Force Fields

Force Fields

Seminar 3 in the series…

G Vriend 15-9-2009

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What is a Force Field ? Wikipedia:• 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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Molecularly speaking, it is about time versus accuracy

• Quantum chemistry• Approximations• Force Fields• Hybrid methods• Self consistent fields• Molecular dynamics and energy calculations (seminar 6)

• Minimisers• 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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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.

This will not come back at the exam

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Quantum chemistry is accurate, but slow

This will not come back at the exam

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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 given the problem you want to solve.

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Next step: 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 YASARA movie in the practical 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

This will not come back at the exam

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Molecular dynamics using Newton’s ‘rules’

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If it is all still too slow, 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 (or many) positions. This, of course, is an approximation relative to methods based on molecular dynamics. Often used in drug design (seminar 5).

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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 (seminar 5).

This will not come back at the exam

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Electrostatic calculations

Electrostatic calculations are based on self-consistent field principles. This field is different from the force fields we have seen so far. It is a distribution of potentials digitized on a grid that covers the space in and around the molecule. Needed for drug design (seminar 5) and some MD (seminar 6).

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Electrostatic calculations

Often physics looks like Chinese typed backwards by a drunken sailor, but when you spend a bit of time, you will see 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):

This will not come back at the exam

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Electrostatic calculationsThe 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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But even this is too time consuming to calculate

So, most MD force fields don’t recalculate atomic charges, at all...

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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 all frequencies:

P(aa,HSR)=P(aa)*P(HSR)

Calibrate the method

Use it by looping over the amino acids in the protein to be tested and multiply all chances P(aa,HSR).

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One ‘serious’ example: Chou and Fasman

Example of Chou and Fasman:We count all amino acids in a dataset of 400 proteins with know structure (they had many fewer proteins available in 1974, but anyway...)

These 400 proteins in total have 102.197 amino acids.

Ala 7.123 = 7.0% Helix 35.017 = 34.3%Cys 1.232 = 1.2% Sheet 27.038 = 26.5%Asp 5.993 etc Rest 40.142 = 39.3%Glu 6.086Phe 4.822Gly 7.339His 989Ile 6.550Lys 8.127

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What is the null-model?

The null-model is the model that assumes that there is no signal in the input data. In case of our Chou-and-Fasman example, the null model assumes that there is no relation between the amino acid type and the secondary structure.

So, if 7% (0.07) of all amino acids are of type Ala, and ~34% (0.34) of all amino acids are in a helix, then 7% of 34% (0.07*0.34) is 2.4% (0.024) of all alanines should be observed in a helix.

And since that isn’t true, we can make a model that differs from the null-model, and thus we can make predictions.

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One ‘serious’ example: Chou and Fasman

These 400 proteins in total have 102.197 amino acids.Ala 7.123 = 7.0% Helix 35.017 = 34.3%Cys 1.232 = 1.2% Sheet 27.038 = 26.5%Asp 5.993 etc Rest 40.142 = 39.3%

(Ala,Helix)predicted=0.07*34.3=2.4% or 2505 Ala-in-helix predicted in the data set of 400 proteins.But we count 3457 Ala-in-helix; that is 1.38 times ‘too many’.So chances are ‘better than random to find an alanine in a helix. How do we quantify this?

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Come to the rescue, one long dead physicist

This is at the basis of:ΔG = ΔH - TΔSΔG = -RTln(K)

And of Vriend’s rule of 10...

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One ‘serious’ example: Chou and Fasman

These 400 proteins in total have 102.197 amino acids.Ala 7.123 = 7.0% Helix 35.017 = 34.3%Cys 1.232 = 1.2% Sheet 27.038 = 26.5%Asp 5.993 etc Rest 40.142 = 39.3%

(Ala,Helix)predicted=0.07*34.3=2.4% or 2505 Ala-in-helix predicted in the data set of 400 proteins.But we count 3457 Ala-in-helix; that is 1.38 times ‘too many’.

So the ‘score’ for (Ala,helix) = Pref(A,H)= ln(observed/predicted) = ln(3457/2505)=ln(1.38)=0.32. The preference parameter Pref(A,H) is positive. So, here positive is good (unlike ΔG or AIDS tests).

And how do we now predict the secondary structure of a protein?

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One ‘serious’ example: Chou and Fasman

One example could be:Loop over the sequence to be predictedGive each residue 3 energy scores (for H, S, and rest, respectively)Find stretches where the energy for one of the three is higher than the other two.

Hundreds of recipes can be thought of, and the best one wins (CASP competition teaches us that the best one is a neural network that uses multiple sequence alignments as input).

SNDALPIVAKGS Nice helix, but for the PIV.TSEAAQIALHSG Aha, P is exception, and V tooTNDLLMAVMRGG and I too, so it IS a helix after all.

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And the other way around

ΔG= -RT.ln(K)

ΔG is just over 1kCal/Mole when K=10 and K is the ratio between two ‘somethings’ (can be anything). Swimming into a gradient of a factor 10 costs 1 kCal/Mole. A pH unit difference must be ‘worth’ a kCal/Mole.

A nice exam question would be to think of an example of this ‘law of 10’ that hasn’t been discussed in the course yet...