SUELOS ESTABILIZADOS “IN SITU” CON CEMENTO

In this section, a lattice model will be used to illustrate using the constraint manifold to enforce constraints. Section 8.1 discusses that the PSD collision avoidance energy can be modified to include enforcing equality constraints.

Figure 7.1 presented the lattice model, this lattice model is an α-carbon chain. It is called “lattice1 small” because the atoms are placed 1 ˚A apart, a shorter distance than the 3.8 ˚A for a real protein.

(a) (b) (c)

Figure 7.1: The lattice structure “lattice1 small”. a) ball and stick model of the starting conformation. b) stick model of the starting conformation. c) stick model of the ending conformation.

Figure 7.1 a) gives the beginning conformation as a “ball and stick” representation, where the balls represent the α-carbons. Figure 7.1 b) gives the beginning conformation using a “stick” representation, where the atoms are not visible. The stick representation allows the orientation of the backbone to be shown. Figure7.1c) is the ending conformation, where the bottom part of the lattice, coloured in black, has undergone a 180◦ rotation.

(a) t = 0.25 (b) t = 0.79 (c) t = 0.80 (d) t = 1.00

Figure 7.2: Transitions generated by Hookean ENI without the constraint manifold for lattice1 small. The final conformation is in the correct orientation after an abrupt flip at t = 0.79.

In order for the ending conformation to be in the correct orientation, the part of the lattice structure coloured in black in Figure 7.1 c) must remain rigid throughout the

to 10, then constraining the bond length between consecutive α carbons, together with constraining the bottom piece colored in black to be rigid requires the following pairs of constraints:

(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8), (8, 9), (9, 10), (6, 8), (6, 9), (6, 10), (7, 9), (8, 10) Figure 7.2 shows the transition generated using Hookean ENI without constraints, but with the steric collision energy. An interesting observation from this interpolation is that at t = 0.79, the bottom piece is abruptly flipped to the other side to arrive at the final conformation. Hookean iENM has generated the same transition and therefore images of the intermediate conformations will be omitted.

(a) t = 0.25 (b) t = 0.60 (c) t = 0.86 (d) t = 1.00

Figure 7.3: Transitions generated by PSD ENI without the constraint manifold for lattice1 small. The final conformation is in the wrong orientation.

Figure7.3shows the transition generated by PSD ENI without the constraint manifold. In this case, because the bottom piece has not been constrained to be rigid, the final conformation generated is in the wrong orientation. PSD iENM has generated the same transition and therefore images of the intermediate conformations will be omitted.

Figure7.4shows the transitions generated by Hookean ENI from moving on the constraint manifold. The intermediate conformations generated in this case has the bottom piece, which is coloured in black in Figure7.4, stay rigid for all intermediate conformations, and therefore, the ending conformation is in the correct orientation.

Note that the top part of the structure in Figure7.4, colored gray, does not stay rigid. Recall from the discussion in Section 2.8.3that when three atoms are collinear, the matrix used by Fast Projection to solve for the Lagrange multipliers becomes singular, therefore the top part of lattice1 small cannot be constrained to be rigid. Since this lattice is not a real protein, we will not explore this issue further in this thesis.

The transitional conformations generated by Hookean iENM are the same as that generated by Hookean ENI, the figures have therefore been omitted. The transitions

(a) t = 0.25 (b) t = 0.50 (c) t = 0.65 (d) t = 0.99

Figure 7.4: Transitions generated by Hookean ENI with the constraint manifold for lattice1 small. The black coloured part of the structure is constrained to be rigid to allow it to rotate 180◦.

generated by PSD ENI, and PSD iENM also show the same principle, and therefore these images are also omitted. In both cases, the final conformation is in the correct conformation, unlike in Figure 7.3.

0 20 _{Conformation Index}40 60 80 100 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Norm of constraint

Hookean_ENI lattice1small_1

(a) Without constraint manifold.

0 20 _{Conformation Index}40 60 80 100 0.10 0.05 0.00 0.05 0.10 Norm of constraint Hookean_ENI lattice1small_1

(b) With constraint manifold

Figure 7.5: k C(−→pt) k for each transitional conformation −→pt generated by Hookean ENI.

The same graph was generated by Hookean iENM.

Figure7.5shows the norm of the constraint, k C(−→p ) k for each transitional conformation generated by Hookean ENI (recall Equation (2.47). Hookean iENM produced the same

being in the correct orientation, the intermediate conformations do not stay on the constraint manifold. Note the peak of the graph around t = 0.79 and t = 0.8, where the abrupt flip happened as shown in Figure7.2. Figure7.5b) shows after enforcing constraints, k C(−→p ) k stays very close to zero for each transitional conformation, with no abrupt changes.

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0.00

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Norm of constraint

PSD_ENI lattice1small_1

(a) Without constraint manifold.

0 20 _{Conformation Index}40 60 80 100 0.10 0.05 0.00 0.05 0.10 Norm of constraint PSD_ENI lattice1small_1

(b) With constraint manifold

Figure 7.6: k C(Pt) k for each transitional conformation Pt generated by PSD ENI. The

same graph was generated by PSD iENM.

Figure 7.6 shows k C(Pt) k for each Pt generated by PSD ENI. Figure 7.6 a) and b)

show respectively that if enforcing constraints is omitted, the intermediate conformations are not guaranteed to be on the constraint manifold.