Self-avoiding random walk , lead

Note It is interesting to compare the discussion in this section with that of Section 6.2.1 on the conformation of random-coil linear polymers. Also in that case a larger molecule, i.e., one consisting of a higher effective number of chain elements is more tenuous. Equation (6.4) reads rm = b(n )v, where rm may be considered proportional to the parameter R in Eq. (13.12) b then would correspond to a in (13.12), and to Np. For a polymer molecule conformation that follows a self-avoiding random walk, the exponent v is equal to 0.6. Rewriting of Eq. (6.4) then leads to n = (rrn/b)1 67, which is very similar to Eq. (13.12) with a fractal dimensionality of 1.67. Depending on conditions, the exponent can vary between about 1.6 and 2.1. 

MODELING LEAD OPTIMIZATION AS A SELF-AVOIDING RANDOM WALK... 

Delaney [26] recently proposed that the temporal trajectory of a lead optimization project in chemical space can be modeled by a self-avoiding random walk (SAW). A random walk (RW) is a mathematical formalization of a trajectory which is created by snccessively moving a point in discrete jumps in a random direction [27]. An SAW is a popular variation of the RW, whereby it is forbidden to visit the same place twice dnring the walk, that is, the SAW trajectory cannot intersect with itself [28]. Indeed, there are several direct analogies between the chemical space trajectory of a lead optimization project and an SAW, for example. 

As in the case of the Ising model, self-avoiding random walks on a lattice have been investigated for short n leading to values for lattice type. They give... 

Figure 6. Log-log plot of the size versus mass relation of the self-avoiding random walk on cubic lattice. The step length is 20 3 lattice points. The vedue of the slope of the straight line which can be drawn through the points leads to the value of the fractal dimension d = 1.96.

The above calculations assume that the gross chain conformations are those of a random walk, which is the case in the melt. However, for an isolated polymer molecule in a dilute solution, the average conformation is affected by excluded-volume interactions between one part of the chain and another. Because the chain must avoid self-intersection, the conformation of the chain will be that of a self-avoiding walk, rather than a random walk, if the solution is athermal—that is, if all interactions are negligible except excluded volume. Self-avoiding walks lead, on average, to more expanded coil dimensions, since expanded configurations are less likely than contracted ones to lead to self-intersection of the chain. Thus, in an athermal solution, the mean-square end-to-end dimension of a polymer molecule scales as... 

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