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Why are bonds larger than their equilibrium value?
#26575 02/21/11 04:47 PM
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Hi,

We have studied a small 4-residue protein with CHARMM's
parameter set 22 (with CMAP correction terms) in explicit
TIP3 water (constant pressure & temperature), SHAKE on
hydrogen bonds, 2fs time-step (leap frog).

We note that the average bond lengths are all slightly
above their equilibrium value of the harmonic potential.
For example, the N-terminal bond +H3N---C(alpha) of the
first residue (Gly) has an average value of 1.50 Ang;
whereas the equilibrium bond length of the CHARMM potential
is 1.48 Ang in the parameter set 22.



When we construct from the trajectory the potential of
mean force we obtain a harmonic potential that is
slightly shifted to larger values, and the force
constant is also slightly increased (250 kcal mol-1 Ang-2
instead of 200 kcal mol-1 Ang-2 of the CHARMM potential).
This is not due to statistical errors as you can see by
the error bars; the bars correspond to the standard
deviation for five independent runs (each them 1.5 microseconds long!).

What could be the reason for this increase in
the equilibrium distance and the force constant?
Is this a general behavior?

Thanks,
Martin

Re: Why are bonds larger than their equilibrium value?
mspichty #26576 02/21/11 05:00 PM
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Why wouldn't they? There are other forces than the bond force action on an atom, and the only way for the bond to create a balancing force is to deviate slightly from its equilibrium length. The equilibrium lengths in the parameter file therefore have to be set to values such that in the end good agreement with observed actual bondlengths is obatined.


Lennart Nilsson
Karolinska Institutet
Stockholm, Sweden
Re: Why are bonds larger than their equilibrium value?
lennart #26594 02/22/11 02:14 AM
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Lennart is correct. In your case there may be repulsive electrostatic interactions that lead to the minimized or PMF based bond length being longer that that specified by the bond parameters in combination with the harmonic bond term alone.

alex


School of Pharmacy
University of Maryland
20 Penn Street
Baltimore, MD, 21201
Re: Why are bonds larger than their equilibrium value?
alex #26598 02/22/11 08:32 AM
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Thanks, Lennart and Alex!

Re: Why are bonds larger than their equilibrium value?
mspichty #26610 02/23/11 09:54 AM
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No guys. There's another larger effect here. It's an entropic effect. There is simply more phase space with the bond stretched than there is when the bond is compressed.

You could run your script on a simple diatomic system using LD, and see a similar stretching even when there are no other forces present.

Re: Why are bonds larger than their equilibrium value?
BRBrooks #26611 02/23/11 10:34 AM
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Maybe for a larger system, but the diatomic is essentially a 1-D harmonic oscillator, and yet you say it would know which direction is expansion and which is contraction, even in the absence of other forces?


Lennart Nilsson
Karolinska Institutet
Stockholm, Sweden
Re: Why are bonds larger than their equilibrium value?
lennart #26612 02/23/11 03:59 PM
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Yes. It's both for large and small system. This is related to the Jacobian.

There are two ways to think about this:

1. If you take a diatomic molecule at rest (i.e. zero energy), and you randomly move one atom by 0.1 A (isotropicaly random), then the bond will more likely increase in length, than decrease. There's simply more volume in the "increase" direction.

2. If you dynamics run a molecule in a vacuum (coupled to a heatbath in some manner -- say Langevin Dynamics), then the pressure on the system must remain at zero. If the kinetic energy is positive, then the virial must be negative on average, and the only way you can get a negative average virial is if the bond is stretched, on average.

For larger molecules, it gets a little more complex, but simply think that if there were not overall attractive forces to hold the molecule together, it would fly apart. The graph plotted by Martin simply reflects those forces that are keeping the molecule from flying apart into atoms.

Re: Why are bonds larger than their equilibrium value?
BRBrooks #26628 02/25/11 03:44 AM
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I can see the entropic effect is real, but is its magnitude really that large? Because minimizing protonated H-Gly-NMe in vacuum gives me a +H3N---C(alpha) bond length of 1.513 due to the effect cited by Lennart and Alex (probably mostly electrostatic interactions). My minimized bond length is ever so slightly larger than Martin's fitted value of 1.498 (admittedly on a different molecule, but still...) It looks like the solvation shortens the bond length in comparison with vacuum (probably due to a combination of pressure and weakening of repulsive electrostatic interactions), and that this "dielectric/pressure effect" is larger in magnitude than the entropic effect.

Re: Why are bonds larger than their equilibrium value?
Kenno #26637 02/26/11 01:31 AM
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I think that it depends on the system and on the simulation setup. Clearly the way to estimate this effect is to look at the same system at different temperatures. I was simply pointing out that as the temperature increases, bond lengths tend to get longer, and not shorter.

Also, running with a large time step (as reported above) should make the effect appear larger, since this reflects motion in a high frequency degree of freedom.

I may have been wrong when I claimed it was a "larger effect", but it is an effect. (Note that when the bonds are rather stiff, the effect is smaller).

Also note that when people calculate a radial potential of mean force, they almost always correct for this same effect.

Re: Why are bonds larger than their equilibrium value?
BRBrooks #26638 02/26/11 01:34 AM
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One more point:

Proteins undergo about a 3% volume increase in heating from 0K to 300K.
This reflects a 1% change in linear dimensions, and some of that is due
to bond length stretching.


Moderated by  alex, lennart, rmv 

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