The Accessibility model for transfer free energies

The Accessibility model revisits the Tanford additive transfer model from first principles, explicitly accounting for the mutual shielding between the peptide backbone and the amino acid side chain. It requires no experimental data beyond what is already used by the AutonBolen model, but redistributes the backbone and side-chain contributions to transfer free energies (TFEs) and denaturation m-values in a way that is consistent, simultaneously, with urea and with eight protecting osmolytes (TMAO, sarcosine, betaine, proline, sorbitol, sucrose, glycerol, and trehalose).

Motivation. In the established (AutonBolen) model, the side-chain TFE of residue aa is obtained by subtracting the full glycine TFE from the amino acid TFE:

\[\text{TFE}^{sc}_{aa} = \text{TFE}_{aa} - \text{TFE}_\text{Gly}.\]

This removes exactly one backbone unit's worth of TFE, which is only correct if the backbone of aa is as exposed as the backbone of glycine. The MoeserHorinek universal-backbone correction fixes this for the backbone term alone, by referencing all backbone contributions to the glycine ASA, but it does not account for the fact that the side chain is itself partly shielded by the backbone. As a result, it reproduces urea denaturation well (once the glycine-activity correction is included) but systematically underestimates the protective effect of protecting osmolytes, for which no analogous activity correction exists.

Construction. The Accessibility model decomposes the TFE of an amino acid into an isolated side chain, an isolated backbone unit, and terminal (capping) groups, weighted by accessibility factors $\alpha^{sc}_{aa}$ and $\alpha^{bb}_{aa}$ ($\alpha=1$: fully exposed; $\alpha=0$: fully shielded):

\[\text{TFE}_{aa} = \alpha^{sc}_{aa}\, \text{TFE}^{i\text{-}sc}_{aa} + \alpha^{bb}_{aa}\, \text{TFE}^\text{bb} + \text{TFE}^\text{TG}.\]

Since glycine consists only of a backbone and terminal groups, $\text{TFE}_\text{Gly} = \text{TFE}^\text{bb} + \text{TFE}^\text{TG}$, which eliminates the (experimentally inaccessible) $\text{TFE}^\text{TG}$ term and gives the TFE of the isolated side chain:

\[\text{TFE}^{i\text{-}sc}_{aa} = \frac{1}{\alpha^{sc}_{aa}} \left[ \text{TFE}_{aa} - \text{TFE}_\text{Gly} + \left(1 - \alpha^{bb}_{aa}\right)\text{TFE}^\text{bb} \right].\]

Using the apparent, activity-corrected transfer free energies of Auton & Bolen ($\text{TFE}^{sc,app}_{aa}$, with the small amino-acid activity corrections $\gamma_{aa}$ and $\gamma_\text{Gly}$, the latter relevant only for glycine in urea):

\[\text{TFE}^{i\text{-}sc}_{aa} = \frac{1}{\alpha^{sc}_{aa}} \left[\text{TFE}^{sc,app}_{aa} + \gamma_{aa} - \gamma_\text{Gly} + \left(1 - \alpha^{bb}_{aa}\right)\text{TFE}^\text{bb} \right].\]

The term $\left(1-\alpha^{bb}_{aa}\right)\text{TFE}^\text{bb}$ corrects for the fact that subtracting $\text{TFE}_\text{Gly}$ removes a full backbone unit, when only the exposed fraction of the backbone of aa should be removed.

Accessibility parameters. $\alpha^{sc}_{aa}$ and $\alpha^{bb}_{aa}$ are obtained from accessible surface area (ASA) calculations over a non-redundant protein structural database (CATH S20):

\[\alpha^{sc}_{aa} = \frac{\text{ASA}^{sc}_{aa}}{\text{ASA}^{i\text{-}sc}_{aa}}, \qquad \alpha^{bb}_{aa} = \frac{\text{ASA}^{bb}_{aa}}{\text{ASA}^\text{bb}}\]

where $\text{ASA}^{sc}_{aa}$ and $\text{ASA}^{bb}_{aa}$ are the side-chain and backbone ASAs of residue aa in its normal (attached) context, and $\text{ASA}^{i\text{-}sc}_{aa}$ and $\text{ASA}^\text{bb}$ are the ASAs of the side chain detached from the backbone and of the isolated backbone unit (equal to the backbone ASA of glycine), respectively:

$\alpha^{bb}_{aa}$ is well below unity for all residues but glycine: the side chain shields the backbone from direct solvent contact. Conversely, $\alpha^{sc}_{aa}$ shows that the backbone shields at least ~20% of even the bulkiest side chains (Trp), an effect that previous models did not consider explicitly.

Backbone accessibility is mechanism-dependent. The geometric ratio $\alpha^{bb}_{aa} = \text{ASA}^{bb}_{aa}/\text{ASA}^\text{bb}$ is appropriate when the cosolvent is excluded from, or interacts non-specifically with, the protein surface — the case for protecting osmolytes. Urea, however, interacts with the backbone through hydrogen bonds, on the face of the peptide unit opposite to the side chain; simulations show that the number of backbone–urea hydrogen bonds is nearly independent of residue type, i.e., the side chain does not shield the backbone from urea. The model therefore allows a mechanism parameter $x$ per cosolvent:

\[\alpha^{bb}_{aa}(x) = \left(\text{ASA}^{bb}_{aa}/\text{ASA}^\text{bb}\right)^{1-x}\]

with $x=0$ recovering the geometric, ASA-based exposure (used for all protecting osmolytes) and $x=1$ giving full backbone accessibility regardless of residue type, $\alpha^{bb}_{aa}=1$ (used for urea). With $x=1$ for urea, the backbone term of the model reduces exactly to the MoeserHorinek universal-backbone term, recovering its predictive accuracy; with $x=0$, side-chain shielding of the backbone is taken at face value, which is appropriate for excluded-volume protectants.

Final model. Combining the isolated side-chain TFE with the universal backbone term gives the transfer free energy of a protein:

\[\text{TFE}^\text{prot} = \sum_{i=1}^{N_r} \left[ \left(\frac{\text{TFE}^{i\text{-}sc}_{aa}}{\text{ASA}^{i\text{-}sc}_{aa}}\right)\text{ASA}^{sc}_{aa} + \left(\frac{\text{TFE}^\text{bb}}{\text{ASA}^\text{bb}_\text{Gly}}\right)\text{ASA}^{bb}_{aa} \right]\]

where the sum runs over the $N_r$ residues of the protein, and $\text{ASA}^{sc}_{aa}$ and $\text{ASA}^{bb}_{aa}$ are the actual (state-dependent) side-chain and backbone ASAs of each residue, computed internally from the structure. The backbone term is identical to that of the MoeserHorinek model; the side-chain term uses the isolated side-chain TFE and ASA, which differ from the quantities used by any of the other models.

The model is available for all cosolvents in the Auton & Bolen parameterization (TMAO, Sarcosine, Betaine, Proline, Sorbitol, Sucrose, Urea, Glycerol, and Trehalose):

using PDBTools
creamer_model = CreamerDenaturedModel(read_pdb(PDBTools.TESTPDB, "protein"))
m = mvalue(creamer_model, "urea"; model=Accessibility)
println("m-value: tot = $(m.tot), bb = $(m.bb), sc=$(m.sc)")

m-value: tot = -1.3231321161399867, bb = -0.6949057404198392, sc=-0.6282263757201474

m = mvalue(creamer_model, "tmao"; model=Accessibility)
println("m-value: tot = $(m.tot), bb = $(m.bb), sc=$(m.sc)")

m-value: tot = 2.2305621684693144, bb = 1.6036284083969474, sc=0.626933760072367

For protecting osmolytes, Accessibility predicts total m-values similar to AutonBolen, but with side-chain contributions that are stabilizing and comparable to, or larger than, the backbone contribution — in contrast to the backbone-dominated picture of AutonBolen. For urea, it reproduces the balanced backbone/side-chain partition of MoeserHorinek. The predictions can differ from the models when the ratio of backbone and side-chain accessibilities differ substantially from averages, or if the exposed surface have very particular amino acid residue compositions.