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Ben-Naim warns that there are pitfalls in pursuing Anfinsen’s thermodynamic hypothesis. We show that the deepest one of his pitfalls is not a pitfall at all and the believing pursuing the minimum of the Gibbs free energy will lead us to the native structure. The “pitfall” came from the flawed inference of Gibbs energy function and the muddiness on concepts such as thermodynamic system and the second law of thermodynamics.

The abstract of [

We will discuss “the most profound one” of Ben-Naim’s pitfalls and at the end show that it is Ben-Naim’s misinterpretation of the Anfinse’s threnodynamic hypothesis that produced the “pitfall”.

First let us recall what is Anfinsen’s thernodynamic hypothesis (it should be called thermodynamic principle, since it is nothing but the second law of thermodynamics once the spontaneousnesss of protein folding is proved). Anfinsen stated in [

Ben-Naim first pointed out in [

“The following two statements are true:

(a) The native stable structure of the protein must be at a minimum of the GEL (Gibbs Energy Landscape).

(b) Upon releasing a constraint within the system, specified by the variables:

“The first statement (a) is essentially Anfinsen’s hypothesis… The second statement (b) is equivalent to the Second Law of thermodynamics.”

Then there is a pitfall: “From the two true statements (a) and (b), people have concluded that the stable state of the protein must be in a global minimum in the GEL. Unfortunately, this conclusion is invalid. The last statement seems to be an absurd; how can a false statement be derived from two solid, true statements? The reason so many people fell into this pitfall is that in making statements (a) and (b), we have not specified the variables with respect to which the Gibbs energy has a minimum.” [

Ben-Naim here clearly tells us that the pitfall comes from correctly specifying the variable of a Gibbs free energy function.

Bennaim’s “the deepest pitfall” [

“(iii) The third pitfall is the deepest, hence, potentially most harmful, or more dangerous to fall into. As stated in Anfinsens hypothesis: the Gibbs free energy of the whole system is lowest. That is certainly correct, but that is also trivially correct for any process or any chemical reaction. It follows from the Second Law of Thermody- namics that when a spontaneous process occurs in a system under constant temperature, pressure and compo- sition, the Gibbs energy will be lowered, and at the new equilibrium state it will be at a minimum. This principle follows from the Second Law, and is equivalent to the statement that in an isolated system the entropy reaches its maximum. The question is, maximum of entropy with respect to what, or minimum of Gibbs energy with respect to what? If you walk along a rough road and you are not careful, you might fall into a pitfall. Why?”

So the problem is still correctly identifying the “whole system” and formulate the Gibbs free energy function with coreect variable. What is this function? Ben-Naim shows his reason in [

“If we start with a system having one particle at a fixed position, say

This argument has a fundamental flaw, because with a little mathematics, it infers that:

This is indeed a very bad function, not only it is discontinuous everywhere, but also it cannot have any finite minimum value. No such function can be an energy function in physics. No matter what is the reason, this inference just points out that something was wrong in Ben-Naim’s argument. Ben-Naim pointed out that the

“The case of protein folding has the same pitfall as in the relatively simple examples discussed above. The GEL function is

So the answer to his question of “minimum of Gibbs energy with respect to what?” is certainly not the conformation

Then one falls into “the deepest pitfall”. The reason is not clearly given, if he thinks that the function of

he is definitely wrong as we have already expained. Except this, he did mention that the second law of the thermodynamics cannot guarantee that the Gibbs free energy will have a global minimum if the variable is the conformation [

Thus we may say that Ben-Naim actually falls into a pitfall: he thinks that only by considering an ensemble of conformations we can apply the second law of thermodynamics.

In fact, the Gibbs free energy function

So the “deepest pitfall” of Ben-Naim claimed comes from flawed inferences, misunderstanding of Anfinsen’s thermodynamic hypothesis, and prejudice about how to apply the second law of thermodynamics.

If Ben-Naim noticed that there are actually two kids of thermodynamic systems, single molecule and ensemble, and if he uses the single molecule one in his point (a) and (b) and apply the second law of thermodynamics, he would not have claimed “the deepest pitfall”.

In fact, to generate the GEL, there should be a Gibbs free energy function

Hence

We will use

where

This is a single molecule version of the Gibbs free energy. If known, then for the ab initio prediction of protein structure,

Such an analytic function

For the meaning of terms in (7) please see [

Now consider a system of

The problem is, nobody knows

Because that any conformation

We have demonstrated that the “deepest pitfall” of Ben-Naim claimed is not a pitfall at all. Because of lack of mathematical skill, misconnect of second law of thermodynamics can only be applied to ensemble systems, and moreover, faille of deriving a Gibbs free energy formula