where, T1 and T2 are the first and second moments of the total time taken for protein-ligand docking.

4 Results and Analysis

4.1 Problems in Validation of Our Model

Before presenting the results, we first discuss the difficulty of experimentally validating our model. Note that we compute the average time for protein-ligand binding in this paper. Existing experimental results are based on estimation of the binding rate of the ligands to a specific protein. We consider the binding of the turkey ovomucoid third domain (OMTKY) ligand to the human leukocyte elastase protein to generate the results. The experimental rate constant of 106M^s-1 as reported in [19] is derived from these rate measurements. Hence, the number of ligands in the cell will affect this estimate of time taken by one single ligand to bind to the protein because the rate of reaction incorporates the ligand concentration as well. However, our model computes the time taken by any particular ligand to bind to the protein which should be independent of the number of ligands in the cell. It is currently very difficult to carry out experiments to track a particular ligand and physically compute the time. Also, the stochastic nature of the binding process suggests that the distribution of the time taken will have a very high variance. In other words, in some cases the ligand requires time in microseconds whereas in other cases it might take as long as 1 second. The results (for the ligand-protein pair identified above) we present in the next section assume that the time taken for any particular OMTKY-human leukocyte elastase binding has a rate constant of 106M^s-1 (as reported in [19]) even though it cannot be a true estimate of this event. Also, note that our model can be easily extended to incorporate the effects of multiple ligands present in the cell on the binding rate as discussed in Section 5.2.

4.2 Numerical Results

In this section, we present the numerical results for the theoretical model derived in the paper. Figs 11-15 present the results for OMTKY-Human leukocyte elastase binding in an average human cell with 20 [im diameter. Also, the results were generated for ns = 8 docking points on the protein/ligand. The different parameters assumed for the numerical results are concisely presented in Table 1. We used actual values from the from the PDB database [7] and some assumptions as reported in [19].

Calculation of Id and wd. To calculate Id we need to know the point of intersection of the straight lines denoting the docking sites of the protein and ligand. Because, we need to estimate the average rotational inertia, we consider two cases: (1) the intersecting point is at the center of the docking site on the ligand and (2) the intersecting point is at the end of the docking site on the ligand. Note that the coordinates of the exact set of docking points and their corresponding points on the protein/ligand backbones have been estimated using the LPC software [24]. Also the density of the ligand molecule is assumed to be 1.44 g/cm3 as the molecular weight of OMTKY is « 6 KDalton (see [23] for details).

The corresponding values for wd (assuming Ef = —7 Kcal/mol, from [19]) are 63.5 x 109 and 31.75 x 109 radians/sec respectively. Note that, [25] reports that the average angular velocity of a protein molecule is in the range « 109 radians/sec, which is very close to our estimate.

Table 1. Parameter Estimation for an average Human Cell


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