Extending Algorithm1 for atomic coordinates in protein structure

As we can see Algorithm1 will return true only if there exists an exact rigid transformation ($R$ ,$T$ ) which when applied to point sets $S_1$ will give $S_2$ or vice-versa. But in the context of protein structure where there is a considerable noise while measuring the coordinates during X-Ray crystallography it does'nt make sense for looking for such exact rigid transformations between the protein sub-structures $P_1$ and $P_2$ , we need an algorithm which can take the coordinates of the protein sub-structures and tell if one sub-structure can be approximately transformed into other sub-structure using some rigid transformation. So we extend the Algorithm1 which can tell if two protein structures can be approximately transformed from one another.
We define Weighted distance($W_{i,j}$ between two sorted vectors $V_i$ and $V_j$ (each of length $n$ )as follows.

$$W_{i,j} = \sum_{k=1}^{n}(n-k)*\sqrt{(V_i[k]-V_j[k])^2}$$

We also define an approximate threshold value $\epsilon$ , whose value is proportional to $n$ the typical values of $\epsilon$ is $1.8$ for $n=20$ , we have determined these from several experimental runs of our program. With these two definitions we give the Algorithm2 which can detect if given two atomic coordinate sets(from protein structure) $P_1$ and $P_2$ can be approximately transformed from one to the other, with an error of $\epsilon$ .

boxed