All energy calculation methods and the derived techniques are one way or another based on a force field. Although it is a FORCE field, and one would expect therefore that it describes forces, the explanation is normally given in terms of ENERGIES. This should not worry you, because The force is just the first derivative of the energy. That means that if the energy gets bigger, the force gets bigger, and vice versa. Also, the order of the terms will not differ too much. If the energy depends with a very high order on the inter-atomic distance, so does the force. The above explanation is not really correct, because there is no energy between two atoms, but just a force. However, the energy that we discuss is the integral of the force over the distance, or in other words, when moving the two atoms with respect to each other in very, very small steps, you add up the forces needed for each of those very, very small steps, and you get the energy. This second explanation is also not yet correct, but without complicated formulas, this is about the best I can do.
In the seminar on energy calculations and molecular dynamics you will get this topic explained to you in a proper fashion. Here we will only highlight some of the basics.
Energy minimization and molecular dynamics calculations today all rely on the basic idea that the intricate balance of forces that govern molecular structures can be separated in ~ N*(N-1)/2 individual forces Fij between all atom pairs (i,j) (with N atoms in the system).
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Figure 18. If there were to exist only two atoms in vacuum, the forces are the electrostatic force and the Van der Waals term. There is an attractive Van der Waals term and a repulsive Van der Waals term. The attractive term behaves roughly with the sixth power of the distance the repulsive one roughly with the twelfth power. |
The other so-called non-bonded interaction is the electrostatic force. If two atoms have the same charge, they push each other away, if they have opposite charges they attract each other. The electrostatic force is a constant times the product of the two charges, divided by the distance.
These interactions are called non-bonded because the atoms between which the force works are neither covalently bound (1-2 bonds), nor bound to the same atom (1-3 bonds), nor do they have two atoms between them (1-4 bonds). This last case, the 1-4 bonded atom pairs are treated differently in many different energy calculation programs.
The other major category of interactions is formed by the so-called through bond interactions, such as direct neighbours, or atom pairs that have one or two atoms between them in the molecule. The figure below summarizes all the simple interactions that most energy calculation programs take into account:
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Figure 19. Bonded versus non-bonded interactions. |
The total force between two atoms is the sum of all individual terms (again, these are actually energies not forces):
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Figure 20. In this formula the first three terms are bonded, and the last big one that runs over atom pairs I,J combines the Van der Waals and the electrostatic terms. |
The 1-2 and 1-3 bonded terms behave quadratically. The torsional term is of course circular, and typically behaves like:
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Figure 21. In ethane the three peaks are equally high, and the three valleys are equally deep, but in proteins there are most of the time pronounced differences between the three minima. Any idea why there are three minima, and not four or seven or...? |
This torsional term is the main reason for the three rotamers from which we all the time have to choose when we build new side chains onto our backbone. Without knowing it, you have been thinking several days about this plot.
o give you an impression of the behaviour of some terms as function of the distance:
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Figure 22. You can see that any term that goes with the third power of the distance does not do much at long distances. Thus, Van der Waals interactions (hydrophobic interactions fall in this category) only work between atoms that more or less touch each other, but electrostatic interactions (salt bridges are the clearest example of these) still work when the groups involved are rather far apart (10 angstrom or more). |
Everything written above is of course wrong. The only half way correct approach to energy calculations is quantum chemistry. Unfortunately, that is prohibited by CPU time limitations, so we must make approximations.
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Figure 23. In the field, specialists are working on shortcuts in quantum chemistry and on improvements in the quality of force fields for MD and EM. Just to give you an impression what would be waiting for you in this field. |
Question 15: How many classes of energy terms do you need for a molecular dynamics simulation computer program? List those classes and group them.
AnswerQuestion 16: For each of those classes, explain what the parameters mean. Explain which parameters are constants and which are a function of the coordinates.
AnswerQuestion 17: We discussed protein mobility. Mention all types of mobility involved in the function of a protease. So, start your story with a protease hungry and a peptide/substrate happily swimming around, and end the story with with the protease still hungry, but the peptide already eaten. All mobilities, from macromolecular dynamics to quantum dynamics should be included in the story.
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