Biophysical characterization requires strict optical cleanliness and instrument hygiene to prevent artifacts:

• Far-UV Quartz Cuvette Hygiene: Far-UV circular dichroism and UV-Vis spectrophotometry below 220 nm require synthetic suprasil quartz cuvettes (optical pathlength 0.1 mm to 10 mm). Never wipe optical faces with abrasive tissues; rinse thoroughly with ultrapure Milli-Q water (18.2 MΩ·cm) and soak periodically in 2% (v/v) Hellmanex III solution. Avoid strong base or chromic acid which etches quartz.
• High-Voltage Gel Safety: Electrophoretic systems operate at hazardous potentials (100–300 V, up to 100 mA). Never touch buffer tanks or leads during runs; ensure safety interlocking lids are engaged before applying current. In unpolymerized acrylamide prep, neurotoxic acrylamide monomer must always be weighed under a certified fume hood with nitrile gloves.
• Class 3B / 4 Laser Precautions: DLS and fluorescence instruments utilize monochromatic solid-state lasers (e.g., 632.8 nm He-Ne or 532 nm green diode lasers). Ensure the sample chamber interlock is fully closed prior to measurement to prevent stray specular reflections into the retina.
• Degassing & Microfiltration of SEC Mobile Phases: Chromatographic columns (Superdex 75/200 Increase) are irreversibly damaged by particulates or outgassing microbubbles. All buffers must be vacuum-filtered through 0.22 μm nylon or PES membranes and sonicated under vacuum for 15 minutes before pumping.

Absorption of near-ultraviolet light by proteins at $lambda = 280text{ nm}$ arises almost exclusively from electronic transitions within the conjugated aromatic side chains of Tryptophan ($text{Trp}, text{W}$) and Tyrosine ($text{Tyr}, text{Y}$) residues, with a minor contribution from disulfide bonds (cystine, $text{C-C}$). Phenylalanine exhibits absorption maxima at shorter wavelengths ($257text{ nm}$) with negligible molar absorptivity at $280text{ nm}$.

According to the Beer-Lambert Law, light absorbance ($A$) is directly proportional to the molar absorption coefficient ($epsilon$, $text{M}^{-1}text{cm}^{-1}$), the analyte molar concentration ($c$, $text{M}$), and the optical pathlength ($l$, $text{cm}$):
$$A_{280} = -log_{10}left(frac{I}{I_0}right) = epsilon_{280} cdot c cdot l$$

For any folded or denatured protein of known primary sequence, the theoretical molar extinction coefficient at $280text{ nm}$ is accurately predicted using the Pace / Edelhoch equation in $6.0text{ M Guanidine Hydrochloride}$:
$$epsilon_{280} = (N_{text{Trp}} times 5500) + (N_{text{Tyr}} times 1490) + (N_{text{Cystine}} times 125) quad (text{M}^{-1}text{cm}^{-1})$$

Nucleic Acid Purity Evaluation: Purine and pyrimidine bases in DNA and RNA absorb maximally at $260text{ nm}$ ($pi to pi^*$ transitions) with an extinction coefficient nearly 50-fold higher per unit mass than proteins. A pure protein solution exhibits an $A_{280}/A_{260}$ ratio of $1.75text{ to }1.85$. An $A_{280}/A_{260}$ ratio below $1.5$ indicates significant nucleotide or nucleic acid contamination, requiring baseline correction formulas (e.g., Warburg-Christian equation: $text{Protein (mg/mL)} = 1.55 cdot A_{280} – 0.76 cdot A_{260}$).

• Spectrophotometer: Shimadzu UV-2600 or Thermo Scientific Evolution 300 dual-beam UV-Vis spectrophotometer with deuterium ($190–350text{ nm}$) and tungsten-halogen lamps.
• Cuvettes: Precision matched synthetic suprasil quartz cuvettes ($1.000text{ cm} pm 0.005text{ cm}$ pathlength, $1.5text{ mL}$ semi-micro volume).
• Protein Standards: Highly purified lyophilized Bovine Serum Albumin (BSA, $ge 98%$, Fraction V, Sigma-Aldrich A7906, $M_r = 66,430text{ Da}$).
• Assay Buffer: $50text{ mM Sodium Phosphate}$, $150text{ mM NaCl}$, $text{pH } 7.4$ (PBS), vacuum-filtered through $0.22 mutext{m}$ PES membrane.
• Denaturing Buffer (for Edelhoch validation): $6.0text{ M Guanidinium Hydrochloride}$ in $20text{ mM Sodium Phosphate}$, $text{pH } 6.5$.

Regression Analysis: Linear regression of $A_{280,text{corr}}$ vs Molar Concentration yields:
$$A_{280} = (44,380text{ M}^{-1}text{cm}^{-1}) cdot c + 0.0003 quad (R^2 = 0.9998)$$

Extinction Coefficient Verification: For BSA ($2text{ Trp}, 20text{ Tyr}, 17text{ Disulfides}$), theoretical Pace $epsilon_{280} = (2 times 5500) + (20 times 1490) + (17 times 125) = 11,000 + 29,800 + 2,125 = mathbf{42,925text{ M}^{-1}text{cm}^{-1}}$ (or $E^{1%}_{1text{cm}} = 6.46$). The observed native $epsilon_{280} = 44,380text{ M}^{-1}text{cm}^{-1}$ reflects local hydrophobic burial and microenvironment dipole enhancement of Trp residues in native folded BSA.

In Sodium Dodecyl Sulfate Polyacrylamide Gel Electrophoresis (SDS-PAGE), the anionic detergent SDS binds to unfolded polypeptide backbones at a constant stoichiometric ratio of approximately $1.4text{ g SDS per 1.0 g protein}$ (roughly one SDS molecule per two amino acid residues). This imparts a uniform, high negative net charge density that overwhelms intrinsic protein charges, producing a constant charge-to-mass ratio ($q/m$).

Consequently, electrophoretic separation occurs solely on the basis of hydrodynamic sieving through the polyacrylamide gel pore matrix, governed by the Ferguson relationship:
$$log_{10}(R_f) = log_{10}(Y_0) – K_R cdot %T$$
where $R_f$ is relative mobility, $Y_0$ is free electrophoretic mobility, $K_R$ is the retardation coefficient (proportional to molecular weight), and $%T$ is total acrylamide concentration. In a fixed-percentage resolving gel ($12%T$), electrophoretic mobility is inversely proportional to the logarithm of molecular mass:
$$log_{10}(M_r) = -m cdot R_f + b$$
where relative mobility $R_f$ is defined as:
$$R_f = frac{text{Distance migrated by protein band (mm)}}{text{Distance migrated by Bromophenol Blue dye front (mm)}}$$

Isotachophoresis Stacking Mechanism: The discontinuous Laemmli system employs a two-gel architecture with differing $text{pH}$ and pore sizes:

1. Stacking Gel ($4%T$, $text{pH } 6.8$ Tris-HCl): Chloride ions ($text{Cl}^-$) possess high mobility and act as the leading ions. Glycine, whose isoelectric point is $text{pI } = 5.97$, exists predominantly as a zwitterion ($>99%$) with very low electrophoretic mobility at $text{pH } 6.8$, acting as the trailing ion. Protein-SDS complexes have intermediate mobilities and are compressed into an ultra-sharp ($<0.1text{ mm}$) starting band between the leading chloride and trailing glycine fronts.
2. Resolving Gel ($12%T$, $text{pH } 8.8$ Tris-HCl): At $text{pH } 8.8$, glycine fully ionizes into glycinate ($text{Gly}^-$), speeding past the proteins directly behind chloride. The unstacked proteins then resolve purely according to molecular sieve sieving.

• Electrophoresis Apparatus: Bio-Rad Mini-PROTEAN Tetra Cell with casting stands, $1.0text{ mm}$ glass plates, and 10-well combs. PowerPac Basic power supply.
• Acrylamide Stock: $30% (w/v)$ Acrylamide / Bis-acrylamide solution ($29:1$ crosslinking ratio).
• Gel Buffers: $1.5text{ M Tris-HCl}$ ($text{pH } 8.8$) for resolving gel; $0.5text{ M Tris-HCl}$ ($text{pH } 6.8$) for stacking gel.
• Running Buffer ($1times$): $25text{ mM Tris}$, $192text{ mM Glycine}$, $0.1% (w/v)text{ SDS}$, $text{pH } 8.3$.
• Sample Loading Buffer ($4times$): $200text{ mM Tris-HCl}$ ($text{pH } 6.8$), $8% (w/v)text{ SDS}$, $40% (v/v)text{ Glycerol}$, $20% (v/v) betatext{-mercaptoethanol}$, $0.04% (w/v)text{ Bromophenol Blue}$.
• Protein Molecular Weight Ladder: Broad-range prestained marker ($14.4–116.0text{ kDa}$: β-galactosidase $116.0text{ kDa}$, BSA $66.2text{ kDa}$, Ovalbumin $45.0text{ kDa}$, Carbonic Anhydrase $29.0text{ kDa}$, Lysozyme $14.4text{ kDa}$).
• Staining & Destaining Solutions: Coomassie Brilliant Blue R-250 ($0.1%$ in $40%$ methanol, $10%$ acetic acid); Destain: $20%$ methanol, $10%$ acetic acid.

Regression Equation:
$$log_{10}(M_r) = -1.350 cdot R_f + 5.281 quad (R^2 = 0.9984)$$

Calculation for Unknown:
$$log_{10}(M_r) = -1.350(0.431) + 5.281 = 4.699 implies M_r = 10^{4.699} = mathbf{50,020text{ Da}} approx mathbf{50.0text{ kDa}}$$

Conclusion: The calculated subunit molecular weight of $50.0text{ kDa}$ matches the predicted open reading frame translation of the target recombinant monomer with $<1.5%$ experimental error.

Size-Exclusion Chromatography (SEC), also termed gel filtration, separates biological macromolecules strictly according to their hydrodynamic volume (effective Stokes radius, $R_s$), rather than molecular weight alone. The stationary phase consists of spherical, highly porous beads with a controlled pore size distribution.

When a sample is pumped through the column:

1. Large macromolecules whose Stokes radius exceeds the bead pore diameter are completely excluded from the internal pore volume and elute first in the Void Volume ($V_0$).

2. Very small molecules (salts, acetone) diffuse freely into the entire solvent volume of the column, eluting last in the Total Liquid Volume ($V_t$).

3. Intermediate proteins partition partially into the pores; larger proteins elute before smaller ones.

The retention of a solute is quantitatively parameterized by the partition coefficient ($K_{av}$):
$$K_{av} = frac{V_e – V_0}{V_t – V_0}$$
where $V_e$ is the elution volume corresponding to the peak maximum. Over the fractionation range of the gel, $K_{av}$ displays an inverse linear relationship with $log_{10}(M_r)$ or Stokes radius ($R_s$):
$$K_{av} = -a cdot log_{10}(M_r) + b$$

Oligomeric State Determination: While SDS-PAGE yields the denatured subunit molecular weight ($M_{r,text{subunit}}$), SEC reveals the native quaternary structure ($M_{r,text{native}}$). Comparing $M_{r,text{native}} / M_{r,text{subunit}}$ distinguishes monomeric, homodimeric, or higher-order multimeric quaternary architectures in solution.

• Chromatography System: ÄKTA Pure 25 FPLC system equipped with UV monitor ($280text{ nm}$ and $254text{ nm}$), conductivity flow cell, and automated fraction collector.
• Column: Superdex 75 Increase 10/300 GL ($24text{ mL}$ bed volume, fractionation range $3,000–70,000text{ Da}$, Cytiva).
• Calibration Standards:
  • Blue Dextran 2000 ($M_r approx 2,000text{ kDa}$, $V_0$ void marker, $1.0text{ mg/mL}$)
  • Bovine Serum Albumin ($66.2text{ kDa}$, $R_s = 3.48text{ nm}$)
  • Ovalbumin ($44.3text{ kDa}$, $R_s = 3.05text{ nm}$)
  • Carbonic Anhydrase ($29.0text{ kDa}$, $R_s = 2.43text{ nm}$)
  • Ribonuclease A ($13.7text{ kDa}$, $R_s = 1.64text{ nm}$)
  • Acetone ($0.5% v/v$, $V_t$ total permeation marker)
• Running Buffer: $20text{ mM Sodium Phosphate}$, $150text{ mM NaCl}$, $text{pH } 7.2$, filtered ($0.22 mutext{m}$) and degassed for 20 min.

Calculation of Native Molecular Weight:
Total available pore volume: $V_t – V_0 = 23.90 – 7.85 = 16.05text{ mL}$.

For Target Protein: $K_{av} = frac{11.10 – 7.85}{16.05} = mathbf{0.2025}$.

Regression fit: $K_{av} = -0.605 cdot log_{10}(M_r) + 3.018 quad (R^2 = 0.9972)$.
$$log_{10}(M_{r,text{native}}) = frac{3.018 – 0.2025}{0.605} = 4.654 implies M_{r,text{native}} = mathbf{45,100text{ Da}} approx mathbf{45.1text{ kDa}}$$

Quaternary Structure Conclusion: SDS-PAGE revealed a denatured subunit mass of $22.5text{ kDa}$. The native SEC mass of $45.1text{ kDa}$ demonstrates that the active protein exists in solution as a stable, monodisperse homodimer ($2 times 22.5text{ kDa} = 45.0text{ kDa}$).

Circular Dichroism (CD) is the differential absorption of left- and right-circularly polarized light by optically active, chiral molecules:
$$Delta A = A_L – A_R = (epsilon_L – epsilon_R) cdot c cdot l = Delta epsilon cdot c cdot l$$
Because differential absorption transforms circularly polarized light into elliptically polarized light, CD is conventionally reported in terms of ellipticity ($theta$) in millidegrees ($text{mdeg}$), where $theta = 32.98 cdot Delta A$.

In proteins, peptide bonds ($-text{CONH}-$) are identical achiral chromophores, but when organized into regular secondary structures, chiral exciton coupling between peptide electronic transitions ($pi to pi^*$ at $sim 190text{ nm}$ and $n to pi^*$ at $sim 222text{ nm}$) generates unique, diagnostic Far-UV CD signatures:

• $alpha$-Helix: Characteristic double negative minima at $208text{ nm}$ ($pi to pi^*$ polarized parallel to helix axis) and $222text{ nm}$ ($n to pi^*$ transition), and a strong positive peak at $193text{ nm}$ ($pi to pi^*$ perpendicular).
• $beta$-Sheet: A single broad negative minimum at $218text{ nm}$ ($n to pi^*$) and a positive maximum at $195text{ nm}$.
• Random Coil / Unfolded: Strong negative band at $198text{ nm}$ and near-zero ellipticity above $220text{ nm}$.

To compare proteins independently of molecular weight, raw ellipticity ($theta_{text{obs}}$, mdeg) is normalized to Mean Residue Ellipticity ($[theta]_{text{MRE}}$, $text{deg}cdottext{cm}^2cdottext{dmol}^{-1}$):
$$[theta]_{text{MRE}} = frac{theta_{text{obs}} times 100}{c times l times N_{text{res}}}$$
where $c$ is protein molar concentration ($text{M}$), $l$ is optical pathlength ($text{cm}$), and $N_{text{res}}$ is total amino acid residue count. Percentage $alpha$-helical content can be estimated directly from $[theta]_{222}$:
$$% alphatext{-helix} = left(frac{-[theta]_{222} – 3,000}{36,000}right) times 100$$

• Spectropolarimeter: Jasco J-815 or Chirascan Far-UV spectropolarimeter equipped with Peltier temperature controller and continuous dry $text{N}_2$ gas purge ($5.0text{ L/min}$).
• Cuvette: Precision synthetic suprasil quartz demountable cuvette with $0.100text{ cm}$ ($1.0text{ mm}$) pathlength.
• Proteins: Equine Skeletal Muscle Myoglobin ($>95%$, predominantly $alpha$-helical) and Bovine Ribonuclease A.
• Far-UV Transparent Buffer: $10text{ mM Potassium Phosphate}$, $text{pH } 7.0$ (Phosphate buffers have minimal Far-UV absorbance down to $185text{ nm}$; chlorides, Tris, and DTT are strictly avoided as they absorb heavily below $210text{ nm}$).

Deconvolution & Secondary Structure Estimation:
Sample parameters: $c = 8.57 times 10^{-6}text{ M}$, $l = 0.1text{ cm}$, $N_{text{res}} = 154text{ residues}$.
$$[theta]_{222} = frac{-36.1 times 100}{8.57 times 10^{-6} times 0.1 times 154} = mathbf{-27,310text{ deg}cdottext{cm}^2cdottext{dmol}^{-1}}$$
$$% alphatext{-helix} = left(frac{27,310 – 3,000}{36,000}right) times 100 = mathbf{67.5%}$$

Thermal Denaturation Fit: Fitting the sigmoidal $[theta]_{222}$ transition vs temperature yields a two-state cooperative melting transition with $T_m = mathbf{78.4^circtext{C}}$ and unfolding enthalpy $Delta H_v = 412text{ kJ/mol}$, demonstrating high conformational stability.

Förster Resonance Energy Transfer (FRET) is a non-radiative, electrodynamic process whereby an excited donor fluorophore ($D$) transfers its excitation energy to an acceptor chromophore ($A$) through long-range dipole-dipole resonance coupling without photon emission or re-absorption.

FRET transfer occurs under three physical conditions:

1. Significant spectral overlap between donor fluorescence emission spectrum ($F_D(lambda)$) and acceptor absorption spectrum ($epsilon_A(lambda)$), quantified by the overlap integral $J(lambda)$:
$$J(lambda) = int_0^infty F_D(lambda) cdot epsilon_A(lambda) cdot lambda^4 , dlambda$$

2. Close spatial proximity ($1–10text{ nm}$) between donor and acceptor.

3. Favorable relative orientation of transition dipoles, described by the orientation factor $kappa^2$ (typically assumed to be $kappa^2 = 2/3$ for rapidly tumbling, isotropic fluorophores).

The FRET efficiency ($E$) exhibits an exquisite inverse sixth-power dependence on inter-fluorophore distance ($r$):
$$E = frac{R_0^6}{R_0^6 + r^6}$$
where $R_0$ is the Förster distance at which energy transfer efficiency is precisely $50%$ ($E = 0.50$):
$$R_0^6 = 8.79 times 10^{-5} cdot (kappa^2 cdot n^{-4} cdot Phi_D cdot J(lambda)) quad (text{in AA}^6)$$
where $n$ is solvent refractive index ($1.333$ for water) and $Phi_D$ is donor fluorescence quantum yield in the absence of acceptor.

Experimentally, efficiency $E$ is determined from donor fluorescence quenching in the presence ($F_{DA}$) and absence ($F_D$) of acceptor:
$$E = 1 – frac{F_{DA}}{F_D} = 1 – frac{tau_{DA}}{tau_D}$$
Because the slope of $E$ vs $r$ is steepest between $0.5 R_0$ and $1.5 R_0$, FRET functions as an ultra-sensitive “Spectroscopic Molecular Ruler” capable of measuring conformational changes on the scale of $0.1text{ to }0.2text{ nm}$ ($1–2 text{AA}$).

• Spectrofluorometer: Horiba Fluorolog-3 or PTI QuantaMaster spectrofluorometer with double excitation/emission monochromators and temperature-controlled cuvette holder ($25^circtext{C}$).
• Cuvette: 4-window optical quartz fluorescence cuvette ($10.0text{ mm} times 10.0text{ mm}$, $1.0text{ mL}$).
• Recombinant FRET Constructs:
  • Construct 1: Uncoupled Donor alone (mCerulean3 CFP, $lambda_{text{ex}} = 433text{ nm}$, $lambda_{text{em}} = 475text{ nm}$)
  • Construct 2: FRET Pair with short 6-amino acid linker ($text{CFP}-text{(GS)}_3-text{YFP}$, Expected $r approx 3.2text{ nm}$)
  • Construct 3: FRET Pair with medium 16-amino acid linker ($text{CFP}-text{(GS)}_8-text{YFP}$, Expected $r approx 4.8text{ nm}$)
  • Construct 4: FRET Pair with extended 32-amino acid linker ($text{CFP}-text{(GS)}_{16}-text{YFP}$, Expected $r approx 7.2text{ nm}$)
• Buffer: $20text{ mM Tris-HCl}$, $150text{ mM NaCl}$, $text{pH } 7.5$, sterile-filtered.

Mathematical Distance Calculation:
For the CFP/YFP pair, experimental calibration yields Förster radius $R_0 = mathbf{4.92text{ nm}}$ ($49.2 text{AA}$).

For Construct 2 ($16text{ aa}$ linker):
$$E = 1 – frac{F_{DA}}{F_D} = 1 – frac{47,200}{100,000} = mathbf{0.528} quad (52.8%)$$
Solving for distance $r$ using the rearranged Förster equation:
$$r = R_0 cdot left(frac{1 – E}{E}right)^{1/6} = 4.92 cdot left(frac{1 – 0.528}{0.528}right)^{1/6} = 4.92 cdot (0.8939)^{0.1667} = mathbf{4.88text{ nm}} quad (48.8 text{AA})$$

Biophysical Conclusion: Because $52.8%$ efficiency lies almost precisely at the $50%$ inflection midpoint ($r approx R_0$), this geometry provides maximal sensitivity to conformational movements of $< 0.15text{ nm}$.

Dynamic Light Scattering (DLS), or Quasi-Elastic Light Scattering (QELS), measures the rate of diffusion of macromolecules undergoing continuous, random Brownian motion in liquid suspension. When illuminated by monochromatic, coherent laser light ($lambda = 633text{ nm}$), particles scatter light that interferes constructively and destructively at a high-speed detector (Avalanche Photodiode, APD).

Because smaller particles move faster than larger particles, their scattered light intensity fluctuates on a much shorter timescale. These temporal fluctuations are quantified by the normalized second-order intensity autocorrelation function ($g^{(2)}(tau)$):
$$g^{(2)}(tau) = frac{langle I(t) cdot I(t + tau) rangle}{langle I(t) rangle^2}$$
where $tau$ is the decay delay (lag) time. Via the Siegert relation, $g^{(2)}(tau)$ relates to the electric field correlation function $g^{(1)}(tau)$:
$$g^{(2)}(tau) = 1 + beta |g^{(1)}(tau)|^2 = 1 + beta exp(-2Gamma tau)$$
where $beta$ is instrument coherence factor and $Gamma$ is the characteristic decay rate:
$$Gamma = D_t cdot q^2$$
where $D_t$ is the translational diffusion coefficient and $q$ is the magnitude of the scattering vector ($q = frac{4pi n}{lambda_0} sin(theta/2)$).

From $D_t$, the hydrodynamic diameter ($D_h$) is calculated via the fundamental Stokes-Einstein relation:
$$D_h = frac{k_B cdot T}{3pi cdot eta cdot D_t}$$
where $k_B$ is the Boltzmann constant ($1.3806 times 10^{-23}text{ J/K}$), $T$ is absolute temperature ($298.15text{ K}$), and $eta$ is solvent dynamic viscosity ($0.8872text{ cP} = 8.872 times 10^{-4}text{ Pa}cdottext{s}$ for water at $25^circtext{C}$).

Polydispersity Index (PDI): Cumulants analysis fits $ln[g^{(1)}(tau)]$ to a polynomial series: $ln[g^{(1)}(tau)] = -bar{Gamma}tau + frac{mu_2}{2}tau^2$. The dimensionless Polydispersity Index ($text{PDI} = mu_2 / bar{Gamma}^2$) measures particle size heterogeneity:

• $text{PDI} < 0.08$: Monodisperse, high purity (suitable for protein crystallization, Cryo-EM, and NMR).
• $0.08 le text{PDI} le 0.25$: Narrow moderately polydisperse distribution.
• $text{PDI} > 0.30$: Broadly polydisperse, aggregate contamination.

• Dynamic Light Scattering Instrument: Malvern Zetasizer Nano-ZS or DynaPro Plate Reader III ($632.8text{ nm}$ He-Ne laser, $173^circ$ Non-Invasive Backscatter NIBS optics).
• Cuvette: Low-volume disposable polystyrene micro-cuvette ($40 mutext{L}$) or optical quartz square cell ($45 mutext{L}$, Zen2112).
• Protein Samples: Native Bovine Serum Albumin (BSA, $M_r = 66.4text{ kDa}$) and Monoclonal IgG antibody ($M_r = 150text{ kDa}$).
• Ultra-Clean Buffer: $20text{ mM Phosphate-Buffered Saline}$ ($text{pH } 7.4$), filtered sequentially through $0.22 mutext{m}$ and $0.02 mutext{m}$ Anotop syringe filters to eliminate particulate dust.

Stokes-Einstein Calculation:
$$D_h = frac{(1.3806 times 10^{-23}text{ J/K}) times (298.15text{ K})}{3 times 3.14159 times (8.872 times 10^{-4}text{ Pa}cdottext{s}) times (6.707 times 10^{-11}text{ m}^2/text{s})} = mathbf{6.48 times 10^{-9}text{ m}} = mathbf{6.48text{ nm}}$$

Stokes Radius ($R_s$): $R_s = D_h / 2 = mathbf{3.24text{ nm}}$, in excellent agreement with size exclusion chromatography ($3.12text{ nm}$).

Structural Sample Conclusion: With a Polydispersity Index of $0.052$ ($< 0.08$) and $>99%$ monomer volume fraction, the preparation is monodisperse, aggregate-free, and suitable for structural determination via X-ray crystallography and Cryo-EM.