# APL@Voro

## Area per Lipid

To calculate the projected area per lipid, the coordinates of certain key atoms of the lipids are projected onto a plane and a Voronoi diagram is calculated using these projected coordinates. Formally the projected area *A(l)* of one lipid *l* present in the Leaflet *L*, represented by one key atom *pl*, is then defined as the area of the corresponding Voronoi cell (the area of the polygon) *A _{VC}(pl,L)* with:

where *N* is the number vertices of the Voronoi cell and *(x _{i},y_{i})* with

*i mod(N)*are the coordinates of the vertices of the Voronoi cell.

The molecular total projected area of a lipid *A(l)* described by *N* key atoms is then defined as:

The properties of the created Voronoi cells heavily depend on the chosen set of key atoms: thus the whole Voronoi diagram is a function of a specific set of key atoms. Depending on the key atoms, the information derived from the diagram can change.

## Bilayer thickness

APL@Voro calculates the interpolated bilayer thickness as the distance between key atoms in the top key atoms in the bottom leaflet. For every selected key atom, the interpolated local thickness is calculated by identifying the nearest vertical neighbors (key atoms in the opposite leaflet). The coordinates of a key atom from one leaflet are projected onto a triangulation of the key atoms of the other leaflet. The interpolated local bilayer thickness is then interpolated between the key atom coordinates of the triangle the projection has fallen in and the coordinates of the projected atom itself. Formally: The equation of the plane used to compute the elevation at any point on the triangle is defined by the three vertices of the triangle the projected atom *p _{a}(x_{a}, y_{a}, z_{a})* falls in:

Where *A, B, C,* and *D* are computed from the triangle coordinates *p _{0}(x_{0}, y_{0}, z_{0}), p_{1}(x_{1}, y_{1}, z_{1}), p_{2}(x_{2}, y_{2}, z_{2})* and the atom coordinates

*p*:

_{a}(x_{a}, y_{a}, z_{a})

## Protein atoms (TPI method)

The calculation of the interpolated bilayer thickness and the projected area per lipid is supported even with embedded proteins. For this, a special procedure is used to include relevant protein atoms. Atoms of embedded proteins are inserted into a triangulation of selected lipid atoms using the TPI (Triangular Prism insertion) method. The TPI method is used to identify the boundary lipids (lipids in direct neighborhood of the protein). A lipid is defined as a boundary lipid if and only if a protein atom falls inside the Triangular Prism defined by a triangle in the triangulation. The Triangular Prism is based on the lifted triangle defined by the R3 coordinates of the lipid atoms present in the triangulation, with a height of the maximum z-distance of the three contributing lipid atoms including their van der Waals radii. This method ensures the insertion of protein atoms while taking the fluctuations of the bilayer thickness into account.