Surface Model¶
Nanoparticle Geometry¶
The nanoparticle (NP) is modeled as a sphere of radius R whose surface is populated by discrete beads representing surface moieties (silanol groups on silica, thiol groups on gold, etc.).
The protein interacts with these surface beads rather than with a continuous analytical surface. This allows localized interactions — some orientations place reactive residues directly above surface beads while others do not — which is physically important for understanding orientation selectivity.
Fibonacci Sphere Bead Placement¶
Bead positions are generated by the Fibonacci sphere algorithm, which distributes N points nearly uniformly on a sphere without poles, clustering, or latitude artifacts.
The algorithm generates the i-th point (0-indexed) as:
where \(\varphi = (1+\sqrt{5})/2 \approx 1.618\) is the golden ratio and \(\theta_i\) is the azimuthal angle. The (x, y) components are:
Each point is then scaled to radius R.
Why Fibonacci Sphere?¶
Alternative approaches (uniform latitude grids, random sampling) produce either polar clustering or non-uniform density. The Fibonacci spiral gives a near-optimal uniform distribution with the minimum pairwise distance between any two points growing as \(\sim 1/\sqrt{N}\) — ensuring consistent bead density regardless of viewing direction.
Bead Density¶
The number of surface beads N is chosen to give approximately 1 bead/nm²:
For R = 71.3 nm: N ≈ 63,800 beads. This density correctly models a silica surface with ~4.7 Si-OH groups/nm² — only one-fifth of those are charged at pH 7.4, but the bead density is set to match the spacing of surface moieties.
Surface Charge: The Grahame Equation¶
The charge per surface bead is calibrated from the experimentally measurable zeta potential (ζ) — a quantity measured routinely by dynamic light scattering (DLS) or electrophoretic light scattering (ELS).
The relationship between ζ and surface charge density σ (C/m²) is given by the Grahame equation, which is the exact solution to the Gouy-Chapman diffuse-layer problem for a planar interface:
where: - \(\varepsilon_0 \varepsilon_r\) = permittivity of water (≈ 6.9 × 10⁻¹⁰ C²/J·m) - \(k_B T\) = thermal energy (≈ 4.28 × 10⁻²¹ J at 310 K) - \(n_\infty\) = bulk ion number density (m⁻³, from ionic_strength_mM) - \(e\) = elementary charge
The bead charge is then:
where \(A_\text{bead} = 4\pi R^2 / N\) is the surface area per bead.
Physical Meaning¶
For silica at pH 7.4 (ζ = −50 mV, 20 mM NaCl), the Grahame equation gives σ ≈ −0.012 C/m². This corresponds to approximately 0.07 elementary charges per nm², or roughly one charge per 14 nm². This is consistent with the known silica surface chemistry: ~14% of surface silanols are deprotonated at pH 7.4.
Why the Grahame equation and not the linear approximation?
The Debye-Hückel linearization \(\sigma \approx \varepsilon_0 \varepsilon_r \kappa \zeta\) is valid only when \(e\zeta \ll 2k_BT\) (i.e., |ζ| ≪ 50 mV). For the common experimental range of |ζ| = 20–60 mV, the full Grahame equation is more accurate. At ζ = −50 mV: Grahame gives σ ≈ −0.012 C/m², while the linear approximation gives −0.011 C/m² — a 10% difference that propagates into ΔG_ads.
Debye-Hückel Screening (Debye Length)¶
All electrostatic interactions in the simulation are screened by the ionic atmosphere of the solution. The characteristic decay length is the Debye length λ_D:
where I is the ionic strength in molar (M) and T is the temperature in Kelvin.
Debye Length at Common Conditions¶
| Ionic strength | λ_D at 298 K | λ_D at 310 K | Context |
|---|---|---|---|
| 20 mM NaCl | 2.15 nm | 2.19 nm | Low salt (in vitro reference) |
| 150 mM NaCl | 0.79 nm | 0.80 nm | Physiological (blood plasma) |
| 1 M NaCl | 0.30 nm | 0.31 nm | High-salt screening |
The Debye length is critical because it determines how far the electrostatic field of the NP surface extends into solution. At 150 mM (physiological), λ_D ≈ 0.8 nm — the electrostatic influence is very short-range. At 20 mM, λ_D ≈ 2.2 nm — the field reaches the protein COM and strongly steers its orientation toward the surface.
Automatic Computation¶
AdsPro computes λ_D automatically from ionic_strength_mM and temperature. The user does not need to set it explicitly unless they want to override it with lambda_D_nm.
Surface Charge Mode¶
Two modes are supported:
grahame (default) — Computes σ from ζ via the Grahame equation. This is the recommended mode because ζ is directly measured from your experimental NP samples.
direct — The user specifies σ directly as sigma_direct_e_per_nm2 (elementary charges per nm²). Use this if you have surface titration data but not a DLS measurement.