We next assume that the colliding ligand molecule must have free energy EAct or greater to overcome the energy barrier and bind to the specific protein molecule. The kinetic energy of approach of the ligand towards the protein with a velocity

U is E = mp^u where mpr = mp-mL = the reduced mass, mr = mass (in

2 ' PL mp +mL ' L v gm) of the ligand molecule and mP = mass (in gm) of the protein. We assume that as the kinetic energy, E, increases above EAct, the number of collisions that result in binding also increases [46]. Thus the probability for a binding to occur because of sufficient kinetic energy of the ligand molecule is given by:

and the overall probability, po, for collision with sufficient energy is given by:

The above equations assumed a fixed relative velocity U for the reaction. We will use the Maxwell-Boltzmann distribution of molecular velocities for a species of mass m given by:

where kB = Boltzmann's constant = 1.381 x 10~23 kg m2/s2/K/molecule and T denoting the absolute temperature. Replacing m with the reduced mass mPL of the ligand and protein molecules, we get, tt2

The term on the left hand side of the above equation denotes the fraction of this specific ligand molecule with relative velocities between U and (U + dU). Summing up the collisions for the ligand molecule for all velocities we get the probability of collision with sufficient energy, pb as follows:

Now, recalling E = mp^u , i.e., dE = m.ppUdU and substituting into Eqn. 35, we get:

Thus we get:

b = r {E ~ EAct)^n2nr2PLAt / 1 c-#rdE JEAct VkBT V 2nkbTrnPL

V V mpL

3 Computing the Time Taken for Protein-Ligand Docking

Now, we are in a position to analytically compute the time taken for ligand-protein docking. This can be divided into two parts: 1) computing the time taken for the ligand to collide with the binding site of the protein molecule with enough activation energy to create a temporary binding and 2) computing the time taken for the rotation of the ligand axis to stabilize the binding to the protein molecule. Note that the first part computes the time for the random collisions until the creation of the precursor state A - B (as shown in Eq. 5) and involves the first two steps in Eq. 5. The second part computes the time taken for the formation of the final docked complex, AB, from A - B.

3.1 Time Taken for the Ligand to Collide with the Binding Site of the Protein Molecule with Enough Activation Energy for Successful Docking

Let At = t = an infinitely small time step. The ligand molecules try to bind to the protein through collisions. If the first collision fails to produce a successful binding , they collide again after t time units and so on.

We can interpret pt as the probability of a successful binding in time t. Thus, the average time for the ligand to collide with the binding site of the protein molecule with enough activation energy for successful docking denoted by T£ is given by:

Pt and the corresponding second moment, T2c, is given by:

=Pt(r2) +Pt(l ~Pt)(2r)2 +Pt( 1 -pt)2(3r)2 + ... = (2 ~ ^

The average and second moment computations follow from the concept that the successful collision can be the first collision or the second collision or the third collision and so on. We find that the time for ligand-protein collisions (which is a random variable denoted by x) follows an exponential distribution for the specific ligand and protein used to generate the results (because the mean and standard deviation are fairly equal as reported in the next section). It should be noted that as we assume t to be quite small, we can approximate the total time measurements of binding using a continuous (exponential in this case) distribution instead of a discrete geometric distribution. Thus as reported later, we find TC « T2c, and hence the pdf of the exponential distribution is given by:

0, otherwise

3.2 Finding the Average Time for Rotation of Ligand Axis

Now to rotate the docking site on the ligand about the axis to reach the final docking configuration, we need to have some rotational energy which is contributed by the total change in free energy in forming the docked complex (denoted by Ef). Thus we have:

where, Id and wd are respectively the average rotational inertia and angular velocity of the docking site of the ligand. Now the estimates of Ef have been reported extensively in the literature, and our goal is to calculate Id and wd.

Calculating the average moment of inertia of the ligand, Id: The moment of inertia calculation can become tricky as we have to consider the axis of rotation as well as its distance from the ligand axis. Fig 9 illustrates the possible orientations of the protein and ligand axis where the dotted line with an arrow signifies the axis of rotation. Note that the protein and ligand axes might not intersect as well in some configurations (Figs 9(b),(c),(d)). In such cases, it becomes imperative to calculate the distance of the ligand axis from the point about which it rotates making the moment of inertia calculation quite cumbersome.

ligand axis protein axis (b)

ligand axis ligand axis protein axis

ligand axis protein axis (d)

Fig. 9. Possible orientations of the protein and ligand axes

We assume that the ligand and protein axes do actually intersect in all cases (i.e. Figs 9(b),(c),(d) can never occur). This is a practical consideration because the ligand physically collides with the protein. We also assume that the ligand axis rotates about this point of intersection. Note that this simplifies the average moment of inertia calculation as the intersection point will always be on the ligand axis (and we do not have to compute the distance of the ligand axis from the axis of rotation).

From section 2.3 we can easily find the equations of the two lines denoting the protein and ligand axes (as the coordinates of at least 3 points on each line is known). Hence the point of intersection can be computed in a straightforward manner. Let the point of intersection be denoted by (Sx,Sy). Also, we can estimate the coordinates of the beginning (denoted by (bx,by)) and end (denoted by (ex,ey)) points on the ligand axis corresponding to the first and last docking points 1 and ns.

Fig. 10. Approximate model of the Ligand molecule

As explained before, the docking sites of the ligand and protein axes are assumed as straight lines, such that the ligand can be approximated as a sphere (of radius rL) with a rubber handle (which is the straight line denoting the docking site on the ligand backbone). Fig 10 explains the model. This rubber handle on the ligand can be approximated as a cylinder with radius rd and length \/(bx — ex)2 + (by — ey)2. Note that in Section 2.5 we had modelled the ligand as a hard sphere. However, the calculation of 0avg and Id requires the docking site of the ligand axis to be a straight line (for ease in computation). Note that, in general, the docking site is quite small compared to the length of the entire ligand, and thus the rubber handle assumption is quite feasible. The collision theory estimate can still treat the entire ligand as a sphere without taking into account the rubber handle part. However, because the docking site is approximated as a rubber handle, only this part rotates to bind to the corresponding site on the protein and hence Id is the rotational inertia of the docking site only. We also assume that the docking site on the ligand has uniform density, pd, and cross-sectional area, Ad = nr2d. Thus we can approximate Id as follows:

([(5. - exf + (6y - eyf]ï + [(^ - bxf + (6y - byf]i) (39)

Calculating T1[: The average time for rotation of the docking site of the ligand axis (denoted by T[) is given by:

However, this does not allow us to compute the second moment of the time for rotation. We assume that the time for rotation follows an exponential distribution and hence the second moment of the time for rotation is given by:

Thus this exponential distribution has both mean and standard deviation as T[ and pdf of the form:

0, otherwise

0 0

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