# [gmx-users] Re: Re: Re: Re: Which/What is the adequate overlap

Ullmann, Thomas thomas.ullmann at mpibpc.mpg.de
Wed Jul 10 14:01:33 CEST 2013

```Hi,

>----------------------
Do I understand it right, that for instance (consider lambda step = 0.5)
ONLY
N(+Delta H(lambda=0) | lambda=0.5) AND N(-Delta H(lambda=0.5) |
lambda=0)
has to overlap, and then
N(+Delta H(lambda=0.5) | lambda=1) AND N(-Delta H(lambda=1) |
lambda=0.5)
has to overlap, to get an accurate estimation of the free energy
difference, and not (essentially) other distributions? An example would
really help me out of my confusion :(
<-----------------

Yes, only directly neighboring states have to overlap. At the moment I don't have an example at hand, but you already have examples yourself. I would think that the data shown in your plots from your previous post should indeed show great overlap if you plot them as described. The overlap is then simply visible as the area that lies under both histogram curves at the same time.

>-------------
And is it right, that N(-Delta H(lambda=y) | lambda=x) equals N(+Delta
H(lambda=y) | lambda=x) if I
simply mirror the distribution at the y-axis?
<--------------

Yes.

>------------------
> See also the original paper of Charles Bennett for an explanation of
> the overlap and how it is measured.
>
I read the original paper a couple of times, and I understand, that the
squared value of the standard deviation (eq 11) measures the overlap.
And according to Fig.5, I understand, that the (complementary) fermi
functions have to show an overlap and that in principle the method
calculates the perturbation between A -> B via a shifted state by a
constant C in between. But I cannot bring it to the context of the g_bar
output
N( Delta H(lambda=x) | lambda=y).

How are the fermi functions and the constant C related to N( Delta
H(lambda=x) | lambda=y) in detail?
Or did I miss the definition in Bennett's paper, where is said, which
distributions in detail have to overlap and how they are defined?
<-------------

Ok, the paper is admittedly not easy to digest. There are also a couple of papers on BAR by Michael Shirts, you could also try those.
The overlap is most clearly visible in the integral on the right-hand side of Eq. (11) of Bennett's paper.
This integral is similar to the so-called Hodgkin index for measuring the similarity between two distributions.
The equation is not directly related to the finite sample of energy difference values from your simulations but to the overlap of the entire equilibrium probability distributions in phase / configuration space.
If you had complete knowledge of those probability distributions, which you don't have for the large systems usually considered in MD, the equation tells you how large the standard deviation of your free energy estimate would be in the limit of many samples.

I used the equation for the reverse purpose, to get an estimate for the overlap from my estimated standard deviation. This standard deviation was computed from a set of free energy estimates which were computed from blocks of my energy difference samples ( block size some thousands or ten-thousands of energy difference samples).

I hope that helped you to see clearer ...

Best,
Thomas.
--------------------------------------------------------------------------------------------------------------------------
R. Thomas Ullmann, PhD
Theoretical & Computational Biophysics
Max Planck Institute for Biophysical Chemistry
Am Fassberg 11
37077 Göttingen, Germany
thomas.ullmann at mpibpc.mpg.de
www.bisb.uni-bayreuth.de/People/ullmannt
--------------------------------------------------------------------------------------------------------------------------

________________________________________
From: gmx-users-bounces at gromacs.org [gmx-users-bounces at gromacs.org] on behalf of Mike Nemec [mike.nemec at stud.uni-due.de]
Sent: Wednesday, July 10, 2013 1:13 PM
To: gmx-users at gromacs.org
Subject: [gmx-users] Re: Re: Re: Re: Which/What is the adequate overlap

Hi Thomas,

> Hi Mike,
>
> Your confusion might stem from a very simple issue.
> To see the overlap of the forward and backward distributions,
> you have to plot
>
>  N(+Delta H(lambda=x) | lambda=y) and
>  N(-Delta H(lambda=y) | lambda=x).
>
Do I understand it right, that for instance (consider lambda step = 0.5)
ONLY
N(+Delta H(lambda=0) | lambda=0.5) AND N(-Delta H(lambda=0.5) |
lambda=0)
has to overlap, and then
N(+Delta H(lambda=0.5) | lambda=1) AND N(-Delta H(lambda=1) |
lambda=0.5)
has to overlap, to get an accurate estimation of the free energy
difference, and not (essentially) other distributions? An example would
really help me out of my confusion :(

> That is, you have to plot the histogram for the negative of the energy
> difference samples in the backward direction.
>
And is it right, that N(-Delta H(lambda=y) | lambda=x) equals N(+Delta
H(lambda=y) | lambda=x) if I
simply mirror the distribution at the y-axis?

> See also the original paper of Charles Bennett for an explanation of
> the overlap and how it is measured.
>
I read the original paper a couple of times, and I understand, that the
squared value of the standard deviation (eq 11) measures the overlap.
And according to Fig.5, I understand, that the (complementary) fermi
functions have to show an overlap and that in principle the method
calculates the perturbation between A -> B via a shifted state by a
constant C in between. But I cannot bring it to the context of the g_bar
output
N( Delta H(lambda=x) | lambda=y).

How are the fermi functions and the constant C related to N( Delta
H(lambda=x) | lambda=y) in detail?
Or did I miss the definition in Bennett's paper, where is said, which
distributions in detail have to overlap and how they are defined?

> Best,
> Thomas.
>
> ---------------------------------------------------------------------------------
>
> R. Thomas Ullmann, PhD
> Theoretical & Computational Biophysics
> Max Planck Institute for Biophysical Chemistry
> Am Fassberg 11
> Göttingen, Germany
> thomas.ullmann at mpibpc.mpg.de
> www.bisb.uni-bayreuth.de/People/ullmannt
> ---------------------------------------------------------------------------------
>
>
>
Thank you for any help,

Mike
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