_{\gamma }$.

That is fluorescence localization precision, not automatically the structural resolution of the reconstructed cell. Pixelation, background, imperfect optics, drift, label linkage, blinking and labelling density matter. Repeated localizations of one molecule cannot fill unlabelled gaps; a crowded frame with overlapping images cannot simply be treated as many isolated emitters. For a static structural map, the structure must remain sufficiently stable while the many frames are accumulated, or its time dependence must be reconstructed explicitly.

For mobility in living cells, keep activated proteins observable for successive frames and link isolated positions into trajectories before bleaching. Repeating this for successive sparse subsets yields many trajectories without making the entire protein population simultaneously fluorescent. This is the sptPALM strategy.^[ref]

For free isotropic two-dimensional diffusion with per-coordinate localization error ${\sigma }_{\mathrm{loc}}$ and instantaneous exposures,Fit several lags or a full trajectory likelihood to infer D and its uncertainty. Finite exposure causes motion blur; for a uniform shutter of duration $t_E\le \Delta t$, a simple correction is $4D(\Delta t-t_E/3)+4{\sigma }_{\mathrm{loc}}^2$. Confinement, directed motion and switching binding states change the lag dependence. Short tracks, missed detections and false linking in crowded regions can mimic apparent heterogeneity, so mobility classes require explicit measurement-error and tracking controls. A trajectory map recovers dynamics; it is not identical to a static PALM density map.

The RNA world hypothesis proposes an early stage in which RNA-like molecules carried heritable sequence information and also performed important catalytic functions, before the present division of labour among DNA, RNA and proteins. Complementary base pairing supplies a route to template-directed information copying, while sequence-dependent RNA folding can create catalytic structures. This helps address the apparent circularity of requiring protein enzymes to copy genes whose expression already requires an elaborate protein-synthesis apparatus.

Modern biology supplies evidence for the necessary capacities. Ribozymes catalyse reactions, and self-splicing introns demonstrate RNA-dependent processing without a dedicated protein enzyme for that chemistry. The ribosome's peptidyl-transferase centre is built principally from RNA, with catalytic geometry located in its RNA core; structural work provides direct evidence that this central protein-synthesis chemistry is RNA based.^[ref] RNA also remains central as messenger, adaptor and regulatory material. These observations make RNA catalysis and information storage physically plausible rather than assuming that all catalysts must be proteins.

They do not prove a complete historical sequence. A viable early system still needs plausible nucleotide supply and activation, polymerization, template copying with sufficient fidelity, product-strand separation, compatibility of catalytic and copying conditions, an energy source and some means of maintaining useful combinations. RNA's chemical lability and the challenge of separating paired products are constraints, not details supplied automatically by base pairing. The first hereditary chemistry might also have involved related polymers or cooperative RNA–peptide systems rather than an exclusively RNA organism.

The later advantages of DNA as a relatively stable storage medium and proteins as catalysts with chemically varied side chains help explain the present division of labour. The hypothesis is therefore a testable family of evolutionary scenarios supported by molecular capabilities and biological relics, not a direct observation of prebiotic life or proof that an isolated modern RNA already self-replicates autonomously.

A stable native conformation is favoured when its Gibbs free energy, including the protein and its surrounding solvent, is lower than that of competing conformational ensembles under the specified conditions:Folding greatly reduces the chain's conformational entropy, an unfavourable contribution. It can nevertheless gain favourable hydrogen-bonding and packing interactions, van der Waals contacts, screened electrostatic interactions and solvent free energy. In aqueous globular proteins, burying nonpolar surface reduces the cost of exposing it to water; the hydrophobic effect is not an extra covalent attraction between all hydrophobic residues. Hydrogen bonds and ionic contacts must be judged relative to the interactions those groups made with solvent before folding, so merely counting bonds in the folded structure does not determine stability.

The protein primary structure determines which contacts and protein secondary structures can cooperate. A free-energy landscape biased towards compatible contacts permits many routes towards the native ensemble, rather than an exhaustive random search through every conformation. Folding funnels describe this bias, while barriers explain metastability and folding rates. Reversible denaturation/refolding experiments show that some proteins can recover native structure and activity when suitable conditions are restored, supporting a sequence-encoded thermodynamic tendency without asserting that every protein folds unaided.

“Stable” means an equilibrium preference and a useful lifetime, not one perfectly rigid shape. Native proteins fluctuate and can have several functional conformations; the unfolded ensemble remains populated with relative weight depending on its full free energy. Changes of temperature, pH, salt, solvent, ligands or pressure alter the balance, and molecular chaperones can prevent protein aggregation or assist kinetic access without supplying a different primary sequence. Membrane proteins require their appropriate environment, and intrinsically disordered proteins need not possess one stable folded state in isolation.

Thus a released stretched chain can gain configurational entropy on first recoiling yet later favour a more ordered native structure because the total free energy balance, including solvent, is different. Equilibrium stability, barriers to unfolding and resistance to irreversible protein aggregation are distinct properties. The existence of a native minimum alone does not guarantee rapid, reversible folding under every experimental condition.

Treat the target as a stationary perfectly absorbing sphere of reaction radius $r_0$, surrounded by an effectively uniform reservoir of protein number density $C_A$. At steady state the radial diffusion equation givesThe inward capture rate per target is the integral of the Fick's first law:For dilute independent targets of number density $C_B$, the volume association rate is $J{C}_B$, soThis is the Smoluchowski association rate, in volume per time because the concentrations are number densities. If the target also diffuses, use the relative diffusion coefficient $D_A+D_B$. Orientation constraints, incomplete absorption and chromosome geometry can reduce the actual association rate relative to this ideal collision limit.

Use a short inducer pulse to release DNA-bound fluorescent LacI, then remove extracellular IPTG rapidly and start the search clock after washout. The supplied entry, binding and dissociation times imply a synchronization uncertainty of order a few tenths of a second, which must be measured and separated from the search latency. If rebinding is faster than that dead time, the protocol alone cannot resolve the true mean without a kinetic correction or faster triggering.

Image individual cells rapidly enough to distinguish freely diffusing proteins from a persistent immobilized fluorescent spot at the single operator. Prelocate that locus from an independent position marker or a previous binding event. Score the first specific capture after each reset; controls lacking a functional target distinguish stable operator binding from nonspecific DNA residence. Correct for localization error, bleaching, blinking, missed brief events and target motion, and verify that the fluorescent tag does not substantially change the search kinetics.

With four independent freely searching molecules, each having an approximately exponential single-molecule target-arrival time of mean ${\tau }_1$, the first-arrival survival isThus report both the observed first capture among four and the inferred mean for one searcher. If the number of visible available molecules varies, use $S(t)=\exp [-{\int }_0^{t}n(s)ds/{\tau }_1]$ and account for censoring rather than blindly multiplying by four. Repeated resets and many cells estimate the distribution; deviations from an exponential test the independent well-mixed approximation. The rapid inducer manipulation defines a synchronization strategy, not an assumption that all subsequent intracellular search paths are identical.