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2.4 Computing Qavg Using a 3-D Coordinate System

As mentioned before, a 3-d coordinate system can be used in a similar way to compute 0avg. However, as this increases the number of unknown variables appreciably, we need to assume that at least 15 docking points of the protein/ligand come within the threshold distance (to solve all the equations). This greatly increases the number of equations that has to be solved as well. Moreover, for small docking sites, the assumption of 15 docking points coming close might not be a practical way of solving the problem. Another disadvantage of the 3-d calculations is that as we need more docking points to come close, the value of 0avg becomes less than what we estimate with the 2-d system, resulting in a further decrease in the estimation of the time for the rotation of the ligand axis. Fig 6 plots the rotational energy required (measured in terms of total change in free energy reported in [19]) for different number of docking points coming within threshold distance (varied from 3 to 15). The results were generated for the protein-ligand pair of human leukocyte elastase and OMTKY3 where the optimal configuration corresponds to 15 docking points coming close (as we will have maximum chance of docking in that case) and the subsequent energy requirements were assumed for lesser number of docking points coming close. We observe that as more docking points come close, the rotational energy required is lesser i.e., the ligand axis has to rotate less to reach the docked conformation

Number of docking points within threshold distance

Fig. 6. Dependence of rotational energy on the number of docking points within threshold distance (the curve derived from the data points of [19])

Number of docking points within threshold distance

Fig. 6. Dependence of rotational energy on the number of docking points within threshold distance (the curve derived from the data points of [19])

indicating that the time required for rotation also decreases. Note that the requirements of 3 docking points coming close for the 2-d system and 15 points for the 3-d system is not specific to any protein-ligand pair and are a requirement of our model to be able to compute 0avg.

As we show later, the total protein-ligand docking time is primarily governed by the collision theory component (i.e., the time required for rotation of the ligand axis is negligible in comparison to the time taken by the ligand to collide with the docking site on the protein), and hence the lesser accuracy of the 2-d based computations is not a deterrent in estimating the total docking time. Also, this reduces the number of equations that need to be solved making the model computationally fast which is a basic requirement for our discrete event-based simulator.

2.5 Calculating pb

We assume that the ligand molecules enter the cell one at a time to initiate the binding. From the principles of collision theory for hard spheres, we model the protein and ligand molecules as rigid spheres with radii rP and rL respectively (Fig 7). We define our coordinate system such that the protein is stationary with respect to the ligand molecule, so that the latter moves towards the protein with a relative velocity U. The ligand molecule moves through space to sweep out a collision cross section A = nrpL (as illustrated in Fig 8), where rPL is the collision radius given by:

Fig. 7. Schematic diagram of protein and ligand molecules
Fig. 8. Volume swept out by the ligand molecule in time At

The number of collisions during a time period At is determined when a ligand molecule will be inside the space that is created by the motion of the collision cross section over this time period due to the motion of the ligand molecule. As mentioned before, pb denotes the probability of collision of the ligand with the protein with enough kinetic energy for the binding to occur successfully. In time At, the ligand molecule sweeps out a volume AV given by:

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