Introduction: Simplified models of interaction

We have reviewed the methods of representing molecules with quantum chemical methods and found that (with some limited exceptions) it is impractical to apply these methods to biomolecules. It is therefore necessary to consider a more simple level of representation where "atom" centres are considered at the nuclear position for each atom and these interact by a variety of effects. The models developed should both aid qualitative understanding of the physical basis of various experimental effects and provide a quantitative insight via their implementation as part of a potential energy function. For convenience we divide the topic into consideration of covalent interactions (namely bonds, bond angles and dihedral angles) and non-bonded interactions (electrostatic interactions, induction and dispersion effects, repulsion terms) which are treated in the next section.

Covalent interactions

Covalent bonds

In simple terms a covalent bond exists between two atoms if they share electrons between them. In contrast, an ionic bond is formed if electrons are transferred between atoms (e.g., in sodium chloride an electron is given up by the sodium atom to form a Na⁺ ion and accepted by the chlorine atom to form a chloride, Cl^- ion) - such formally charged interactions will be dealt with later). A single bond is formed when one pair of electrons is involved and a double bond when two pairs are involved. In quantum chemical terms such a picture is overly simplistic as the although a bonding orbital results in an increase in electron density between the atoms it also spreads over the rest of the molecule. This is particularly in the case of delocalized bonding. The "classical" example of delocalized bonding is the benzene molecule - which can be described as resonance hybrid between a number of alternate structures:

(Click on icon to proper sized image).

If you are unfamiliar with this then look at any high school chemistry text book or even better have a look under "benzene" and "resonance" in Atkins. Delocalized bonding is important in protein structure: it is why the peptide bond is planar (in Section 3) and it occurs in phenylalanine, tryptophan, glutamic acid, arginine side chains.

The standard way to approximate the potential energy for a bond in a protein and most other molecules is to use a Hooke's law term: E_bond = K_r (r-r_eq)<sup2

where r is the length of the bond (i.e., the distance between the two nuclei of the atoms between which the bond acts), r_eq is the equilibrium bond length and K_r is a spring constant. This basically represents the bond as a spring linking the two atoms.

A few points can be made about this representation:

The shape of the potential energy well will be parabolic and the motion will therefore tend to be harmonic (see a high school physics book if you can't remember why!).
Note that when the bond is at its equilibrium length i.e., r = r_eq the potential energy is assigned to be zero and when r approaches large values (i.e. the bond breaks) the energy goes infinite. This kind of approach does not attempt to reflect the energy of formation of the bond - it only seeks to reflect the energy difference on a small motion about the equilibrium value. You may wish to see a mathematical justification of why a harmonic function can do this.
A much more accurate representation is based on the application of the Morse potential which has an anharmonic potential energy well. This is not greatly used for applications in which the intention is to look at structural details but is necessary if one is interested in spectroscopic applications. With this approach it is also easy to avoid classical approximation in which energy for the bond can take a value from a continuous range as opposed to the correct quantum view in which only certain values are allowed.
If one is interested in treating changes in which bonds are made or broken a quantum mechanical treatment is necessary. A consequence of this is that workers interested in using molecular dynamics methods to look at folding tend to avoid proteins which have disulphide bonds!

Typical values for bond constants and equilibrium bond lengths taken from the AMBER potential energy function:

Atom pair	r_eq in Å	K_r in kcal/(molÅ²)
C = O	1.229	570
C - C2	1.522	317
C - N	1.335	490
C2 - N	1.449	337
N - H	1.010	434

These are the bond parameters necessary in representing a glycine residue and its connections to neighbouring residues. The atom types are:

O a carbonyl oxygen

C a sp2 carbon (such as that attached to an O)

N a main chain nitrogen atom

H a hydrogen atom attached to the N

C2 a "united atom" CH_2 group

(united atom means that instead of representing the carbon atom and the two hydrogen atoms which are bonded to it separately only one centre is considered - further explanation and picture)

Small molecule X-ray crystal structures are typically used to obtain r_eq values. The spring constants K_r are found by performing normal mode calculations and comparing the results with experimental microwave frequencies (as done in the case of AMBER). Parameters can also be obtained from ab initio quantum chemical calculations.

Click here if you would like to see a graph of the energy for a typical bond.

Bond angles

A bond angle theta

between atoms A-B-C is defined as the angle between the bonds A-B and B-C:

As bond angles are found (experimentally and theoretically) to vary around a single value it is sufficient in most applications to use a harmonic representation (in a similar manner to the bond potential): E_angle = K_th (th-th_eq)< sup2

Typical values for equilibrium bond angles and bond angle constants taken from the AMBER potential energy function:

Angle theta_eq in degrees K_theta in kcal/(mol.degrees²)

C-N-H 119.8 35.0

C2-N-C 121.9 50.0

C2-N-H 118.4 38.0

C-C2-N 110.3 80.0

C2-C-O 120.4 80.0

C2-C-N 116.6 70.0

O-C-N 122.9 80.0

(explanation of atom types)

These are the bond angle parameters necessary in representing a glycine residue and its connections to neighbouring residues. A bond angle around 109 degrees means that the central atom is tetrahedral (with four other atoms bonded to it):

The angle around the C beta atom of an alanine residue - showing the three hydrogen atoms bonded to the carbon. In contrast an angle around 120 degrees indicates a flat (sp2) central atom with three other atoms bounded to it:

This shows the angles made around a main chain nitrogen atom are all approximately equal to 120 degrees: consequently the group is planar.

The source of bond angle parameters is the same as for bonds: high resolution small molecule X-ray structures for eqilibrium values and either spectroscopic data or ab initio calculations for force constants.

Click here

if you would like to see a graph of the energy for a typical bond angle.

Dihedral angles

In case you are not too familiar with the what a dihedral angle is why not revise the Peptide Torsion Angles section that you saw earlier in the course. Formally the dihedral angle

(also known as a torsion angle) between four atoms A-B-C-D is defined as the angle between the the planes ABC (marked by red lines) and BCD (marked in purple):

Thus a dihedral angle of zero is a cis conformation and 180 degrees is a trans conformation:

(Click on icon to proper sized image).

The standard functional form for representing the potential energy for a torsional rotation was introduced by Pitzer (Disc. Faraday Soc. 107:4519-4529, 1951):

$E_dihe= sum_(i=1)<supn (V_n/2)*[1+cos{n.phi-gamma}]$

V_n gives the energy barrier to rotation, n the number of maxima (or minima) in one full rotation and gamma determines the angular offset. The use of the sum allows for complex angular variation of the potential energy (in effect a truncated fourier series is used). Barriers for dihedral angle rotation can be attributed to the exchange interaction of electrons in adjacent bonds (see Pauling). Steric effects can also be important (see next section).

Click here if you would like to see a graph of the potential energy terms used by the AMBER PEF to keep the peptide bond planar.

Dihedral angle rotation is also restricted by steric effects

In the 1960's when potential energy functions for proteins were first developed (notably by Harold Scheraga and co-workers at Cornell University) it was found the Pitzer potential was insufficient to give a full represention of the energy barriers of dihedral angle change. Modern potential energy functions normally model the dependence of the energy on dihedral angle change by a combination of Pitzer potential terms and non-bonded effects. The AMBER field half weights 1-4 non-bonded contacts (i.e. those between atoms A and D in a dihedral angle A-B-C-D) justifying this by pointing out that the representation of the repulsive part of the van der Waals contact by a R^{-12} term is too steep compared to the more correct exponential term (used in the Buckingham potential) and that there would be significant charge redistribution at such short distances. Potential energy functions differ in whether they calculate the torsion energy for every set of four bonded atoms, as in AMBER, or whether only one angle (the principal angle) is considered for each bond.

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O	a carbonyl oxygen
C	a sp2 carbon (such as that attached to an O)
N	a main chain nitrogen atom
H	a hydrogen atom attached to the N
C2	a "united atom" group

Angle	in degrees	in kcal/(mol.degrees²)
C-N-H	119.8	35.0
C2-N-C	121.9	50.0
C2-N-H	118.4	38.0
C-C2-N	110.3	80.0
C2-C-O	120.4	80.0
C2-C-N	116.6	70.0
O-C-N	122.9	80.0