Re: A question on Jarkynski equation

From: Wang Yi (
Date: Tue Jun 14 2011 - 08:11:46 CDT

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.


Yi (Yves) Wang
Duke University

在 2011-6-14,上午4:43, Jérôme Hénin 写道:

> 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:
> Cheers,
> Jerome
> On 14 June 2011 05:01, Jun Zhang <> 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

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