diffab/modules/common/geometry.py

import torch
import torch.nn.functional as F

from diffab.utils.protein.constants import (
    BBHeavyAtom, 
    backbone_atom_coordinates_tensor,
    bb_oxygen_coordinate_tensor,
)
from .topology import get_terminus_flag


def safe_norm(x, dim=-1, keepdim=False, eps=1e-8, sqrt=True):
    out = torch.clamp(torch.sum(torch.square(x), dim=dim, keepdim=keepdim), min=eps)
    return torch.sqrt(out) if sqrt else out


def pairwise_distances(x, y=None, return_v=False):
    """
    Args:
        x:  (B, N, d)
        y:  (B, M, d)
    """
    if y is None: y = x
    v = x.unsqueeze(2) - y.unsqueeze(1)  # (B, N, M, d)
    d = safe_norm(v, dim=-1)
    if return_v:
        return d, v
    else:
        return d


def normalize_vector(v, dim, eps=1e-6):
    return v / (torch.linalg.norm(v, ord=2, dim=dim, keepdim=True) + eps)


def project_v2v(v, e, dim):
    """
    Description:
        Project vector `v` onto vector `e`.
    Args:
        v:  (N, L, 3).
        e:  (N, L, 3).
    """
    return (e * v).sum(dim=dim, keepdim=True) * e


def construct_3d_basis(center, p1, p2):
    """
    Args:
        center: (N, L, 3), usually the position of C_alpha.
        p1:     (N, L, 3), usually the position of C.
        p2:     (N, L, 3), usually the position of N.
    Returns
        A batch of orthogonal basis matrix, (N, L, 3, 3cols_index).
        The matrix is composed of 3 column vectors: [e1, e2, e3].
    """
    v1 = p1 - center    # (N, L, 3)
    e1 = normalize_vector(v1, dim=-1)

    v2 = p2 - center    # (N, L, 3)
    u2 = v2 - project_v2v(v2, e1, dim=-1)
    e2 = normalize_vector(u2, dim=-1)

    e3 = torch.cross(e1, e2, dim=-1)    # (N, L, 3)

    mat = torch.cat([
        e1.unsqueeze(-1), e2.unsqueeze(-1), e3.unsqueeze(-1)
    ], dim=-1)  # (N, L, 3, 3_index)
    return mat


def local_to_global(R, t, p):
    """
    Description:
        Convert local (internal) coordinates to global (external) coordinates q.
        q <- Rp + t
    Args:
        R:  (N, L, 3, 3).
        t:  (N, L, 3).
        p:  Local coordinates, (N, L, ..., 3).
    Returns:
        q:  Global coordinates, (N, L, ..., 3).
    """
    assert p.size(-1) == 3
    p_size = p.size()
    N, L = p_size[0], p_size[1]

    p = p.view(N, L, -1, 3).transpose(-1, -2)   # (N, L, *, 3) -> (N, L, 3, *)
    q = torch.matmul(R, p) + t.unsqueeze(-1)    # (N, L, 3, *)
    q = q.transpose(-1, -2).reshape(p_size)     # (N, L, 3, *) -> (N, L, *, 3) -> (N, L, ..., 3)
    return q


def global_to_local(R, t, q):
    """
    Description:
        Convert global (external) coordinates q to local (internal) coordinates p.
        p <- R^{T}(q - t)
    Args:
        R:  (N, L, 3, 3).
        t:  (N, L, 3).
        q:  Global coordinates, (N, L, ..., 3).
    Returns:
        p:  Local coordinates, (N, L, ..., 3).
    """
    assert q.size(-1) == 3
    q_size = q.size()
    N, L = q_size[0], q_size[1]

    q = q.reshape(N, L, -1, 3).transpose(-1, -2)   # (N, L, *, 3) -> (N, L, 3, *)
    p = torch.matmul(R.transpose(-1, -2), (q - t.unsqueeze(-1)))  # (N, L, 3, *)
    p = p.transpose(-1, -2).reshape(q_size)     # (N, L, 3, *) -> (N, L, *, 3) -> (N, L, ..., 3)
    return p


def apply_rotation_to_vector(R, p):
    return local_to_global(R, torch.zeros_like(p), p)


def compose_rotation_and_translation(R1, t1, R2, t2):
    """
    Args:
        R1,t1:  Frame basis and coordinate, (N, L, 3, 3), (N, L, 3).
        R2,t2:  Rotation and translation to be applied to (R1, t1), (N, L, 3, 3), (N, L, 3).
    Returns
        R_new <- R1R2
        t_new <- R1t2 + t1
    """
    R_new = torch.matmul(R1, R2)    # (N, L, 3, 3)
    t_new = torch.matmul(R1, t2.unsqueeze(-1)).squeeze(-1) + t1
    return R_new, t_new


def compose_chain(Ts):
    while len(Ts) >= 2:
        R1, t1 = Ts[-2]
        R2, t2 = Ts[-1]
        T_next = compose_rotation_and_translation(R1, t1, R2, t2)
        Ts = Ts[:-2] + [T_next]
    return Ts[0]


# Copyright (c) Meta Platforms, Inc. and affiliates.
# All rights reserved.
#
# This source code is licensed under the BSD-style license found in the
# LICENSE file in the root directory of this source tree.
def quaternion_to_rotation_matrix(quaternions):
    """
    Convert rotations given as quaternions to rotation matrices.
    Args:
        quaternions: quaternions with real part first,
            as tensor of shape (..., 4).
    Returns:
        Rotation matrices as tensor of shape (..., 3, 3).
    """
    quaternions = F.normalize(quaternions, dim=-1)
    r, i, j, k = torch.unbind(quaternions, -1)
    two_s = 2.0 / (quaternions * quaternions).sum(-1)

    o = torch.stack(
        (
            1 - two_s * (j * j + k * k),
            two_s * (i * j - k * r),
            two_s * (i * k + j * r),
            two_s * (i * j + k * r),
            1 - two_s * (i * i + k * k),
            two_s * (j * k - i * r),
            two_s * (i * k - j * r),
            two_s * (j * k + i * r),
            1 - two_s * (i * i + j * j),
        ),
        -1,
    )
    return o.reshape(quaternions.shape[:-1] + (3, 3))


# Copyright (c) Meta Platforms, Inc. and affiliates.
# All rights reserved.
#
# This source code is licensed under the BSD-style license found in the
# LICENSE file in the root directory of this source tree.
"""
BSD License

For PyTorch3D software

Copyright (c) Meta Platforms, Inc. and affiliates. All rights reserved.

Redistribution and use in source and binary forms, with or without modification,
are permitted provided that the following conditions are met:

 * Redistributions of source code must retain the above copyright notice, this
   list of conditions and the following disclaimer.

 * Redistributions in binary form must reproduce the above copyright notice,
   this list of conditions and the following disclaimer in the documentation
   and/or other materials provided with the distribution.

 * Neither the name Meta nor the names of its contributors may be used to
   endorse or promote products derived from this software without specific
   prior written permission.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR
ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON
ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
"""
def quaternion_1ijk_to_rotation_matrix(q):
    """
    (1 + ai + bj + ck) -> R
    Args:
        q:  (..., 3)
    """
    b, c, d = torch.unbind(q, dim=-1)
    s = torch.sqrt(1 + b**2 + c**2 + d**2)
    a, b, c, d = 1/s, b/s, c/s, d/s

    o = torch.stack(
        (
            a**2 + b**2 - c**2 - d**2,  2*b*c - 2*a*d,  2*b*d + 2*a*c,
            2*b*c + 2*a*d,  a**2 - b**2 + c**2 - d**2,  2*c*d - 2*a*b,
            2*b*d - 2*a*c,  2*c*d + 2*a*b,  a**2 - b**2 - c**2 + d**2,
        ),
        -1,
    )
    return o.reshape(q.shape[:-1] + (3, 3))


def repr_6d_to_rotation_matrix(x):
    """
    Args:
        x:  6D representations, (..., 6).
    Returns:
        Rotation matrices, (..., 3, 3_index).
    """
    a1, a2 = x[..., 0:3], x[..., 3:6]
    b1 = normalize_vector(a1, dim=-1)
    b2 = normalize_vector(a2 - project_v2v(a2, b1, dim=-1), dim=-1)
    b3 = torch.cross(b1, b2, dim=-1)

    mat = torch.cat([
        b1.unsqueeze(-1), b2.unsqueeze(-1), b3.unsqueeze(-1)
    ], dim=-1)  # (N, L, 3, 3_index)
    return mat


def dihedral_from_four_points(p0, p1, p2, p3):
    """
    Args:
        p0-3:   (*, 3).
    Returns:
        Dihedral angles in radian, (*, ).
    """
    v0 = p2 - p1
    v1 = p0 - p1
    v2 = p3 - p2
    u1 = torch.cross(v0, v1, dim=-1)
    n1 = u1 / torch.linalg.norm(u1, dim=-1, keepdim=True)
    u2 = torch.cross(v0, v2, dim=-1)
    n2 = u2 / torch.linalg.norm(u2, dim=-1, keepdim=True)
    sgn = torch.sign( (torch.cross(v1, v2, dim=-1) * v0).sum(-1) )
    dihed = sgn*torch.acos( (n1 * n2).sum(-1).clamp(min=-0.999999, max=0.999999) )
    dihed = torch.nan_to_num(dihed)
    return dihed


def knn_gather(idx, value):
    """
    Args:
        idx:    (B, N, K)
        value:  (B, M, d)
    Returns:
        (B, N, K, d)
    """
    N, d = idx.size(1), value.size(-1)
    idx = idx.unsqueeze(-1).repeat(1, 1, 1, d)      # (B, N, K, d)
    value = value.unsqueeze(1).repeat(1, N, 1, 1)   # (B, N, M, d)
    return torch.gather(value, dim=2, index=idx)


def knn_points(q, p, K):
    """
    Args:
        q: (B, M, d)
        p: (B, N, d)
    Returns:
        (B, M, K), (B, M, K), (B, M, K, d)
    """
    _, L, _ = p.size()
    d = pairwise_distances(q, p)  # (B, N, M)
    dist, idx = d.topk(min(L, K), dim=-1, largest=False)  # (B, M, K), (B, M, K)
    return dist, idx, knn_gather(idx, p)


def angstrom_to_nm(x):
    return x / 10


def nm_to_angstrom(x):
    return x * 10


def get_backbone_dihedral_angles(pos_atoms, chain_nb, res_nb, mask):
    """
    Args:
        pos_atoms:  (N, L, A, 3).
        chain_nb:   (N, L).
        res_nb:     (N, L).
        mask:       (N, L).
    Returns:
        bb_dihedral:    Omega, Phi, and Psi angles in radian, (N, L, 3).
        mask_bb_dihed:  Masks of dihedral angles, (N, L, 3).
    """
    pos_N  = pos_atoms[:, :, BBHeavyAtom.N]   # (N, L, 3)
    pos_CA = pos_atoms[:, :, BBHeavyAtom.CA]
    pos_C  = pos_atoms[:, :, BBHeavyAtom.C]

    N_term_flag, C_term_flag = get_terminus_flag(chain_nb, res_nb, mask)  # (N, L)
    omega_mask = torch.logical_not(N_term_flag)
    phi_mask = torch.logical_not(N_term_flag)
    psi_mask = torch.logical_not(C_term_flag)

    # N-termini don't have omega and phi
    omega = F.pad(
        dihedral_from_four_points(pos_CA[:, :-1], pos_C[:, :-1], pos_N[:, 1:], pos_CA[:, 1:]), 
        pad=(1, 0), value=0,
    )
    phi = F.pad(
        dihedral_from_four_points(pos_C[:, :-1], pos_N[:, 1:], pos_CA[:, 1:], pos_C[:, 1:]),
        pad=(1, 0), value=0,
    )

    # C-termini don't have psi
    psi = F.pad(
        dihedral_from_four_points(pos_N[:, :-1], pos_CA[:, :-1], pos_C[:, :-1], pos_N[:, 1:]),
        pad=(0, 1), value=0,
    )

    mask_bb_dihed = torch.stack([omega_mask, phi_mask, psi_mask], dim=-1)
    bb_dihedral = torch.stack([omega, phi, psi], dim=-1) * mask_bb_dihed
    return bb_dihedral, mask_bb_dihed


def pairwise_dihedrals(pos_atoms):
    """
    Args:
        pos_atoms:  (N, L, A, 3).
    Returns:
        Inter-residue Phi and Psi angles, (N, L, L, 2).
    """
    N, L = pos_atoms.shape[:2]
    pos_N  = pos_atoms[:, :, BBHeavyAtom.N]   # (N, L, 3)
    pos_CA = pos_atoms[:, :, BBHeavyAtom.CA]
    pos_C  = pos_atoms[:, :, BBHeavyAtom.C]

    ir_phi = dihedral_from_four_points(
        pos_C[:,:,None].expand(N, L, L, 3), 
        pos_N[:,None,:].expand(N, L, L, 3), 
        pos_CA[:,None,:].expand(N, L, L, 3), 
        pos_C[:,None,:].expand(N, L, L, 3)
    )
    ir_psi = dihedral_from_four_points(
        pos_N[:,:,None].expand(N, L, L, 3), 
        pos_CA[:,:,None].expand(N, L, L, 3), 
        pos_C[:,:,None].expand(N, L, L, 3), 
        pos_N[:,None,:].expand(N, L, L, 3)
    )
    ir_dihed = torch.stack([ir_phi, ir_psi], dim=-1)
    return ir_dihed


def apply_rotation_matrix_to_rot6d(R, O):
    """
    Args:
        R:  (..., 3, 3)
        O:  (..., 6)
    Returns:
        Rotated 6D representation, (..., 6).
    """
    u1, u2 = O[..., :3, None], O[..., 3:, None] # (..., 3, 1)
    v1 = torch.matmul(R, u1).squeeze(-1)    # (..., 3)
    v2 = torch.matmul(R, u2).squeeze(-1)
    return torch.cat([v1, v2], dim=-1)


def normalize_rot6d(O):
    """
    Args:
        O:  (..., 6)
    """
    u1, u2 = O[..., :3], O[..., 3:]     # (..., 3)
    v1 = F.normalize(u1, p=2, dim=-1)   # (..., 3)
    v2 = F.normalize(u2 - project_v2v(u2, v1), p=2, dim=-1)
    return torch.cat([v1, v2], dim=-1)
    

def reconstruct_backbone(R, t, aa, chain_nb, res_nb, mask):
    """
    Args:
        R:  (N, L, 3, 3)
        t:  (N, L, 3)
        aa: (N, L)
        chain_nb:   (N, L)
        res_nb:     (N, L)
        mask:       (N, L)
    Returns:
        Reconstructed backbone atoms, (N, L, 4, 3).
    """
    N, L = aa.size()
    # atom_coords = restype_heavyatom_rigid_group_positions.clone().to(t) # (21, 14, 3)
    bb_coords = backbone_atom_coordinates_tensor.clone().to(t)  # (21, 3, 3)
    oxygen_coord = bb_oxygen_coordinate_tensor.clone().to(t)    # (21, 3)
    aa = aa.clamp(min=0, max=20)    # 20 for UNK

    bb_coords = bb_coords[aa.flatten()].reshape(N, L, -1, 3)    # (N, L, 3, 3)
    oxygen_coord = oxygen_coord[aa.flatten()].reshape(N, L, -1)  # (N, L, 3)
    bb_pos = local_to_global(R, t, bb_coords)   # Global coordinates of N, CA, C. (N, L, 3, 3).

    # Compute PSI angle
    bb_dihedral, _ = get_backbone_dihedral_angles(bb_pos, chain_nb, res_nb, mask)
    psi = bb_dihedral[..., 2]   # (N, L)
    # Make rotation matrix for PSI
    sin_psi = torch.sin(psi).reshape(N, L, 1, 1)
    cos_psi = torch.cos(psi).reshape(N, L, 1, 1)
    zero = torch.zeros_like(sin_psi)
    one = torch.ones_like(sin_psi)
    row1 = torch.cat([one, zero, zero], dim=-1)     # (N, L, 1, 3)
    row2 = torch.cat([zero, cos_psi, -sin_psi], dim=-1) # (N, L, 1, 3)
    row3 = torch.cat([zero, sin_psi, cos_psi], dim=-1)  # (N, L, 1, 3)
    R_psi = torch.cat([row1, row2, row3], dim=-2)       # (N, L, 3, 3)

    # Compute rotoation and translation of PSI frame, and position of O.
    R_psi, t_psi = compose_chain([
        (R, t), # Backbone
        (R_psi, torch.zeros_like(t)),       # PSI angle
    ])
    O_pos = local_to_global(R_psi, t_psi, oxygen_coord.reshape(N, L, 1, 3))

    bb_pos = torch.cat([bb_pos, O_pos], dim=2)  # (N, L, 4, 3)
    return bb_pos
    

def reconstruct_backbone_partially(pos_ctx, R_new, t_new, aa, chain_nb, res_nb, mask_atoms, mask_recons):
    """
    Args:
        pos:    (N, L, A, 3).
        R_new:  (N, L, 3, 3).
        t_new:  (N, L, 3).
        mask_atoms: (N, L, A).
        mask_recons:(N, L).
    Returns:
        pos_new:    (N, L, A, 3).
        mask_new:   (N, L, A).
    """
    N, L, A = mask_atoms.size()

    mask_res = mask_atoms[:, :, BBHeavyAtom.CA]
    pos_recons = reconstruct_backbone(R_new, t_new, aa, chain_nb, res_nb, mask_res) # (N, L, 4, 3)
    pos_recons = F.pad(pos_recons, pad=(0, 0, 0, A-4), value=0) # (N, L, A, 3)

    pos_new = torch.where(
        mask_recons[:, :, None, None].expand_as(pos_ctx),
        pos_recons, pos_ctx
    )   # (N, L, A, 3)

    mask_bb_atoms = torch.zeros_like(mask_atoms)
    mask_bb_atoms[:, :, :4] = True
    mask_new = torch.where(
        mask_recons[:, :, None].expand_as(mask_atoms),
        mask_bb_atoms, mask_atoms
    )

    return pos_new, mask_new