How to obtain FES from well-tempered metadynamics?

[cptnm...@gmail.com]

I found the following movie demonstrating how non-tempered and well-tempered metadynamics works:
https://sites.google.com/site/giovannibussi/gallery

But I do not understand how the error of well-tempered metadynamics is calculated. The text mentions that the bias potential does not fully compensate the FES, but rather just reduces the energy barrier. The resulting FES (blue and yellow area in the movie) does not converge to a flat profile. Therefore the barriers in the FES obtained from sum_hills are slightly too small. Where does the green line in the movie come from?

I found the same effect in my own simulations. The barriers from well-tempered metadynamics are systematically smaller than the ones from non-tempered metadynamics and get larger when increasing the bias factor. How can I obtain a reliable estimate for the FES from well-tempered metadynamics, that is equal to the one from non-tempered metadynamics and does not depend on the bias factor? I cannot find any reference to this in the original paper.

Also, how does reweighting fit into this? The movie was made in 2008, long before the reweighting algorithm was published, so that cannot be the solution. When I reweight my own simulations, the barriers get even smaller and still depend on the bias factor in the same way. Without reweighting the deviation from non-tempered metadynamics is about 5%, with reweighting it is almost 10%. As far as I understand, reweighting should recover the correct FES, no matter how the bias looks like (as long as it is does not change significantly with time).

[Giovanni Bussi]

Hi,

"Therefore the barriers in the FES obtained from sum_hills are slightly too small"

this is not correct. What is shown in the movie as a green line is the difference between the real FES and the one obtained with sum_hills. For well tempered metadynamics, sum_hills does not compute the negative bias potential but rather the negative bias potential multiplied by a suitable factor (larger than one) that should converge to the FES.

So, if you see different barriers there should be another reason. It is difficult to judge without knowing more, but something that I can guess is that with well-tempered metadynamics you use smaller hills in the long run, so that the hysteresis in the bias potential is smaller and you see smaller barriers. These barriers should be closer to reality than those obtained with non-well-tempered. However, these is also a drawback: if the hills get too small your simulation will be stuck. This means that the CV is not good enough to driver the transition when the bias becomes quasi-static. With non-well-tempered this does not happen (the bias is always modified at the same speed) so that you typically see a lot of transitions, but the system could go through a path that is farther from equilibrium and then show an apparently larger barrier.

Does this make sense for your case?

Concerning reweigthing, I am aware of three methods that are expected to work with well-tempered metadynamics:

- Bonomi et al JCC 2009

- Branduardi et al JCTC 2012

- Tiwary and Parrinello JPCB 2015

The second and the third at least can be used with PLUMED 2 and should provide comparable results. You can check more in this list to find discussions about differences and similarities.

Giovanni

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[cptnm...@gmail.com]

Hi Giovanni,

thanks a lot for your answers. In the movie, I was thinking about additive constants, which do not matter in this context. But of course a multiplicative constant does matter and increases the height of the barrier. That is where the green line comes from.

So you are saying my well-tempered and non-tempered simulations disagree, because the hills in the non-tempered case are too big and therefore the system does not find the optimal path? I do not think this applies in my case, because I am using a single CV and therefore there is only a single path between the states. To test this hypothesis, I repeated the non-tempered simulations and decreased the height of the hills by a factor of 50. That is about the final height from the well-tempered simulations. I also decreased the deposition frequency by a factor of 10. After the simulation converged, there was no significant change in the barrier height from the previous runs. So it seems the height of the hills is not the problem. Am I missing something?

What do you mean by "if the hills get too small your simulation will be stuck"? Once the simulation converged, the system will be able to move in the relevant area of CV space and many transitions will occur. This behavior will remain for the entire duration of the simulation, no matter how small the hills get. How can the system get stuck at this point? Of course, if the bias factor is too small to begin with, the system will not reach this state an get stuck in some minimum.

For the reweighting question, I should have been more specific. I am using the method from the third reference you mentioned by using the REWEIGHT_METAD, HISTOGRAM and CONVERT_TO_FES actions. The reweighting is performed on the same CV that was used for biasing, so the resulting FES should match the one from sum_hills.
I have now applied the same procedure also to the non-tempered simulations and it seems the barriers are again slightly smaller than the ones from sum_hills. The same thing happened when reweighting the well-tempered simulations. Is it a general feature of this reweighting method, that the barriers get slightly smaller? I thought this might be due to the system finding different paths between the states, but since there is only a single CV, this is probably not the case.

Alex

[Alessandro Crnjar]

Hi all,

I found online this conversation and I thought about posting here my doubt. Is there a reference for the algorithm used by sum_hills in order to generate the fes.dat file? I didn't find anything like it on its plumed webpage. Is it the same of applying the Pratyush reweighting? Is there a problem in using sum_hills for getting the FES of a well-tempered metadynamics run?

Thanks a lot!

Alessandro Crnjar

[Giovanni Bussi]

Hi,

sum_hills is just doing the following:

- sum the Gaussians on a grid. This is identical to what PLUMED does internally on the fly to put the Gaussian on a grid, except that (a) the grid spacing  and range might be different and (b) PLUMED additionally multiply by a biasfactor-related term to obtain the bias potential.

- change the sign

- possibly integrate over some of the dimensions (with --idw)

This is not the same as Tiwary & Parrinello reweighting, and is not expected to give the same result.

It is instead very close to Branduardi et al reweighting (i.e. "final bias"), except for details related to the smoothing given by the Gaussians.

It is safe, with the caveats discussed above in this thread. One thing that should NOT be done is reporting the "convergence" obtained with sum_hills (which shows the convergence of the bias) as a convergence in free energy estimates. Another suggestion is that, since reweighting schemes are based on a weighted histogram, you can use them to discard the initial part of the simulation. This should give a more robust estimate. And it allows you to do block analysis, as you can see in some of our masterclasses.

I hope this helps!

Giovanni

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[Alessandro Crnjar]

Dear Giovanni,

thanks a lot! This actually correct a wrong belief I had for years on the plumed package. I always though sum_hills would execute a Pratyush reweighting and thus would be safe to use in any circumstance. 

Best,

Alessandro