# Re: A question on Jarkynski equation

From: Jun Zhang (coolrainbow_at_yahoo.cn)
Date: Wed Jun 15 2011 - 00:25:17 CDT

Hello Yi,

I have read your letter and got your ideas. I think it is the solvent molecular motions which constitue a huge phase space that give the opportunity to get a zero average.  I have just calculate the work of a SMD simulation, containing a ligand far away from receptor moving in solvent, and I found a typo in my tcl script which  loses the sign of the force so leading to a wrong result. Force does show an oscillation around zero and the work approaches zero.

But I am uncertain about my formula to calculate the work. Assuming the SMD output force is {fx, fy, fz}, with the norm f = sqrt(fx^2+fy^2+fz^2), the work is :

W(t) = v \int_{0}^{t} dt' f(f')

It is right? I think the molecular is not necessarily always moving along the SMDdir direction, so it can deviate the path and I think the correct formula should be the dot product between the force and the displacement.

I am glad to disscuss with you.

Good luck!

Jun Zhang
Nankai University
coolrainbow_at_yahoo.cn

--- 11年6月14日，周二, Wang Yi <dexterwy_at_gmail.com> 写道：

Hi Jun,

Dr. Hénin has pointed out the general concerns regarding SMD and JE. But for processes like moving in water, you might not have to worry too much about that. What I mean is:

The work done for dragging a ligand in a water box is mostly against the "friction" from solvent. Since water molecules move in all directions, the "frictions" the ligand is experiencing changes all the time. Thus, you will notice the SMD force fluctuating roughly around zero (albeit with large swings). Then after integrating the force curve, the positive work and negative work will mostly cancel each other. And after applying JE, although there exists the issue of "heavey-weight rare events", the final value would be pretty small (compared to thermo energy). That's especially true in a unbinding process simulation, where the unbinding part has a much larger magnitude than the "free moving in water" part.

That is based on my understanding.

Best,

___________________________

Yi (Yves) Wang
Duke University

Hi,

You put your finger on a very interesting question. The Jarzynski
equality is formally exact, hence the estimator will converge towards
zero, at least on paper. As you note, since the reversible work is
zero, all measured work is irreversible: we expect it to be positive
basically all the time. Then how can the average be zero?

Actually, statistical mechanics dictates that a few trajectories will
give negative work values, i.e. negative irreversible work, i.e. a net
decrease in entropy. This is why such situations are sometimes called
'second law violations'. They are not really violations, just a
reminder that the second law applies to macroscopic systems only (even
if we in the molecular modeling community often stretch the notion of
macroscopic a little far). In the case of a solute moving through a
solvent, the 'negative work' case would be a trajectory where random
fluctuations in the solvent happen to push the solute along its path,
instead of slowing it down in a normal frictional behavior.

Such negative work values will be exceedingly rare, but they have a
huge weight in the exponential average of the Jarzynski formula. That
is why a few negative values are enough to make the average zero even
though almost all values are positive. Because these events are so
rare, numerical convergence will be awful, so it is unlikely that you
will manage to get a zero free energy value from a numerical
simulation. To some extent, the same can be said of any application of
the Jarzynski estimator, and is also true of FEP calculations with the
exponential formula: these averages are dominated by rare events,
which results in various degrees of convergence problems.

For more details on 'second law violations', see for example:
http://arxiv.org/abs/cond-mat/0401311

Cheers,
Jerome

On 14 June 2011 05:01, Jun Zhang <coolrainbow_at_yahoo.cn> wrote:

Hello Everyone:

I want to use Jarkynski's equation combined with SMD to compute the binding free energy of a protein and its ligand (eg. JCP, 120, 5946). However, I was puzzled by some theoretical issues.

For example, a system composed of water and a ligand. if we move a single ligand in aqueous for some distances, the work done cannot be zero, but the free energy change should be zero since the state of the ligand has not changed.

It seems somewhat strange, and may be a naive question. But I am really puzzled by it, so I'm looking for help. Thank you in advance!

Cheers up!

Jun Zhang

Nankai University

coolrainbow_at_yahoo.cn

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