Force Fields

Force Fields

G Vriend 20-9-2005

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Force Fields

What is a Force Field ?• A force field is a set of equations and parameters which when

evaluated for a molecular system yields an energy • A force field is a specific type of vector field where the value of a

given force is defined at each point in space. Examples include gravitational fields  and electrostatic fields

• In the fictional Star Trek  universe, force shields are the defenses most commonly used to protect a starship. The physics of a shield is extracted from the physics of a force field ….. etc.

• The space around a radiating body within which its electromagnetic oscillations can exert force on another similar body not in contact with it

• Force field analysis evaluates non-monetary factors, just as cost-benefit analysis evaluates monetary factors

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Force Fields

It is all about time versus accuracy

• Quantum chemistry• Approximations• Force Fields• Hybrid methods• Self consistent fields• Molecular dynamics and energy calculations• Minimizers• Yasara-Nova

We will first travel from quantum chemistry to brownian motion and after that we will look at a series of other Force Fields.

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Force Fields

Quantum chemistry is accurate, but slow

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Force Fields

Quantum chemistry is accurate, but slow

The largest ‘thing’ that can realistically be worked-out using the Schödinger equation is hydrogen. Other applications are the particle in a box that is mainly of theoretical importance, the postulates of quantum chemistry, etc.

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Force Fields

Quantum chemistry is accurate, but slow

Actually, pure quantum chemistry cannot be applied in our (protein) world. Which is good, because quantum chemistry is much too difficult (for me). But many of the results are very useful. For example, all atoms in proteins display sp2 - sp3 hybridization.

Pictures obtained from Clifford J Creswell

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Force Fields

Approximations, faster, less accurate

Approximations can make quantum chemistry software faster, but at the cost of accuracy. A major part of all efforts in quantum chemistry is to think about short-cuts that have an optimal price/ performance ratio.

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Force Fields

So, we will use Newtonian mechanics

If we want to calculate on molecules that contain thousands of atoms, we have to totally abandon quantum chemistry, and use Newton’s laws of motion, treating atoms as macroscopical particles instead of quantum chemical entities. The following (YASARA) movie will explain how this is done.

ΔH wants to go down ΔS wants to go up ΔCp cannot be calculated

H(T) = H(Tref) + Tref

T

Cp .dT

S(T) = S(Tref) + Tref

T

(Cp /T).dT

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Force Fields

What can we do with EM and MD?Despite all its shortcomings, MD can be used to calculate binding constants of ligands in active site pockets with reasonable accuracy.This is done with so-called thermodynamic integration which works because binding a ligand is a state-function (the path is not important, only the end-points; so non-realistic paths are allowed):

Take any closed cycle. Calculate the easy differences, and since the cycle is closed, you obtain also the value of the difficult transitions.

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Force Fields

We can turn the thing inside-outOther approaches are also possible. Rather than calculating the energy lost or gained to actually move an atom somewhere, we can calculate the potential energy for atoms at a certain position. This, of course, is again an approximation relative to the thermodynamic integration method. Examples: LUDI or GRID.

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Force Fields

And one more approximation step....

Lets go yet one step further. Assume we have a series of docked molecules. We superpose them, and determine what they have in common. The next drug should have those same characteristics. This approximation step is known as QSAR (more precise in DD course).

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Force Fields

Other force fields

So far we discussed molecular dynamics force fields and ‘approximated’ them into ‘experience based’ drug design.Many other force fields exist. For example, many force fields exist for the purpose of validating protein structures or models. Example ProSa:

1) Measure Cα distances2) Score good proteins3) Normalize the scores4) Score protein of

interest

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Force Fields

Electrostatic calculations

Electrostatic calculations are based on self-consistent field principles. This field is not a force field like we have seen so far, but a distribution of charges over a grid that covers the space in and around the molecule.

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Force Fields

Electrostatic calculations

Often physics looks like Chinese typed backwards by a drunken sailer, but when you spend a bit of time, you will that things actually are easy. Take the Poisson Bolzmann equation that is used for electrostatic calculations:

which can be converted into:

This looks clearly impossible, but after a few days of struggling, it becomes rather trivial (next slide):

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Force Fields

Electrostatic calculations

The Poisson Boltzman equation is worked out digitally, i.e., make a grid, and give every voxel (grid-box) a charge and a dielectricum. Now make sure neighbouring grid points have the correct relations. If a voxel has ‘too much charge’ it should give some charge to the neighbours. This is done iteratively till self-consistent.

The same technology is used to design nuclear bombs, predict the weather (including the future path of

tornados), design the hood of luxury cars, predict how water will flow in the Waal, optimize catalysts in mufflers, optimize the horse powers of a car given a certain amount of gasoline (turbo chargers), etc.

And the function is very simple!

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Force Fields

Other force fields

Force fields do not need to be based on atoms. A very different concept would be a secondary structure evaluation force field:

Take many different proteins and determine their secondary structure.Determine how many residues in total are H, S, or R, and do the same for each residue type. Determine preference parameters.

P(aa,HSR)=P(aa)*P(HSR)Pref(aa,HSR)=Ln (observed/predicted)

observed is simply counting (aa,HSR) in the 4000 proteinspredicted is P(aa,HSR) * (total number of aa in the 4000 proteins)Callibrate the method with a Jack-Knife procedureLoop over the aa in the protein to be tested and add up all Pref(aa,HSR). Express outcome in energy or standard deviations.

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Force Fields

Force Fields

So, what is a force field? There are so many different ones for totally different things: car design, electrostatics, nuclear bombs, tornados, …

A force field is a set of rules that can predict the ‘optimal constellation’ of a system in the absence of external forces.

So, in case of electrostatic calculations, the field can be calculated in the absence of molecular motion. But for a weather forecast one can only take small steps in a dynamic system as the sun adds energy to the system, so every time unit everything has to be recalculated days in advance.

Most force fields can be used to optimize/minimize the system, and here we run into the multiple minimum problem.

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Force Fields

Multiple minimum problem

But this is a very simple, one-dimensional case. How many minima do you think can be found in crambin (326 heavy atoms)?

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Force Fields

Back to proteins and MD/EM

During an MD simulation atoms don’t move very far.

A) Because molecules normally aren’t very flexibleB) Because we cannot run the simulations long enoughC) Because the forcefields are far from precise enough

We can use this to do MD differently....

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Force Fields

Back to proteins and MD/EM

We have seen that the few forces that we (think that we) understand mainly are of the form Q=k*(x-x0)In this equation x0 is known with great precision, while k can easily be wrong by a factor of two or more. Can we use the precision of x0?

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