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Fig. 3. The rotation of the ligand axis axis by an angle 0, where (0 < 0 < 2n). However, as we will see in Section 4, this angle is often quite small ranging between (0 < 0 < Also, we must have:

where, y is the threshold distance between any two binding points of A and B respectively for docking to occur. Note that y can be estimated from the structural properties of the protein/ligand interaction [45].

2.2 Assumptions

1. Only the ligand rotates, to reach the final docked conformation whereas the protein remains fixed. In particular, we consider the relative rotation of the ligand axis with respect to the protein axis.

2. The docking point extends out of the ligand/protein backbones at an angle to the corresponding axis. In the analytical model, we have included both the cases when this angle is equal to f- and otherwise. The subsequent numerical results have been generated assuming an angle equal to f as this is not yet reflected in the biological databases.

3. The docking site on the ligand/protein backbones are approximated as straight lines for ease in calculations. Note that the first step is to find the average angle (in radians) that the binding site of the ligand axis has to rotate to reach the final docked conformation. We assume that the binding site of the ligand behaves like a rubber handle extending out of the spherical ligand structure. This allows us to compute the average time taken for the rotation of the ligand axis easily.

4. At least 3 docking points in the ligand has to come within the range of the threshold distance of the corresponding 3 docking points in protein A for a successful binding to occur.

5. We consider a 2-d coordinate system to estimate our results. A 3-d coordinate system can be used following the same concept but the equations become quite complicated to solve as discussed later. If 3 docking points are considered, it is always feasible to have the three points on the same plane where the other points are contributing to reduce the rotational threshold energy required for binding for these three 2-d points. Thus a 2-d assumption is appropriate for the model.

6. The docking points extend out of the protein/ligand backbones in a straight line.

The requirement of at least 3 docking points to come within the threshold distance of y allows us to calculate the average angle of rotation, 0avg, that the ligand axis has to rotate for successful docking with Protein A as discussed below.

2.3 Finding 6, avg

It should be noted that in the subsequent discussion all references to the ligand/protein backbones actually applies to only the docking site of the corresponding backbones (which are assumed as straight lines). Fig 4 shows the

Ligand B

Ligand B

Fig. 4. Ligand and Protein coming within threshold distance of 3 docking points

scenario when the ligand and the protein come within a distance of y for at least 3 docking points.

Conventions

1. There are a total of ns docking points.

2. The docking points on the protein are labelled as (gix,giy) to denote the x and y coordinates respectively of the ith docking point.

3. The points on the amino acid backbone of the protein corresponding to the ith docking points are denoted by (g'ix, g'iy ).

4. The docking points on the ligand are labelled as (hix, hiy) to denote the x and y coordinates respectively of the ith docking point.

5. The points on the amino acid backbone of the ligand corresponding to the ith docking points are denoted by (h'ix, h'iy).

6. The origin of our 2-d coordinate system is at (gix, giy ), i.e, (gix, giy ) = (0, 0).

7. The distance between the ith docking point and the corresponding point on the protein backbone is given by dgi.

8. The distance between the ith docking point and the corresponding point on the ligand backbone is given by dhi.

9. The angle between the straight line connecting the ith docking point and the protein backbone and the straight line denoting the protein backbone is denoted by

10. The angle between the straight line connecting the ith docking point and the ligand backbone and the straight line denoting the ligand backbone is denoted by

11. The docking site on the protein backbone (assumed to be a straight line) is parallel to the x-axis of the 2-d coordinate system. Thus the equation of this straight line is y = -(dg1 )sin^1.

12. The distance between the points on the protein backbone corresponding to the ith and jth docking points is denoted by Dgij.

13. The distance between the points on the ligand backbone corresponding to the ith and jth docking points is denoted by Dhij.

The angles (Vi) are measured from the protein axis to the straight line extending out of the axis carrying the docking point in an anti-clockwise direction as shown in Fig 5. Similarly, the angles (Vi) are also computed.

Calculating the coordinates of the docking points of the protein backbone. Its fairly easy to compute the coordinates of all the ns docking points and their corresponding contact points on the protein backbone. We will sim-plistically sketch the process in this section.

The first docking point on the protein backbone, (g1x, g1y) is considered to be the origin of our coordinate system. Also, because the equation of the straight line denoting the protein axis is known, we can write:

From, Eq 8 we can readily calculate (g'1 x,g' y). Next, we can compute (g'ix,g'y), (1 < i < ns) by solving the following set of equations:

(gix - g'1 x)2 + (giy - giy)2 = (Dg1i)2; 2 < i < ns (10)

Next, we can estimate the coordinates of the docking points of the protein (gix,giy), (2 < i < ns) by solving the following equation pair:

(g'x - gix)2 + (g'y - giy)2 = (dgi)2; giy = g'y + (dgi)sin^i (11)

Calculating the coordinates of any three docking points on the ligand.

The angle 0 as shown in Fig 4 denotes the angle made by the docking sites of the ligand backbone with the protein backbone (and equivalently the x-axis). As mentioned before, we assume that any three docking points on the ligand come within the threshold distance of the corresponding docking points of the protein. Without loss of generality, let us assume that these 3 docking points are denoted by (hix,hiy), (hjx,hjy) and (hkx,hky) corresponding to the docking points on the protein denoted by (gix,giy), (gjx,gjy) and (gux,9ky), where 1 < i,j,k < ns and i = j = k. Thus we can write:

Next, we can find the distance between the docking points (hix, hiy) and their corresponding points of attachment to the ligand axis (h'ix, h'iy) denoted by dhi (from the PDB database [7]) and hence:

The distances between the corresponding points on the ligand axis can also be estimated (from the PDB database) and we have:

Also, our assumption that the docking points extend out of the ligand backbone in a straight line allows us to formulate the slope of these lines as ,

\3y-h,v and • And because the corresponding angles of these lines with the ligand axis can be estimated, we have:

hiy-hiy rr--m tan i>j = ,"T'T ,, , for ?/>•; ^

mhzv_h7 = -1, for tfii tan = " h-'-H'. » for + \

hky-h'ky tan Vfc = '"'^-„i , for V>fc ^ f .

jy jy where, m is the slope of the straight line denoting the ligand axis. Note that, in Section 4, we assume an angle of f to generate the results as the corresponding angles are not reported in the biological databases. Finally, because the points (h'ix, h'iy), (hjx, hjy) and (h'kx,h'ky) lie on the same straight line (i.e, the ligand backbone), we can write:

h x - hix h '■ — hi h'ky ~ hjy = (h'kx ~ hjx)~j~, _ 7 / (24)

Thus, in Equations 12-24, we have 13 equations to solve for the following 13 unknown variables: hix, hy, h.x, h.y, hkx, hky, h'ix, h'iy, hjx, hjy, h'kx, h'ky and m. Note that, we need at least 3 docking points to form sufficient number of equations for solving all the unknown variables. To calculate 0 from m, we observe that the slope of the ligand axis is given by tan (0), such that we have:

Note that the slope can be both positive or negative resulting in clockwise or anticlockwise rotations of the ligand axis. However, because we are interested in computing the time for rotation of the ligand axis, the direction of rotation is not important for us. Also, because the equations are nonlinear and involve inequalities, we can only make an approximate estimate of the coordinates of the docking points on the ligand.

Calculating Qavg from 0. The next step is to estimate the average angle of rotation, 0avg. We will find the angle 0 (as outlined above) considering any 3 docking points out of the possible ns points. This requires a total of ('3) iterations.

We next find the average angle of rotation considering 3 docking points, 0^vg, from the different 03's (1 < i < ("3)) calculated (where, 03 denotes the angle computed using the above equations for the ith combination of 3 docking points). Assuming uniform probability for all these cases, we have:

Note that if greater number of docking points come within the threshold distance, 03avg (4 < j < ns) will continue to decrease. We next consider the case when more than 3 docking points come within the threshold distance. If 4 points come within the distance, we will have an extra 4 variables to solve (hmx,hmy,h'mx,h' ). Note that our assumptions for this coordinate system is only valid if all of these four points are on the same plane. We will have another 4 equations by adding the equations corresponding to this new point to the Eqs 12-14, Eqs 15-17, Eqs 18-19 and Eqs 20-22 respectively as follows:

(hmx gmx) + (hmy gmy) (hmx — hmx) + (hmy — hmy) = (dhm) (hix — hmx) + (hiy — hmy) = (Dhim)

1+m7

Next we can calculate the average angle of rotation considering 4 docking points, 94vg, in the same way as discussed above assuming uniform probability for all the (™s) different cases as follows:

0 0

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