Backbone orientations/translations

In this tutorial, we learn how to obtain backbone orientations and translations from protein structures using protstruc. For each residue, a 3x3 backbone orientation matrix is defined by a Gram-Schmidt orthonormalized basis of two vectors: N-Ca and C-Ca vectors. Meanwhile, a three-element translation vector is simply defined as a coordinate of the Ca atom.

In [1]:

import protstruc as ps
import matplotlib.pyplot as plt
import numpy as np

import protstruc as ps import matplotlib.pyplot as plt import numpy as np

In [2]:

pdb_ids = ['1REX', '4EOT']
sb = ps.StructureBatch.from_pdb_id(pdb_ids)

pdb_ids = ['1REX', '4EOT'] sb = ps.StructureBatch.from_pdb_id(pdb_ids)

Extracting backbone orientations and translations from StructureBatch object¶

In [3]:

orientations = sb.backbone_orientations(a1='N', a2='CA', a3='C').numpy()
translations = sb.backbone_translations(atom='CA').numpy()

orientations.shape, translations.shape

orientations = sb.backbone_orientations(a1='N', a2='CA', a3='C').numpy() translations = sb.backbone_translations(atom='CA').numpy() orientations.shape, translations.shape

Out[3]:

((2, 184, 3, 3), (2, 184, 3))

Orthogonality of residue frames (orientations)¶

Let's check whether vectors forming residue frames (which represent the orientation of residues) are orthogonal for each others.

In [4]:

from itertools import combinations
for i, j in combinations([0, 1, 2], 2):
    inner_products = (orientations[:, :, :, i] * orientations[:, :, :, j]).sum(axis=2)

    assert np.allclose(inner_products[~np.isnan(inner_products)], 0.0, atol=1e-6)

from itertools import combinations for i, j in combinations([0, 1, 2], 2): inner_products = (orientations[:, :, :, i] * orientations[:, :, :, j]).sum(axis=2) assert np.allclose(inner_products[~np.isnan(inner_products)], 0.0, atol=1e-6)

In [5]:

fig = plt.figure()
ax = fig.add_subplot(projection='3d')

prt_idx = 1
chain_idx = sb.get_chain_idx()

for idx in [0, 1]:
    chain_mask = chain_idx[prt_idx] == idx
    # plot backbone CA atoms 
    ax.plot(
        translations[prt_idx, chain_mask, 0],
        translations[prt_idx, chain_mask, 1],
        translations[prt_idx, chain_mask, 2],
    )

fig = plt.figure() ax = fig.add_subplot(projection='3d') prt_idx = 1 chain_idx = sb.get_chain_idx() for idx in [0, 1]: chain_mask = chain_idx[prt_idx] == idx # plot backbone CA atoms ax.plot( translations[prt_idx, chain_mask, 0], translations[prt_idx, chain_mask, 1], translations[prt_idx, chain_mask, 2], )

Visualizing both the orientations and translations of residues¶

In [6]:

sb = ps.StructureBatch.from_pdb('4EOT.pdb')
t = sb.backbone_translations()[0].numpy()
o = sb.backbone_orientations()[0].numpy()

fig = plt.figure(figsize=(10, 10))
ax = fig.add_subplot(111, projection='3d')

prt_idx = 0
chain_idx = sb.get_chain_idx()

for idx in [0, 1]:
    chain_mask = chain_idx[prt_idx] == idx
        
    x, y, z = t[chain_mask, 0], t[chain_mask, 1], t[chain_mask, 2]
    o_x, o_y, o_z = o[chain_mask, :, 0], o[chain_mask, :, 1], o[chain_mask, :, 2]

    lw, length = 0.75, 1.5
    ax.quiver(x, y, z, o_x[:, 0], o_x[:, 1], o_x[:, 2], color='C0', linewidth=lw, length=length)
    ax.quiver(x, y, z, o_y[:, 0], o_y[:, 1], o_y[:, 2], color='C1', linewidth=lw, length=length)
    ax.quiver(x, y, z, o_z[:, 0], o_z[:, 1], o_z[:, 2], color='C2', linewidth=lw, length=length)

    ax.plot(x, y, z, c='C7', linewidth=0.5)
    ax.view_init(30, 30)

sb = ps.StructureBatch.from_pdb('4EOT.pdb') t = sb.backbone_translations()[0].numpy() o = sb.backbone_orientations()[0].numpy() fig = plt.figure(figsize=(10, 10)) ax = fig.add_subplot(111, projection='3d') prt_idx = 0 chain_idx = sb.get_chain_idx() for idx in [0, 1]: chain_mask = chain_idx[prt_idx] == idx x, y, z = t[chain_mask, 0], t[chain_mask, 1], t[chain_mask, 2] o_x, o_y, o_z = o[chain_mask, :, 0], o[chain_mask, :, 1], o[chain_mask, :, 2] lw, length = 0.75, 1.5 ax.quiver(x, y, z, o_x[:, 0], o_x[:, 1], o_x[:, 2], color='C0', linewidth=lw, length=length) ax.quiver(x, y, z, o_y[:, 0], o_y[:, 1], o_y[:, 2], color='C1', linewidth=lw, length=length) ax.quiver(x, y, z, o_z[:, 0], o_z[:, 1], o_z[:, 2], color='C2', linewidth=lw, length=length) ax.plot(x, y, z, c='C7', linewidth=0.5) ax.view_init(30, 30)

Initializing a new StructureBatch object from residue orientations and translations¶

In [7]:

pdb_ids = ['1REX', '4EOT']
sb = ps.StructureBatch.from_pdb_id(pdb_ids)

orientations = sb.backbone_orientations(a1='N', a2='CA', a3='C')
translations = sb.backbone_translations(atom='CA')

orientations.shape, translations.shape

pdb_ids = ['1REX', '4EOT'] sb = ps.StructureBatch.from_pdb_id(pdb_ids) orientations = sb.backbone_orientations(a1='N', a2='CA', a3='C') translations = sb.backbone_translations(atom='CA') orientations.shape, translations.shape

Out[7]:

(torch.Size([2, 184, 3, 3]), torch.Size([2, 184, 3]))

In [8]:

fig = plt.figure(figsize=(6, 4))
ax = fig.add_subplot(121, projection='3d')
ax2 = fig.add_subplot(122, projection='3d')

prt_idx = 1
chain_idx = sb.get_chain_idx()

for idx in [0, 1]:
    chain_mask = chain_idx[prt_idx] == idx
    # plot backbone CA atoms 
    ax.plot(
        translations[prt_idx, chain_mask, 0].numpy(),
        translations[prt_idx, chain_mask, 1].numpy(),
        translations[prt_idx, chain_mask, 2].numpy(),
    )
ax.set_title('Original')

# Reinitialize StructureBatch only using backbone orientations and translations
sb2 = ps.StructureBatch.from_backbone_orientations_translations(
    orientations, translations, chain_idx, sb.get_chain_ids(), sb.get_seq()
)

t = sb2.backbone_translations()

for idx in [0, 1]:
    chain_mask = chain_idx[prt_idx] == idx

    ax2.plot(
        t[prt_idx, chain_mask, 0].numpy(),
        t[prt_idx, chain_mask, 1].numpy(),
        t[prt_idx, chain_mask, 2].numpy(),
    )
ax2.set_title('Reinitialized')

fig = plt.figure(figsize=(6, 4)) ax = fig.add_subplot(121, projection='3d') ax2 = fig.add_subplot(122, projection='3d') prt_idx = 1 chain_idx = sb.get_chain_idx() for idx in [0, 1]: chain_mask = chain_idx[prt_idx] == idx # plot backbone CA atoms ax.plot( translations[prt_idx, chain_mask, 0].numpy(), translations[prt_idx, chain_mask, 1].numpy(), translations[prt_idx, chain_mask, 2].numpy(), ) ax.set_title('Original') # Reinitialize StructureBatch only using backbone orientations and translations sb2 = ps.StructureBatch.from_backbone_orientations_translations( orientations, translations, chain_idx, sb.get_chain_ids(), sb.get_seq() ) t = sb2.backbone_translations() for idx in [0, 1]: chain_mask = chain_idx[prt_idx] == idx ax2.plot( t[prt_idx, chain_mask, 0].numpy(), t[prt_idx, chain_mask, 1].numpy(), t[prt_idx, chain_mask, 2].numpy(), ) ax2.set_title('Reinitialized')

Out[8]:

Text(0.5, 0.92, 'Reinitialized')

In [ ]: