Re: Calculating 2D PMF from 1D eABF simulations using CZAR estimator.

From: Abhishek Acharya (
Date: Wed Nov 11 2020 - 16:13:39 CST

Hello Jerome,

Thank you for your quick response.

Well it does look simpler that I thought. Pardon my ignorance; I am a
biologist by training, so mathematics is not a strong suit (how I wish it

I have another follow up question. Looks like I can collect the joint
histogram for all the walkers using the histogram method. Easiest way is to
combine all of these to obtain the full joint histogram, calculate the
marginal and use the final full 1D PMF to obtain the 2D distribution. But I
am concerned that sampling artefacts at the window edges may cause
problems. A better way would be to calculate the 2D PMF separately for each
window, drop the values at the edges and perform an interpolation to obtain
the complete 2D PMF.

A third way I thought is to define slightly broader (and overlapping)
window ranges. This way I can simply combine the data across windows (for
both 1D and 2D cases) after dropping the problematic bins at both ends.
This would be slightly expensive but perhaps a bit cleaner.

Do you have any suggestions in this regard?


On Wed, Nov 11, 2020 at 8:53 PM Jérôme Hénin <> wrote:

> Hello Abhishek,
> you're almost there, but it's simpler than you think. If you can collect
> the joint histogram P(z1, z2), biased along z1, then you can obtain the 2d
> PMF by reweighting using the 1d CZAR PMF:
> A(z1, z2) = -kT ln( P(z1, z2) / P(z1) ) + A_CZAR(z1)
> where P(z1) is the observed (biased) histogram in z1 (aka the z1 marginal
> of the 2d histogram).
> Best,
> Jerome
> ----- On 11 Nov 20, at 20:04, Abhishek Acharya
> wrote:
> > Hello,
> >
> > We are trying to run 1D eABF simulations on our system of interest.
> > Specifically, the simulation is divided into 3 windows and we use 5
> walkers
> > per window for sampling.
> >
> > Obtaining the 1D PMF looks straightforward to do. Just to be sure we,
> > 1) Combine data for walkers using the inputPrefix directive of ABF to get
> > combined outputs for each window.
> > 2) Combine the gradients for each window to obtain the full gradient,
> > taking care of the edge values.
> > 3) Integrate the gradient to obtain 1D PMF.
> >
> > However for better insights and comparison with methods used previously,
> we
> > would like to obtain a PMF projected along an additional CV to obtain a
> 2D
> > picture. So, the idea is to apply ABF bias along, say z1 CV, but also
> > obtain samples along z2 (unbiased and defined without the Extended
> > Largrangian directive) and somehow combine these to obtain the 2D PMF.
> > Naively, I thought that maybe we can do a post-hoc estimation using the
> > CZAR estimator; essentially obtain the biased 2D histogram, P(z1, z2) and
> > the z1-averaged forces from the CV values and total forces printed out to
> > the colvar traj file, and finally using the 1D CZAR expression to obtain
> > the 2D PMF.
> >
> > 1. Is this at all a correct strategy? If not, some hints would be
> helpful.
> >
> > 2. Fu et al, *J. Chem. Theory Comput.* 2016 indicates that accurate
> > post-hoc analysis would require printing out the CV and force values
> every
> > timestep. For the aforesaid procedure, I thought that maybe we can use
> the
> > histogram directive to obtain the final 2D counts and combine it with the
> > 1D averaged forces.
> >
> > Any suggestions would be highly appreciated.
> >
> > Thanks in advance.
> >
> > Sincerely,
> > Abhishek Acharya

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