question related to the reweighting

[qinghua Liao]

Dear Plumed users,

I have one question about the reweighting for metadynamics.

According to my understanding, the free energy is determined as the minus of the sum up of all the added Gaussian bias potential for metadynamics simulation,
while the reweighting is used to calculate the unbiased probabilities from the biased probabilities along the CV of metadynamics simulation. The unbiased probabilities
can then be used to determine the free energy.

Then I would consider the free energy calculated by summing up all the bias potential be the same as the free energy calculated by the unbiased probabilities, if simulation is
converged, they are two ways of calculating the free energy. Is my understanding correct? Please correct me if I am wrong. Thanks very much!


All the best,
Qinghua

[Giovanni Bussi]

Hi,

To be slightly more precise, there are different ways discussed in the literature to reweight well-tempered metadynamics simulations. I know at least three of them.

1. Bonomi, Barducci, and Parrinello, JCC 2009. This is slightly more complicated to implement than the next two since it requires histogramming. I never used it directly, perhaps it is better if Max comments on it.

2. Branduardi, Bussi, and Parrinello, JCTC 2012. This prescribes the use of the final bias potential from metadynamics to compute the reweighting factor. This estimator can be shown to be identical to the usual formula linking V to F within the limit of infinitely narrow Gaussians (that is usually reasonable). So, agreement between this estimator and the formula using V cannot tell you anything interesting!

3. Tiwary and Parrinello, JPCB 2014. This paper gives a slightly different prescription for reweighting factors. I tried on model systems and results are very similar to 2. However, there might be cases where 3 is better than 2 or 2 is better than 3. Since equivalence with the formula using V is not anymore automatic, differences in the result could indicate a problem. I am sure Pratyush can comment more about this.

Finally, in the case of non-well-tempered metadynamics there are less possibilities discussed in the literature. For sure one way to analyze bias-exchange MD is discussed in Marinelli et al PLoS Comput Biol 2009. I am (almost) sure that if you apply the prescription there to a "single replica bias exchange" simulation (that is a normal MetaD simulation) you would again get a result identical to the usual -V formula. So, this should be similar to the approach 2 discussed above.

Certainly the methods above all work correctly for an infinitely long and converged simulation. Thus, if they give different results, this might suggest some convergence problem. However, I don't think there is any guarantee that if they agree the result is correct.

The right way to compute statistical error is not to compare the same simulation analyzed with different approaches, but rather to compare simulations that are as independent as possible (or, as in block analysis, consecutive parts of the same simulation) analyzed with the same approach.

Giovanni

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[Alessandro Laio]

Indeed a procedure for  reweighing standard metadynamics calculations is described in Marinelli et al PLoS Comput Biol. This procedure is valid also if only one replica is used, but of course using a bias exchange scheme helps convergence. I recall that in non-tempered metadynamics the error  can be estimated by performing standard block analysis on  the time average of the bias potential, or on any observable computed by reweighing. Indeed, the process generated by non-tempered metadynamics is stationary in the extended space of the coordinates and of the bias potential (if appropriate boundary conditions are used).
Alessandro

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[Massimiliano Bonomi]

> On May 16, 2017, at 08:46, Giovanni Bussi <giovann...@gmail.com> wrote:
>
> Hi,
>
> To be slightly more precise, there are different ways discussed in the literature to reweight well-tempered metadynamics simulations. I know at least three of them.
>
> 1. Bonomi, Barducci, and Parrinello, JCC 2009. This is slightly more complicated to implement than the next two since it requires histogramming. I never used it directly, perhaps it is better if Max comments on it.

If you are interested in using it, I think it is not difficult to re-adapt the old code provided with PLUMED v 1.3 to the new PLUMED v 2.0 syntax.
However, I would use method 3, which rather than working with histograms (and evolving them along with the update of the bias potential)
gives you directly the “correction weight” for each frame. Or method 2, which assumes that all points have been sampled under a constant bias
potential. This is obviously not true, but as Giovanni said, for very long and converged simulation should give results consistent with method 3 (and 1).

Max

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[qingh...@gmail.com]

Dear Prof. Bussi,

Thanks very much for your detailed comments on my question. It helps me understand the reweighting better. I will continue reading these papers you mentioned.

Also, thanks for Prof. Laio and Max's complementary.


All the best,
Qinghua