Protein Coarse-Graining¶

One Bead Per Residue (Cα Model)¶

AdsPro represents each protein as a linear chain of spherical beads — one bead per amino acid residue — centered at the Cα (alpha-carbon) atom position from the PDB structure.

Each bead carries four properties:

Property	Source
Position	Cα coordinate from PDB
Radius	Residue-specific van der Waals radius (HPS model)
Mass	Residue molecular weight (Da)
Charge	Henderson-Hasselbalch at simulation pH

This resolution — commonly called the Cα or one-bead-per-residue model — is standard in the protein physics literature for studying adsorption and folding at surfaces. It reduces the ~20,000-atom all-atom representation of Lysozyme to 129 beads, enabling nanosecond-timescale MD trajectories that are computationally intractable at full atomic resolution.

Henderson-Hasselbalch Charge Assignment¶

The partial charge on each titratable residue is computed from the Henderson-Hasselbalch equation using experimental pKa values:

For acidic residues (ASP, GLU, CYS, TYR) that lose a proton upon deprotonation:

\[ q(\text{pH}) = \frac{-1}{1 + 10^{(\text{p}K_a - \text{pH})}} \]

For basic residues (LYS, ARG, HIS) that gain a proton upon protonation:

\[ q(\text{pH}) = \frac{+1}{1 + 10^{(\text{pH} - \text{p}K_a)}} \]

These give fractional charges between −1 and +1, reflecting the equilibrium protonation state at the simulation pH. The N-terminus is treated as basic (pKa ≈ 8.0) and the C-terminus as acidic (pKa ≈ 3.1).

pKa Reference Values¶

Residue	pKa	Charge at pH 7.4
ASP	3.65	−1.00
GLU	4.25	−1.00
HIS	6.00	+0.04
CYS	8.14	−0.64
TYR	10.46	0.00
LYS	10.67	+1.00
ARG	12.10	+1.00
N-term	8.00	+0.20
C-term	3.10	−1.00

Source: Thurlkill et al. (2006), Protein Science 15:1214.

The net charge of the protein is the sum of all residue charges. For Lysozyme at pH 7.4: net charge ≈ +8e (strongly cationic), consistent with its known pI of ~11.

Why fractional charges?

The Henderson-Hasselbalch result is a thermal average over the protonation ensemble. A residue with charge 0.04 is 96% deprotonated at any given moment, but in the CG model it appears as a bead with charge +0.04e. This correctly captures the average electrostatic contribution without requiring explicit proton transfer events.

Gō Native Contact Map¶

The protein's internal energy is governed by a structure-based (Gō) model. The native contact map is computed from the PDB structure before any simulation begins.

Contact Detection¶

Two residues (i, j) form a native contact if: 1. Their separation in sequence is |i − j| ≥ 4 (excludes local backbone) 2. Their Cα–Cα distance in the PDB is within a cutoff: r_ij ≤ 4.5 Å

For Lysozyme (129 residues), this yields approximately 280–300 native contact pairs.

Contact Energy¶

Each native contact pair is assigned a Lennard-Jones 12-6 attraction:

\[ V_{G\bar{o}}(r) = \varepsilon_{G\bar{o}} \left[ \left(\frac{r_0}{r}\right)^{12} - 2\left(\frac{r_0}{r}\right)^{6} \right] \]

where: - \(r_0\) = native Cα–Cα distance from the PDB (nm) — the energy minimum - \(\varepsilon_{G\bar{o}}\) = well depth (uniform across all pairs) - \(V_{G\bar{o}}(r_0) = -\varepsilon_{G\bar{o}}\) — energy at native contact distance

Non-native residue pairs interact only through a WCA (purely repulsive) potential — there is no attraction between residues that are not in contact in the native structure. This design makes the native PDB fold the global energy minimum by construction.

Why the Gō Model?¶

The Gō model is particularly powerful for adsorption studies because it captures the key competition:

Surface Morse attraction pulls specific residues toward the surface, stretching and distorting native contacts
Gō internal energy resists this distortion, penalizing contact stretching

This competition determines whether the protein adsorbs in a native-like (small ΔE_internal) or deformed (large ΔE_internal) state. Models without internal flexibility (rigid-body models) cannot capture this and systematically overestimate adsorption affinity.

Bond Inventory¶

Before and after simulation, AdsPro catalogues all non-covalent bonds within the protein using geometry-based criteria:

Bond type	Detection criterion
Hydrogen bond	Donor-acceptor distance < 0.35 nm
Salt bridge	Opposite-charge residues within 0.4 nm
Electrostatic (attractive)	Same-sign charge pair within DH cutoff
van der Waals	Residue pair within sum of radii × 1.5

The change in bond inventory upon adsorption (from initial to final state) quantifies how many intra-protein bonds are broken or formed during adsorption — a measure of structural disruption.

Protein-surface bonds are catalogued separately using the same distance criteria but checking residue–NP bead pairs instead of residue–residue pairs.