Theory: Differentiable Physics in ResonanceFlow
ResonanceFlow treats protein structure prediction as a differentiable optimization problem. The total training loss is a weighted sum of physical and experimental constraint terms:
$$\mathcal{L}{total} = \mathcal{L}{steric} + \lambda_{bond}\,\mathcal{L}{bond} + \mathcal{L}{RDC} + \mathcal{L}_{NOE}$$
Because every term is implemented in JAX, gradients $\nabla_\theta \mathcal{L}_{total}$ flow back through the Transformer weights $\theta$, allowing the model to learn physically valid protein geometries end-to-end.
Steric Clash Penalty
A harmonic repulsion penalises any pair of atoms whose van der Waals radii overlap. Given atoms $i$ and $j$ with radii $r_i$, $r_j$ at positions $\mathbf{x}_i$, $\mathbf{x}_j$, the pairwise overlap is:
$$o_{ij} = \max!\left(0,\; (r_i + r_j) - |\mathbf{x}_i - \mathbf{x}_j|\right)$$
The total loss sums over all non-excluded pairs:
$$\mathcal{L}{steric} = \frac{1}{2} \sum{i \neq j,\; |i-j|>k} o_{ij}^2$$
where $k$ is the exclude_bonded_range parameter (0 = self only;
1 = AMBER 1-2 exclusion; 2 = 1-2 and 1-3 exclusion). Periodic boundary
conditions are supported via jax-md.
Bond Length Constraints
Backbone geometry is constrained by a mean-squared error loss on consecutive C$\alpha$–C$\alpha$ virtual bond distances. The canonical C$\alpha$–C$\alpha$ distance in a peptide chain is 3.80 ± 0.02 Å (Engh & Huber, Acta Crystallogr. A, 1991):
$$\mathcal{L}{bond} = \frac{1}{N-1} \sum{i=1}^{N-1} \left(|\mathbf{x}{i+1} - \mathbf{x}_i| - d{ideal}\right)^2, \quad d_{ideal} = 3.8\,\text{Å}$$
Note: The ideal distance is the virtual bond between consecutive alpha-carbons spanning the full peptide unit, not the C–C covalent bond length (1.52 Å).
Residual Dipolar Couplings (RDCs)
RDCs report the average orientation of an internuclear bond vector relative to the external magnetic field, providing long-range global restraints that complement short-range NOE distances (Bax & Tjandra, J. Biomol. NMR, 1997).
Saupe Tensor Formulation
The RDC for an internuclear vector $\hat{\mathbf{v}} = (x, y, z)$ is:
$$D = D_{max} \left[ S_{xx}(x^2 - z^2) + S_{yy}(y^2 - z^2) + 2S_{xy}\,xy + 2S_{xz}\,xz + 2S_{yz}\,yz \right]$$
where $\mathbf{S}$ is the Saupe order matrix — a traceless, symmetric $3 \times 3$ tensor with five independent components (Saupe, Z. Naturforsch., 1964). Writing $\mathbf{s} = [S_{xx}, S_{yy}, S_{xy}, S_{xz}, S_{yz}]^\top$, the equations for all $N$ bond vectors are the linear system $\mathbf{A}\mathbf{s} = \mathbf{D}$.
Differentiable Fitting
ResonanceFlow fits $\mathbf{S}$ at every forward pass via the
differentiable least-squares solver jnp.linalg.lstsq (SVD-based,
with a ridge penalty $r_{cond} = 10^{-5}$ for stability). The loss is
the MSE of the residual:
$$\mathcal{L}{RDC} = \frac{1}{N}\sum{i=1}^N \left(D_i^{calc} - D_i^{obs}\right)^2$$
Gradient flow through the solver allows the model to learn backbone orientations that are consistent with the measured RDCs.
Quality Factor
The Q-factor (Cornilescu, Marquardt, Ottiger & Bax, JACS, 1998) quantifies structural quality analogously to the crystallographic R-factor:
$$Q = \frac{\sqrt{\langle (D^{calc} - D^{obs})^2 \rangle}}{\sqrt{\langle (D^{obs})^2 \rangle}}$$
A high-quality structure has $Q \leq 0.20$. To detect overfitting, $Q_{free}$ should be computed on a held-out subset of RDCs not used during fitting (Clore & Garrett, JACS, 1999).
N-H Proxy Vectors (Cα-only Models)
Because the current model predicts only C$\alpha$ coordinates, true N–H internuclear vectors are not directly available. ResonanceFlow uses the anti-parallel virtual-bond approximation:
$$\hat{\mathbf{v}}i^{proxy} = \frac{\mathbf{x}{i-1} - \mathbf{x}{i+1}}{|\mathbf{x}{i-1} - \mathbf{x}_{i+1}|}$$
This proxy correlates with the amide N-H orientation in both
$\alpha$-helices and $\beta$-strands (Zweckstetter & Bax,
J. Am. Chem. Soc., 2000). A full-atom model should supply true N–H
vectors directly to rdc_loss.
NOE Distance Restraints
Nuclear Overhauser Effect (NOE) cross-peaks provide upper bounds on inter-proton distances, typically in the range 1.8–6.0 Å (Wüthrich, NMR of Proteins and Nucleic Acids, 1986). ResonanceFlow applies a flat-bottomed harmonic penalty only to upper-bound violations (no lower-bound penalty, since an NOE cross-peak is only observed when protons are close):
$$\mathcal{L}{NOE} = \frac{1}{M} \sum{k=1}^{M} \max!\left(0,\; d_k - d_k^{upper}\right)^2$$
This form matches the standard restraint potential used in CYANA / DYANA (Güntert, Mumenthaler & Wüthrich, J. Mol. Biol., 1997).
Pseudo-Torsion Angles
Because ResonanceFlow often predicts only C$\alpha$ coordinates, traditional $\phi/\psi$ Ramachandran plots cannot be constructed. Instead, the pseudo-torsion angle ($\alpha$) formed by four consecutive C$\alpha$ atoms ($i-1, i, i+1, i+2$) is used to assess structural plausibility (Oldfield & Hubbard, Proteins, 1994).
Predicted structures should cluster within the following canonical ranges: - $\alpha$-helix: ~ +50° - $\beta$-strand: ~ ±180°
Deviations from these clusters (e.g., angles near 0°) typically indicate unphysical backbone geometry or "zigzagging" that satisfies bond constraints but violates steric or folding laws.
References
| Citation | Role in ResonanceFlow |
|---|---|
| Saupe, Z. Naturforsch. 19a, 161 (1964) | Definition of the alignment (Saupe) tensor |
| Engh & Huber, Acta Crystallogr. A 47, 392 (1991) | Ideal Cα–Cα bond distance (3.8 Å) |
| Wüthrich, NMR of Proteins and Nucleic Acids (1986) | NOE distance restraints |
| Bax & Tjandra, J. Biomol. NMR 10, 289 (1997) | RDC structure determination |
| Cornilescu, Marquardt, Ottiger & Bax, JACS 120, 6836 (1998) | RDC Q-factor definition |
| Clore & Garrett, JACS 121, 9008 (1999) | Q_free cross-validation |
| Güntert, Mumenthaler & Wüthrich, J. Mol. Biol. 273, 283 (1997) | NOE restraint potential (CYANA) |
| Zweckstetter & Bax, JACS 122, 3791 (2000) | Cα-based N-H proxy vectors |
| Ottiger & Bax, JACS 120, 12334 (1998) | d_max for ¹⁵N-¹H at 600 MHz |
| Oldfield & Hubbard, Proteins 19, 366 (1994) | Cα pseudo-torsion distribution |