Dynamic Light Scattering (DLS): Hydrodynamic Diameter ($D_h$) & Polydispersity Index (PDI)
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.
DLS is exceptionally sensitive to dust because Rayleigh scattering intensity scales with particle diameter to the 6th power ($I propto d^6$). Centrifuge protein stock at $16,000 times g$ for 20 minutes at $4^circtext{C}$. Filter working buffer through a $0.02 mutext{m}$ syringe filter. Pipette $50 mutext{L}$ of $1.0text{ mg/mL}$ protein sample into dust-free cuvette using clean aerosol-barrier tips.
Insert cuvette into Zetasizer sample holder. Set temperature controller to precisely $25.0^circtext{C}$ and allow thermal equilibration for 120 seconds. Enter solvent parameters: Water viscosity $eta = 0.8872text{ cP}$, refractive index $n = 1.330$, protein refractive index $1.450$.
Collect 3 independent measurement runs, each consisting of 12 sub-runs of 10 seconds. The correlator accumulates raw photon count rates (target $200–400text{ kcounts/s}$). Inspect raw correlation curve: a clean exponential drop from $1.0$ to $0.0$ between $1 mutext{s}$ and $1000 mutext{s}$ confirms high data fidelity.
Analyze data via both Cumulants analysis (Z-average diameter and PDI) and the CONTIN regularized inverse Laplace transform algorithm to resolve multi-modal distributions (intensity, volume, and number distributions).
| Measurement Run | Translational D_t (cm²/s) | Z-Average Diameter D_h (nm) | PDI (Polydispersity) | % Volume (Monomer Peak) | Quality Criterion |
|---|---|---|---|---|---|
| Run 1 | 6.71 × 10−⁷ | 6.48 nm | 0.051 | 99.4% | Passed (Monodisperse) |
| Run 2 | 6.68 × 10−⁷ | 6.51 nm | 0.048 | 99.1% | Passed (Monodisperse) |
| Run 3 | 6.73 × 10−⁷ | 6.46 nm | 0.056 | 99.2% | Passed (Monodisperse) |
| Mean ± SD | 6.71 ± 0.02 × 10−⁷ | 6.48 ± 0.03 nm | 0.052 ± 0.004 | 99.2 ± 0.2% | Crystallography Grade |
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.
