25.1.1. The Lennard-Jones potential

When the interaction energies are calculated, the influence of Lennard-Jones interactions, electrostatic effects and hydrogen bonds are all considered. In the Lennard-Jones potential:

ELJ = A/(d**12) - B/(d**6)

the interacting atoms are a distance d apart, and the Energy Variables A and B are calculated from the Van der Waals radius, polarizability and effective number of electrons of the atoms. These values are tabulated in datafile GRUB, and are assembled correctly by the preceding Programme GRIN.

25.1.2. The hydrogen bond potential

The standard hydrogen bond interaction is computed from:

EHB = [ C/(d**8) - D/(d**6) ] * [f(U,U',U'',,,) ] * [f'Q]

where f and f' are functions; U,U',U'' etc are angles and distances defining the geometrical arrangement of the atoms engaged in hydrogen bonding and their neighbours; and Q depends on the charges of the interacting atoms. Energy Variables C and D are computed from the hydrogen bond radii and hydrogen bond energies of the atoms, which are tabulated in Datafile GRUB.

The hydrogen-bonding functions U,U',U'' etc were devised in order to compute the interaction between explicit hydrogen-bonding atoms, and these functions have been described in the previous sections of this User Guide under the main heading for Programme GRIN. The functions ensure that only the appropriate number of hydrogen bonds with the correct hybridization are selected for inclusion in the EHB term.

25.1.3. The electrostatic potential

The electrostatic interaction is:

EQ = p * q * K * [ (1/d) + (M-W)/[ (M+W)*(SQRT(d*d+4*P*Q)) ] ] / M

where p and q are the electrostatic charges on the Probe group and the pairwise Target atom, and K is a combination of geometrical factors and natural constants. The macromolecular Target and the surrounding water have dielectrics of M and W respectively, and the depth of the charges p and q in the Target phase is P and Q. For small molecules the Target phase is effectively absent, and P and Q are both zero.

25.1.4. Interactions with water

In some situations a water molecule may form a Bridge between Target and Probe. Such Water Bridges can significantly stabilise the overall Target-Probe interaction, and a Directive (LEAU) may be used in order to simulate this effect. Setting LEAU=1 instructs Programme GRID continually to monitor nearby sites on the Target at which a water molecule might be firmly bound. It does this whilst determining favoured binding sites for the Probe on the Target in the normal way. Whenever it finds a water site, GRID determines whether that water could form a Hydrogen Bonding Bridge between Target and Probe, and explicitly takes account of the Water Bridge if it would be energetically worthwhile. Setting LEAU=2 instructs GRID to take account of one or two Water Bridges while the Probe is at each grid point.

It sometimes happens that the chosen Probe may interact favourably with the Target at a certain position, but that a water molecule would interact better at the same place. The net interaction energy of that particular Probe at that grid point would then be unfavourable, and GRID will detect regions where this happens if directive LEAU is set equal to 3. In this case some additional enthalpic and entropic terms are then needed in the GRID Energy function, and these are discussed below under the heading COMPETITION BETWEEN PROBE AND WATER.

25.1.5. The entropy terms

There was no specific entropy term in early Versions of GRID, and the interaction between Probe and Target was solely described by the three enthalpic components ELJ, EHB and EQ. However an entropic term S was needed for the hydrophobic Probe which was introduced in Version 14 of the Programme. Entropic terms are also required for conformationally flexible Targets (See Index under Directive "MOVE") and for the detection of selectively unfavourable sites (See below under the heading COMPETITION BETWEEN PROBE AND WATER). An entropic term S is therefore included in the GRID Energy function.

25.1.6. The total interaction potential

The interaction of the Probe group with the Target is computed at sample positions (the grid points) distributed throughout and around the molecule. With the Probe at each GRID point in turn, the interaction is calculated from:

EXYZ = Sum[ELJ] + Sum[EHB] + Sum[EQ] + [S]

in which Sum[ ] indicates pairwise energy summation between the Probe at its grid point and every appropriate atom of the Target (including predicted water molecules if LEAU has been set), and [S] is the appropriate entropic term at the grid point. The overall energy EXYZ is then assigned to that point and written to file.

Summary output tables are also prepared, describing the molecular environment surrounding each favoured grid point. The number and size of these tables can be adjusted by the User, who may also use Directive POSI in order to call for a table at any position of interest.