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Multi-Modal Observables

Scientific background for the Cryo-EM and SAXS observables generated by synth-pdb.

Cryo-EM Density Simulation

Cryo-Electron Microscopy (Cryo-EM) measures the Coulomb potential of a biological sample. In structural biology, we represent this as a 3D "density map" where voxel values are proportional to the expected electron density.

Gaussian Approximation

At medium to low resolutions (3–10 Å), the contribution of an individual atom to the density map can be accurately modeled as a 3D Gaussian distribution:

\[\rho(r) \propto \exp\left(-\frac{r^2}{2\sigma^2}\right)\]

where \(r\) is the distance from the atomic center and \(\sigma\) is the width of the Gaussian.

Resolution vs. Sigma

The relationship between the target resolution (\(R\)) and the Gaussian width (\(\sigma\)) is defined by the point-spread function. A common convention is that the resolution corresponds to the frequency where the signal power drops significantly. We approximate this as:

\[\sigma \approx \frac{R}{3}\]
  • 3.0 Å Resolution: \(\sigma = 1.0\) Å. The map is sharp enough to distinguish some side-chain features.
  • 8.0 Å Resolution: \(\sigma \approx 2.7\) Å. Only the overall shape and secondary structure elements (helices) are visible.

Ensemble Averaging & Thermal Blurring

Real Cryo-EM maps are the result of averaging thousands of individual particle images. In synth-pdb, we simulate this by generating a conformational ensemble and averaging their individual densities.

Regions of high flexibility (loops, termini) will appear "smeared" or blurred in the averaged map, accurately reflecting the local resolution variations seen in experimental data. This makes synthetic ensembles ideal for benchmarking refinement algorithms that try to resolve heterogeneity.

Small-Angle X-ray Scattering (SAXS)

SAXS is a low-resolution technique that measures the scattering of X-rays by molecules in solution. Unlike Cryo-EM, SAXS provides 1D information (\(I(q)\) vs. \(q\)) but is highly sensitive to the overall shape and global dimensions of the protein.

The Debye Formula

The fundamental equation for computing the scattering intensity \(I(q)\) from atomic coordinates is the Debye formula (Debye, 1915):

\[I(q) = \sum_{i} \sum_{j} f_i(q) f_j(q) \frac{\sin(q \cdot r_{ij})}{q \cdot r_{ij}}\]
Symbol Meaning
\(q\) Scattering vector magnitude (\(\text{\AA}^{-1}\)), \(q = \frac{4\pi\sin\theta}{\lambda}\)
\(r_{ij}\) Distance between atoms \(i\) and \(j\)
\(f(q)\) \(q\)-dependent atomic form factor

Atomic Form Factors

The scattering efficiency of an atom depends on its element and the scattering angle. We use a sum-of-Gaussians approximation for the form factors (Waasmaier & Kirfel, 1995):

\[f(s) = c + \sum_{i=1}^{4} a_i \exp(-b_i s^2), \quad s = \frac{q}{4\pi}\]

synth-pdb includes coefficients for C, N, O, S, P, and H, ensuring high-fidelity scattering curves.

Solvent Contrast

In solution, we measure the excess scattering of the protein relative to the solvent (usually water). We must account for the displaced solvent volume by subtracting its contribution from the atomic form factors:

\[f_{\text{effective}}(q) = f_{\text{vacuum}}(q) - \rho_{\text{sol}} \cdot V_a \cdot \exp\left(-\frac{q^2 V_a^{2/3}}{4\pi}\right)\]

where \(\rho_{\text{sol}}\) is the electron density of water (\(0.334 e/\text{\AA}^3\)) and \(V_a\) is the atomic volume.

Visualizing SAXS Data

Synthetic scattering curves are often easier to interpret when transformed:

  1. Kratky Plot (\(q^2 \cdot I(q)\) vs \(q\)): Highlights protein compactness. A bell-shaped curve indicates a folded, globular protein. A curve that continues to rise at high \(q\) indicates a disordered or flexible ensemble.
  2. Guinier Plot (\(\ln I(q)\) vs \(q^2\)): Used to estimate the Radius of Gyration (\(R_g\)). The slope of the linear fit at low \(q\) is directly related to \(R_g^2/3\).

Why SAXS is Powerful for Ensembles

SAXS is uniquely capable of characterizing Intrinsically Disordered Proteins (IDPs) and large complexes. Because the scattering curve is an ensemble average:

\[\langle I(q) \rangle = \frac{1}{N} \sum_{k=1}^{N} I_k(q)\]

Synthetic SAXS profiles from synth-pdb ensembles allow researchers to validate whether a proposed set of disordered conformations is consistent with experimental solution data.

Key References

  1. Debye, P. (1915). Zerstreuung von Röntgenstrahlen. Annalen der Physik, 351, 809–876.
  2. Waasmaier, D. & Kirfel, A. (1995). Atomic scattering factors for free atoms and ions and their Gaussian approximations. Acta Crystallographica Section A, 51, 416–431.
  3. Svergun, D. I., et al. (1995). CRYSOL - a Program or Evaluating X-ray Solution Scattering of Biological Macromolecules from Atomic Coordinates. J. Appl. Cryst., 28, 768-773.
  4. Frank, J. (2006). Three-Dimensional Electron Microscopy of Macromolecular Assemblies. Oxford University Press.
  5. Levy, E. D. (2010). A Simple Definition of Scattering Surface Area that Explains the Burial of Hydrophobic and Polar Residues in Proteins. J. Mol. Biol., 403, 660–670.