After the helix lengths and shapes are predicted, they are placed in an experimental GPCR bundle template, which is defined by a system of coordinates shown in Figure 15. Each experimental template has 42 degrees of freedom: x, y, z, θ, φ and η values for each of the seven TM helices. The helices
have kinks and bends, so the helical axis is defined as its moment of least inertia. The hydrophobic center is the residue that crosses z = 0, which is defined as the plane that runs through the center of the lipid bilayer, and it is either calculated from the protein’s hydrophobic profile or by homology. The x-axis is defined along the axis from the center of TM3, which is in the middle of the bundle, to the center of TM2 in the mid-plane (z = 0). The definitions of the x-axis and z-axis implicitly define the y-axis. The x and y values of the helices (where the
hydrophobic centers cross z = 0), are defined by the experimental template, and are not sampled by SuperBiHelix. The degrees of freedom that are sampled are
θ, the tilt angle of the helix; φ, the sweep angle of the helix; and η, the rotation of
the helix around the helical axis.
Figure 15. The coordinates used to describe the orientation of the seven helices in a GPCR bundle.
The SuperBiHelix program takes an input GPCR bundle file, and determines its template. It then varies the θ, φ and η values. However, even if just sampling
a small number of angles for each degree of freedom, for example three, that would lead to (3*3*3)7≈ 1010 possible configurations for which to predict the side
However, if the energy is approximated to be made up of only interactions between two helices, the calculation is more tractable. A seven-helix GPCR TM bundle has twelve strongly interacting pairs: TM1-TM2, TM1-TM7, TM2-TM3, TM2-TM4, TM2-7, TM3-TM4, TM3-TM5, TM3-TM6, TM3-TM7, TM4-TM5, TM5- TM6 and TM6-TM7. For each of these twelve interacting pairs, θ, φ and η are
sampled with the other helices absent in order to get the bihelical energies. SCREAM is used to predict the side chain placements, then the side chains are minimized for 10 steps with the backbone fixed. This procedure is illustrated in Figure 16.
Figure 16. Diagram of the SuperBiHelix method, in which the seven-helix TM bundle is split into twelve helix pairs, and the θ, φ and η values for each helix in
the pair is sampled with the other helices not present.
Once the bihelical energies have been determined for all possible combinations of θ, φ and η, the energy of the entire bundle for each possible
intrahelical and interhelical components. The energy of the entire complex is then calculated as ! Eintra
(
" ,# ,$)
= 1 Ni Ei, intraij j=Ji,1,j>i Ji,Ni%
i=1 7%
(
"i,"j,#i,#j,$i,$j)
, Einter(
" ,# ,$)
= Einterij "i,"j,#i,#j,$i,$j(
)
j=Ji,1,j>i Ji,Ni%
i=1 7%
,Etotal
(
" ,# ,$)
=Eintra(
" ,# ,$)
+Einter(
" ,# ,$)
,!
" = "
(
1,"2,"3,"4,"5,"6,"7)
,# = #
(
1,#2,#3,#4,#5,#6,#7)
,$ = $
(
1,$2,$3,$4,$5,$6,$7)
,where Ni is the number of helices interacting with helix i, and Ji,k is the kth
neighbor of helix i. Although the calculation of the energy of a complex based on its bihelical energies is very fast, the calculation of all possible configurations is still too computationally expensive. In practice, the smallest number of conformations sampled would be three values of θ, five values of φ and five
values of η, which would lead to (3*5*5)7≈1013 total bundle conformations. Thus,
a procedure must be developed to determine which conformations for each helix are most favorable, so that fewer total bundle energies have to be calculated.
In order to determine the best conformations for each helix that will lead to the lowest energy bundles, the seven-helix bundle is partitioned into three quadhelix bundles, as shown in Figure 17.
Figure 17. In order to determine best conformations for each helix that will lead to the lowest energy bundles, the seven helix bundle is partitioned into three quadhelix bundles: TM1-TM2-TM3-TM7, TM2-TM3-TM4-TM5 and TM3-TM5- TM6-TM7.
Next, the total bihelical energies of the three quadhelix bundles are calculated. This is feasible because only 3*(3*5*5)4≈ 108 bundle energies must be calculated.
The 2000 structures with the lowest energy for each quadhelix are listed by increasing energy. Then, the conformations are ranked for each helix, using the following protocol:
• For TM1, each unique TM1 conformation (η1,θ1,φ1) in the
TM1-TM2-TM3-TM7 bundle list is taken.
• For TM2, each unique TM2 conformation (η2,θ2,φ2) in the
TM1-TM2-TM3-TM7 bundle list is alternated with each unique TM2 conformation in the TM2-TM3-TM4-TM5 bundle list.
• For TM3, each unique TM3 conformation (η3,θ3,φ3) in the
TM1-TM2-TM3-TM7 bundle list is taken in alternation with each unique TM3 conformation in the TM2-TM3-TM4-TM5 bundle list and each unique TM3 conformation in the TM3-TM5-TM6-TM7 bundle list.
• For TM4, each unique TM4 conformation (η4,θ4,φ4) in the
TM2-TM3-TM4-TM5 bundle list is taken.
• For TM5, each unique TM5 conformation (η5,θ5,φ5) in the
TM2-TM3-TM4-TM5 bundle list is alternated with each unique TM5 conformation in the TM3-TM5-TM6-TM7 bundle list.
• For TM6, each unique TM6 conformation (η6,θ6,φ6) in the
TM3-TM5-TM6-TM7 bundle list is taken.
• For TM7, each unique TM7 conformation (η7,θ7,φ7) in the
TM1-TM2-TM3-TM7 bundle list is alternated with each unique TM7 conformation in the TM3-TM5-TM6-TM7 bundle list.
Finally, from each individual helical conformation list, the best 36 conformations for each helix are used to calculate the energy of 367
≈ 8 x 1010 full bundles, and
output the 1000 best energy structures from this procedure.
In a procedure called SuperComBiHelix, these top 1000 helical bundles are built and the side chains are reassigned, given that they will take different conformations than in the bihelical mode. Then the structure is minimized for 10 steps. The energy ranking will be different in SuperComBiHelix than SuperBiHelix because all seven helices are present instead of just two at a time. This procedure results in an ensemble of low-lying structures. Examination of the low-lying structures shows which helices are flexible, and may give insight into activation.
Extensive testing on these methods, shown in the Validation section, lead to several improvements in the procedure. During the side chain prediction steps in
SuperBiHelix and SuperComBiHelix, SCREAM must be used with a 0.5 Å resolution library instead of the 1.0 Å resolution library that is the default for SCREAM. Additionally, the best results arise from alanizing the final two residues of the C- and N-termini for each helix during the SuperBiHelix step. Then, before the SuperComBiHelix step, the alanized residues are mutated back to their original residues for the building of the full bundles. This step reduces artificial long range electrostatic interactions between charged groups that would be located in the polar head group region of the lipid bilayer.