# Using Force-Field Grids for Sampling Translation/Rotation of Partially Rigid Macromolecules
## 2 The Algorithm Proposed  
This note describes an algorithm that allows the movement of the rigid part while maintaining the grids in their original positions, giving rise to the same savings as obtained for the single molecule system.  
Figure 1 shows the schematics of the system considered. The rigid parts of the macromolecules are represented by the oblongs labeled with R\({}_{1}\) and R\({}_{2}\) and the flexible parts are represented by the loops F\({}_{1}\) and F\({}_{2}\). Macromolecule 2 will be held in place with only the F\({}_{2}\) part moving while macromolecule 1 is allowed to translate/rotate whilst also changing the conformation of the part represented by the atoms forming F\({}_{1}\).  
The key to the algorithm proposed is the simultaneous updating of two equivalent copies of the flexible parts, labeled F\({}_{1}\)\({}^{\prime}\) and F\({}_{2}\)\({}^{\prime}\) in Figure 2. The difference between the primed and unprimed versions is just a translation/rotation of the rigid part; their conformations are identical. The initial position of macromolecule 1 is R\({}_{1}\) and its position after rigid translation/rotation is R\({}_{1}\)\({}^{\prime}\). Such a transformation would also require the transformation of grid G\({}_{1}\) into G\({}_{1}\)\({}^{\prime}\), as shown in Figure 2, but the proposed algorithm avoids this. Instead, the algorithm introduces the alternative position of macromolecule 2, R\({}_{2}\)\({}^{\prime}\). It is constructed by translating/rotating the whole complex in such a way that the rigid part of macromolecule 1 is overlaid on its initial position/orientation. Note that, in reality, only the coordinates of the flexible part F\({}_{2}\) are involved in the calculation.  
Since \(E(\mathrm{F_{1}}\)\({}^{\prime}\),G\({}_{1}\)\({}^{\prime})=E(\mathrm{F_{1}}\),G\({}_{1}\)) and \(E(\mathrm{F_{2}}\),G\({}_{1}\)\({}^{\prime})=E(\mathrm{F_{2}}\)\({}^{\prime}\),G\({}_{1}\)) where \(E(\mathrm{F}\),G\()\) represents the energy of flexible part F interacting with grid G, this arrangement allows the calculation of the energy of a new complex as follows:  
\[E=E(\mathrm{F_{1}}\),G\({}_{1})+E(\mathrm{F_{2}}\),G\({}_{2})+E(\mathrm{F_{1}}\)\({}^{\prime}\),G\({}_{2})+E(\mathrm{F_{2}}\)\({}^{\prime}\),G\({}_{1})+E(\mathrm{F_{1}}\)\({}^{\prime}\),F\({}_{2}) \tag{1}\]  
where \(E(\mathrm{F_{1}}\)\({}^{\prime}\),F\({}_{2})\) is the energy of the interaction between flexible parts F\({}_{1}\)\({}^{\prime}\) and F\({}_{2}\).  
Two kinds of changes are considered: translate/rotate macromolecule 1 in the current conformation of its flexible part, F\({}_{1}\), or change the conformation of the flexible part of one of the macromolecules, F\({}_{1}\) or F\({}_{2}\). When macromolecule 1 is translated/rotated, the coordinates of F\({}_{1}\)\({}^{\prime}\) and F\({}_{2}\)\({}^{\prime}\) have to be updated. When the flexible part of macromolecule 1 is changed, the coordinates of F\({}_{1}\) and F\({}_{1}\)\({}^{\prime}\) have to be updated. When the flexible part of macromolecule 2 is changed, the coordinates of F\({}_{2}\) and F\({}_{2}\)\({}^{\prime}\) have to be updated.  
Note that in Equation (1), the energy of the interaction between the two rigid parts, being separated by the layers of the flexible part, is assumed to be negligible. For the Van der Waals interactions, this assumption holds to a good approximation. As for the electrostatic interactions, it is possible to include them, at negligible computational expense, using a multipole representation of the charge  
Figure 1: Schematics of the system considered. R\({}_{1}\), R\({}_{2}\): rigid part of macromolecules 1 and 2, respectively; F\({}_{1}\), F\({}_{2}\): flexible part of macromolecules 1 and 2, respectively.  
distribution since the two rigid parts can be enclosed into nonoverlapping spheres, ensuring the validity of the multipole representation.