HDX Theory

Linderstrøm-Lang Model

Hydrogen-Deuterium Exchange (HDX) measures the rate at which amide hydrogens in a protein exchange with the solvent. The process is modeled as a two-step kinetic scheme:

\[\text{Closed} \xrightleftharpoons[k_{cl}]{k_{op}} \text{Open} \xrightarrow{k_{int}} \text{Exchanged}\]

In the common EX2 regime (\(k_{cl} \gg k_{int}\)), the observed rate \(k_{obs}\) is proportional to the equilibrium opening constant \(K_{op} = k_{op}/k_{cl}\) (Hvidt & Nielsen, 1966):

\[k_{obs} = K_{op} \cdot k_{int}\]

The Protection Factor (PF) is defined as:

\[PF = \frac{1}{K_{op}} = \frac{k_{int}}{k_{obs}}\]

Deuterium Uptake Kinetics

The time-dependent fractional deuterium uptake \(D(t)\) for a single residue is given by:

\[D(t) = 1 - \exp(-k_{obs} \cdot t)\]

In diff-hdx, we model the Protection Factor as a function of local structural environment (SASA and H-bonding) using separate scaling coefficients:

\[\ln PF \approx \beta_{asa} \cdot (1 - SASA) + \beta_c \cdot E_{hb}\]

where \(\beta_{asa}\) weights the burial contribution and \(\beta_c\) weights the hydrogen-bond contribution independently. Both default to 1.0, recovering the original single-\(\beta\) form. When fitting to experimental protection factors, \(\beta_{asa}\) and \(\beta_c\) should be treated as independent free parameters.

SASA Approximation

We use a differentiable Gaussian occlusion model to estimate the Solvent Accessible Surface Area (SASA). The accessibility of atom \(i\) is reduced by the proximity of neighboring atoms:

\[\text{Accessibility}_i = \frac{1}{1 + \sum_{j \neq i} \exp\!\left(-r_{ij}^2 \,/\, 2\sigma_{eff}^2\right)}\]

where \(\sigma_{eff} = \sigma + r_{probe}\) combines the Gaussian smoothing width with the solvent probe radius \(r_{probe}\) (default 1.4 Å, matching the water probe used in Shrake–Rupley SASA).

Note: This is a differentiable surrogate, not a true Shrake–Rupley SASA. It lacks per-atom van-der-Waals radii and returns dimensionless values in \((0, 1]\). It is appropriate as a smooth proxy for gradient-based refinement but should not be used to report physical SASA values in Å².