All-atom coordinates from backbone frames and torsion angles (AlphaFold)

Algorithms 24 and 25 of the Supplementary Information to Jumper et al. (2021), pp. 30-31. Side chain atoms are placed by walking the torsion hierarchy: each torsion frame is its parent frame composed with a literature transform and a rotation about the torsion axis, which the construction places on the x-axis. Bond lengths and angles come from the caller-supplied literature tables, so only the torsions are free.

Usage

Alfschn(frames, angles, littf, parent, litx, frameof)

Arguments

frames: List of backbone frames, each list(R, t).
angles: Torsion angles, an n x nframe x 2 array of unnormalised (cos, sin) pairs as the network emits them.
littf: List of literature transforms into the parent frame.
parent: Parent frame index per torsion frame; 0 means the frame hangs directly off the backbone frame.
litx: Matrix of idealised atom positions, natoms x 3.
frameof: Torsion frame index for each atom.

Value

A list with atom coordinates x, the composed torsion frames, estimate, n and method.

Details

This implements the algebra, not the amino-acid tables: the idealised geometry is an argument, never baked in.

References

Jumper et al (2021) Nature 596:583-589, Suppl. Algorithms 24-25

Examples

rot <- function(a) { q <- c(1, a) / sqrt(1 + sum(a^2))
  rbind(c(1 - 2 * (q[3]^2 + q[4]^2), 2 * (q[2] * q[3] - q[1] * q[4]), 2 * (q[2] * q[4] + q[1] * q[3])),
        c(2 * (q[2] * q[3] + q[1] * q[4]), 1 - 2 * (q[2]^2 + q[4]^2), 2 * (q[3] * q[4] - q[1] * q[2])),
        c(2 * (q[2] * q[4] - q[1] * q[3]), 2 * (q[3] * q[4] + q[1] * q[2]), 1 - 2 * (q[2]^2 + q[3]^2))) }
fr <- list(list(R = rot(c(0.1, 0, 0.2)), t = c(1, 0, 0)))
lit <- list(list(R = diag(3), t = c(0, 1, 0)), list(R = rot(c(0.2, 0, 0)), t = c(1, 1, 0)))
ang <- array(c(1, 0.6, 0, 0.8), c(1, 2, 2))      # (cos, sin) per torsion
sc <- Alfschn(fr, ang, lit, parent = c(0, 1), litx = rbind(c(1, 0, 0), c(0, 1, 1)), frameof = c(1, 2))
round(sc$x[1, , ], 3)
#>       [,1]  [,2]  [,3]
#> [1,] 1.543 1.286 0.229
#> [2,] 1.484 1.305 1.474