The Rosetta All-Atom Energy Function for Macromolecular Modeling and Design

Abstract

Over the past decade, the Rosetta biomolecular modeling suite has informed diverse biological questions and engineering challenges ranging from interpretation of low-resolution structural data to design of nanomaterials, protein therapeutics, and vaccines. Central to Rosetta's success is the energy function: a model parametrized from small-molecule and X-ray crystal structure data used to approximate the energy associated with each biomolecule conformation. This paper describes the mathematical models and physical concepts that underlie the latest Rosetta energy function, called the Rosetta Energy Function 2015 (REF15). Applying these concepts, we explain how to use Rosetta energies to identify and analyze the features of biomolecular models. Finally, we discuss the latest advances in the energy function that extend its capabilities from soluble proteins to also include membrane proteins, peptides containing noncanonical amino acids, small molecules, carbohydrates, nucleic acids, and other macromolecules.

Figure 1. Van der Waals and electrostatics energies

Comparison between pairwise energies of non-bonded atoms computed by Rosetta and the form computed by traditional molecular mechanics force fields. Here, the interaction between the backbone nitrogen and carbon are used as an example. (A) Lennard-Jones van der Waals energy with well-depths ε_Nbb = 0.162 and ε_Cbb = 0.063 and atomic radii r_Nbb = 1.763 and r_Cbb = 2.011 (red) and Rosetta fa_rep (blue). (B) Lennard-Jones van der Waals energy (red) and Rosetta fa_atr (blue). As the atom-pair distance approaches 6.0 Å, the fa_atr term smoothly approaches zero and deviates slightly from the original Lennard-Jones potential. (C) Coulomb electrostatics energy with a dielectric constant ε = 10, and partial charges p_Nbb = −0.604 and q_Cbb = 0.090 (red) compared with Rosetta fa_elec (blue). The fa_elec model is shifted to reach zero at the cutoff distance 6.0 Å.

Figure 2. A two component Lazaridis-Karplus solvation model

Rosetta uses two energy terms to evaluate the desolvation of protein side chains: an isotropic ( fa_sol) and anisotropic ( lk_ball_wtd) term. (A) and (B) demonstrate the difference between isotropic and anisotropic solvation of the NH2 group by CH3 on the asparagine side chain. The contours vary from low energy (blue) to high energy (yellow). The arrows represent the approach vectors for the pair potentials shown in C-E. In the bottom panel, we compare fa_sol, lk_ball and lk_ball_wtd energies for the solvation of the NH2 group on asparagine for three different approach angles: (C) in line with the 1HD2 atom, (D) along the bisector of the angle between 1HD1 and 1HD2 and (E) vertically down from above the plane of the hydrogens (out of plane).

Figure 3. Orientation-dependent hydrogen bonding model

(A) Degrees of freedom evaluated by the hydrogen bonding term: acceptor—donor distance, d_HA, angle between the base, acceptor and hydrogen θ_EAH, angle between the acceptor, hydrogen, and donor, θ_AHD, and dihedral angle corresponding to rotation around the base—acceptor bond, ϕ_B2BAH. (B) Lambert-azimuthal projection of the $E_{\text{hbond}}^{B_2\mathit{BAH}}$ energy landscape for an sp² hybridized acceptor. (C) $E_{\text{hbond}}^{B_2\mathit{BAH}}$ energy landscape for an sp³ hybridized acceptor. Example energies for the histidine imidazole ring acceptor hydrogen bonding with a protein backbone amide: (D) energy vs. the acceptor—donor distance, $E_{\text{hbond}}^{HA}$ (E) energy vs. the acceptor-hydrogen-donor angle, $E_{\text{hbond}}^{\mathit{AHD}}$ (F) energy vs. the base-acceptor—hydrogen angle, $E_{\text{hbond}}^{\mathit{BAH}}$.

Figure 4. Orientation-dependent disulfide bonding model

(A) Degrees of freedom evaluated by the disulfide bonding energy: sulfur—sulfur distance, d_SS, angle between the β-carbon and two sulfur atoms, θ_CSS, dihedral corresponding to rotation about the α -Carbon and sulfur bond ϕ_CαS, and dihedral corresponding to rotation about the S—S bond ϕ_SS. (B) $E_{\text{dslf}}^{SS}$, (C) $E_{\text{dslf}}^{\mathit{CSS}}$ (D) $E_{\text{dslf}}^{C_{\alpha }{C}_{\beta }SS}$ (E) $E_{\text{dslf}}^{C_{\beta }{\mathit{SSC}}_{\beta }}$.

Figure 5. Backbone torsion energies

The backbone-dependent torsion energies are demonstrated for the lysine residue. (A) The ϕ angle is defined by the backbone atoms C_i–1 – N – C_a – C and the ψ angle is defined by N – C_a – C – N_i+1. (B) rama_prepro energy of lysine without a proline at i-1. (C) rama_prepro energy of lysine with a proline at i-1. (D) p_aa_pp energy of lysine.

Figure 6. Energies for side-chain rotamer conformations

The Dunbrack rotamer energy, fa_dun, is dependent on both the ϕ and ψ backbone torsions and the χ side-chain torsions. Here, we demonstrate the variation of fa_dun when the backbone is fixed in an α-helical conformation with ϕ = −57° and ψ = −47°, and the χ values can vary. χ₁ is shown in blue, χ₂ shown in red and χ₂ shown in green. (A) χ-dependent Dunbrack energy of methionine with an sp³-hybridized terminus (B) χ-dependent energy of glutamine with an sp²-hybridized χ₂ terminus. χ₁, χ₂ and χ₂ of methionine and χ₁ and χ₂ of glutamine express rotameric behavior while χ₂ of the latter expresses broad non-rotameric behavior.

Figure 7. Special case torsion energies

Rosetta implements three additional energy terms to model torsional degrees of freedom with acute preferences. (A) Omega torsion corresponding to rotation about C-N (B) Proline secondary omega torsion corresponding to rotation about C-N related to the C-δ in the ring. (C) Tyrosine terminal χ torsion. (D) Omega energy (E) Proline closure energy (F) Tyrosine planarity energy.

Figure 8. Structural model of the HIV-1 protease bound to the T4V mutant RT-RH derived peptide

(A) Structural model of the native HIV-1 peptidase (teal and dark blue), bound to the native peptide (gray) superimposed onto the T4V mutant peptide (magenta). (B) Contributions greater than + 0.1 kcal/mol to the ΔΔG of mutation for T4V. The remaining contributions are: dslf_fa13 = 0 kcal/mol, hbond_lr_bb = −0.09 kcal/mol, hbond_bb_sc = −0.05, hbond_sc = −0.0104, fa_intra_rep = 0.01, fa_intra_sol = −0.07, and yhh_planarity = 0. (C) Hydrophobic patch of residues surrounding position four on the RT-RH peptide.

Figure 9. Using energies to discriminate docked models of West Nile Virus and the E16 neutralizing antibody

(A) Comparison of the native E16 antibody (purple) docked to the lowest RMS model of the West Nile Virus envelope protein and several other random models of varying energy to show sampling diversity (gray, semi transparent). (B) Change in the interface energy relative to the unbound state versus RMS to native. Models at low RMS to the native interface have a low overall interface energy due to favorable van der Waals contacts, electrostatic interactions, and side-chain hydrogen bonds, as reflected by the Δ fa_atr, Δ fa_elec, and Δ hbond_sc energy terms.