Electrostatic
Opposite charges attract. For example, Mg2+ ions associate with the negatively charged phosphates of nucleotides and nucleic acids. Within proteins, salt bridges can form between nearby charged residues, for example, between a positively charged amino group and a negatively charged carboxylate ion. These electrostatic interactions make an especially large contribution to the folded structure of nucleic acids, because the monomers each carry a full negative charge.
where E is the electrostatic energy in kcal/mol, i and j are the two interacting charges, q is the charges’ magnitude, r is the distance between them, and epsilon is the dielectric of the medium in which they exist. Note that a positive sign for E represents an unfavorable (repulsive) interaction, whereas a negative sign represents a favorable (attractive) interaction.
Van der Waals interactions (see Figure)
It represent the attraction of the nuclei and electron clouds between different atoms. The nucleus is positively charged, while the electrons around it are negatively charged. When two atoms are brought close together, the nucleus of one atom attracts the electron cloud of the other, and vice versa. If the atoms are far apart (a few atomic radii away) from each other, the van der Waals force becomes insignificant, because the energy of the interaction varies with the 12th power of distance. If the atoms come closer together (so that their electron clouds overlap) the van der Waals force becomes repulsive, because the like charges of the nucleus and electron cloud repel each other. Thus, each interaction has a characteristic optimal distance. For two identical atoms, the optimal distance is d=2r, where r is atom radius. Within a biomolecule, these interactions fix the final three-dimensional shape. While van der Waals interactions individually are very weak, they become collectively important in determining biological structure and interactions.
Van-der Waals interactions actually have two component; the attractive interaction between the atoms, which results from the induced dipoles, and a repulsive interaction, which results from overlap of the electron clouds of the two atoms, when they get too close to each other. The total energy of van-der Waals interactions can be approximated by the Lennard-Jones expression:
the detailed formula description
The Lennard-Jones potential
The Lennard-Jones 12-6 potential is given by the expression ![\begin{displaymath}
\phi_{\rm LJ} (r) = 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12}
- \left(\frac{\sigma}{r}\right)^{6} \right]\end{displaymath}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_s0QV4xZLghfxgPQUtWwVdC3gU5UmNUrlsKv5vlohQ7rdpenl70XKdUPEnVW3QBxS-I4148OcAKxBm-phMD28L1mIK03rRzCCdSk0ZYvFs0ZcSEH9cN1hEwULo=s0-d) |
(5) |
This potential has an attractive tail at large r, it reaches a minimum around 1.122
, and it is strongly repulsive
at shorter distance, passing through 0 at 
and increasing
steeply as r is decreased further.The term
, dominating at short distance, models
the repulsion between atoms when they are brought very close to
each other. Its physical origin is related to the Pauli principle:
when the electronic clouds surrounding the atoms starts to overlap,
the energy of the system increases abruptly.
The exponent 12 was chosen exclusively on a practical basis:
equation (2.3) is particularly easy to compute.
In fact, on physical grounds an exponential behavior would
be more appropriate.
The term
, dominating at large distance, constitute
the attractive part. This is the term which gives cohesion to
the system. A 1/r6 attraction is originated by van der Waals
dispersion forces, originated by dipole-dipole interactions
in turn due to fluctuating dipoles.
These are rather weak interactions, which however dominate the
bonding character of closed-shell systems, that is,
rare gases such as Ar or Kr.
Therefore, these are the materials that a LJ potential could
mimic fairly well.
The parameters 
and 
are chosen to fit
the physical properties of the material.
On the other hand, a LJ potential is not at all adequate to model situations with open shells, where strong localized bonds may form (as in covalent systems), or where there is a delocalized ``electron sea'' where the ions sit (as in metals). In these systems the two-body interactions scheme itself fails very badly. Potentials for these systems will be discussed in chapter 4.
However, regardless of how well it is able to model actual materials, the LJ 12-6 potential constitutes nowadays an extremely important model system. There is a vast body of papers who investigated the behavior of atoms interacting via LJ on a variety of different geometries (solids, liquids, surfaces, clusters, two-dimensional systems, etc). One could say that LJ is the standard potential to use for all the investigations where the focus is on fundamental issues, rather than studying the properties of a specific material. The simulation work done on LJ systems helped us (and still does) to understand basic points in many areas of condensed matter physics, and for this reason the importance of LJ cannot be underestimated.
When using the LJ potentials in simulation, it is customary to work in a system of units where
and 
.The example codes accompanying these notes follow this convention.
ps. Link
Non-polar ‘forces’
Uncharged atoms also interact with each other due to attractive non-polar forces (i.e. the “hydrophobic effect“). This interaction does not result from an actual force between the interacting species, but rather from their tendency to avoid water. Thus, water ‘pushes’ the non-polar atoms towards each other:
Non-polar interaction between hydrophobic entities (red). Water molecules are represented by the blue spheres
The energy of non-polar interactions (Enp, in kcal/mol) has been
found empirically to depend on the size of the water-accessible surface
area, lost upon the interaction (ΔSA):
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