16.1. The favourable components WENT and ELJ

Ordered water is commonly observed in the X-ray crystal structures of proteins, but that water is usually making obvious hydrogen bonds to polar groups on the macromolecular surface. One may sometimes find ordered water in a completely hydrophobic region, but this is by no means common and the ordering which is thought to be responsible for the hydrophobic effect cannot, in general, be observed on an X-ray time scale. Some other type of ordering on a faster time scale must be postulated, and the present model is constructed on the assumption that it is the hydrogen bonds of water which are ordered at the undisturbed hydrophobic surfaces of the Target.

Many of the hydrogens of bulk water are making hydrogen bonds, but the present model assumes that a still higher proportion of hydrogens in the hydration shell are engaged in hydrogen bonding. However, this hydration shell is not uniform, because some parts may overlay a polar surface of the Target, while other parts overlay a hydrophobic region. When the Target atoms are hydrophobic they cannot make hydrogen bonds with water molecules, and those waters in the shell must therefore make their hydrogen bonds from one water molecule to another water. These water-water hydrogen bonds generate an ordered "structure of water", and it is this ordering which is disturbed by the interference of polar Target atoms.

In order to estimate the size of WENT it may be noted that there are enough hydrogens in water to have four hydrogen bonds per oxygen atom, and the assumption is made that every hydrogen in the hydration shell does make its hydrogen bond. This is of course a limiting assumption, and in reality such perfect order is not to be expected. We also assume that only three hydrogen bonds are made by each oxygen in bulk water, instead of the four which are theoretically possible in an ideal tetrahedral arrangement. Once again this seems to be an extreme limit, and the actual proportion may well be greater than 3/4.

The four possible hydrogen bonds of a bulk water molecule may be numbered 1,2,3 and 4. Then the three assumed bonds could be selected as numbers 1,2,3 or 2,3,4 or 3,4,1 or 4,1,2. There would be four permutations, and the corresponding contribution to the Free Energy of the water would then be:

WENT = -(1.386*308*1.987)/1000 = -0.848 Kcal/Mole

where Ln(4)=1.386 and 308 Kelvin is taken as body temperature and 1.987 cal/mole.K is the gas constant.

On this model, the hydrogens of water are ordered in hydrophobic regions of the hydration shell, in the sense that all hydrogen bonds are actually made and they are all made from one water molecule to another. On the other hand the hydrogens are more disorganised in bulk water, and are also disorganised wherever the surface of the Target is polar.

When two hydrophobic surfaces come together there will be a favourable induction/dispersion interaction between the molecules, and ELJ is used in order to estimate this component. The displacement of ordered water molecules from the hydrophobic surfaces into bulk water also makes a favourable contribution WENT. Thus (ELJ+WENT) is used to estimate the favourable part of the hydrophobic interaction.

It is not claimed that WENT=-0.848 Kcal/Mole measures the "correct" entropic component, whatever "correct" may mean. One could argue that -0.848 Kcal/Mole may be a high estimate because it was calculated on the basis of limiting assumptions which may tend to overestimate the result. However, one must also bear in mind that water would be displaced from the surfaces of both interacting molecules, and this could be a counter argument for doubling the entropic contribution. These factors tend to oppose each other, and we have therefore retained the original estimate WENT=-0.848 Kcal/Mole for the present computations.

It is interesting to compare the relative magnitudes of ELJ, WENT and EHB in the Grid Force Field. The entropic term (-0.848 Kcal/Mole) is bigger than a typical Lennard-Jones energy between two atoms which is about -0.2 Kcal/Mole for carbons, oxygens and nitrogens. On the other hand it is less than many hydrogen bond energies which are characteristically -2 -3 or even -4 Kcal/Mole. Thus WENT tends to dominate the hydrophobic effect, but is itself overwhelmed by polar interactions (See below).

Of course the actual size of the entropic contribution (-0.848 Kcal/Mole) would be altered if different assumptions were made, and a more realistic partition function were used. However, altered assumptions about the partition function would not change the difference between the computed hydrophobic energy at one grid point, and the computed energy at another. On the present model this difference depends upon the particular properties of the Target surface. The entropic contribution is assumed to be constant at an undisturbed hydrophobic surface.

16.2. Unfavourable hydrophobic components

Even in a predominantly hydrophobic region, there may still be a few polar atoms on the surface of the Target, and those polar atoms will often make hydrogen bonds to water in the hydration shell. They can influence the hydrophobic effect in two distinct ways:

1. They disrupt the local ordering of water in the hydration shell. This diminishes the favourable entropic component.

2. Any hydrogen bonds by polar atoms of the Target to water may be broken when the hydrophobic surfaces come together, because the water will be displaced and the polar atoms will then, in general, face the opposing hydrophobic surface. This breaking of hydrogen bonds is enthalpically unfavourable.

The energy EHB of these hydrogen bonds is therefore set against the favourable components (ELJ+WENT), and the overall hydrophobic interaction energy of the Grid Probe is finally computed as:

ELJ + WENT - EHB

in which each term is negative. A few simple examples will now show how this expression is evaluated.

16.3. Output tables for the hydrophobic probe

Lineprinter output files are produced as usual when the hydrophobic Probe is used. However, some interactions of this Probe cannot be described as atom-atom effects between the Probe itself and one explicit Target atom. These include the entropic component associated with ordered water in the hydration shell, and its modification by polar interactions between water and hydrogen-bonding Target atoms. The tables in a GRIDLONT file may therefore have an extra line which shows "ADJUSTMENTS FOR ENTROPY AND FOR OTHER INTERACTIONS". This extra line is only displayed when the hydrophobic Probe is being used.

16.4. The hydrophobic probe with flexible targets

A special procedure is used for the hydrophobic Probe when parts of the Target are flexible and directive MOVE > 0. For example, the side-chain of lysine has hydrophobic methylene groups and a terminal polar nitrogen cation:

--CH2--CH2--CH2--CH2--NH3+

It is assumed when directive MOVE > 0, that the methylene groups will tend to move in towards the hydrophobic group represented by the hydrophobic Probe, but the polar nitrogen will tend to move in the opposite direction towards the aqueous environment. What actually happens on this model, depends upon the overall balance between these two effects.

16.5. Hydrophobic summary

In this model it is important to note that the whole hydration shell of the Target molecule is regarded as being a special region. The same value (WENT = -0.848 Kcal/Mole) is accepted in principle for the entropic contribution at all undisturbed parts of the shell, and the hydrophobic energy is computed as (WENT+ELJ-EHB).

WENT is therefore responsible for much of the so-called hydrophobic effect, but may not be wholly responsible because the induction/dispersion interaction energy (ELJ) between the hydrophobic molecules must not be forgotten. Some workers regard ELJ as part of the hydrophobic interaction, while others regard ELJ as something completely different. However, that distinction may be a matter of nomenclature if one is working at a constant temperature, and is only trying to find where two hydrophobic molecules would tend to interact favourably with each other in an aqueous environment.

The name of the hydrophobic Probe is DRY, and you can use this name in order to set up a GRID run in exactly the same way that you would use (for example) the name OH for an aromatic hydroxy Probe. You will find DRY in the Probe Menu of Programmes GREAT and GREATER, or you can type DRY into your command file for GRID.