To resolve the protein folding problem, that is: predicting the native structure and describing the folding dynamics, we must work with the fundamental physical law that directly governs protein folding process. That law is the Thermodynamic Principle of Protein Folding [1], it is just the Second Law of Thermodynamics since Anfinsen and others already shown that the folding process is spontaneous. In the protein folding case, the second law is that the Gibbs free energy achieves a minimum at the native structure.

Therefore, we have to figure out what is the Gibbs free energy. The question is, is there a Gibbs free energy function whose variables are all possible conformations of a given protein molecule? Or is it only a Gibbs free energy difference between the folded ensemble of protein molecules and its counterpart, the unfolded ensemble? The former is a single molecule view coming from contemplating a protein molecule comes out of ribosome and changes its conformations until it achieves its native structure; the latter is an ensemble view coming from staring at a tube of purified protein solution and trying to figure out the collective behaviours of the protein molecules in the solution while the solution is going towards equilibrium.

Via quantum statistics applied to a tiny thermodynamic system S X which is tailor made for the conformation X and its immediate physiological environment E U , we have derived the conformational Gibbs free energy (CGFE) function G ( X ; E U , U ) for globular proteins U [2] [3] [4].

Applying CGFE function we translate the thermodynamic principle of protein folding into the Single Molecule Thermodynamic Hypothesis (SMTH) of Protein Folding: putting a protein molecule U in an environment E , a stable conformation X E (may not be unique) of U must be a minimizer (local or global) of the CGFE function G ( X ; E , U ) . In particular, the gradient vanishes at X E , ∇ G ( X E , E , U ) = 0 .

Another big question in protein folding is that is there a folding force? Leventhal in 1969 [5] has shown by contradiction that there must be a folding force, otherwise if the folding process were only random, it would have taken a time span longer than the Earth’s age.

The CGFE function also gives us the deterministic part of the folding force F i acting on an atom a i of U . It is F i ( X ) = − ∇ x i G ( X ; E U , U ) .

A scientific hypothesis has to give verifiable predictions to let people confirm or refute it. We suggest two verifiable predictions of the single molecule thermodynamic hypothesis.

1) ab initio predictions of native structures of globular proteins: the native structure X U of the protein U is a (local or global) minimizer of the CGFE function:

G ( X U ; E U , U ) = min X ∈ U ⊂ X U G ( X ; E U , U ) ,   forsomeneighbourhood U   of X U . (1)

Especially, since G ( X ; E U , U ) is smooth, ∇ G ( X U ; E U , U ) = 0 . In case X U is a global minimizer, then

G ( X U ; E U , U ) = min X ∈ X U G ( X ; E U , U ) . (2)

Here X denotes a conformation and X U is the set of all possible conformations of U .

Therefore, for a globular protein U (we know the formula of G ( X ; E U , U ) ), the prediction of native structure is reduced to a pure mathematical problem, the minimization problem of a known smooth function. A mathematical theorem guarantees that this problem has a solution, i.e., mimimizers always exist, the real task is to find them and determine which is the native structure, a hard programming problem.

2) For a globular protein U , starting from any initial conformation X 0 , there is a folding path X ( t ) = X ( t ; X 0 ) satisfying X ( t 0 ; X 0 ) = X 0 , and for t ≥ t 0 , the following Langevin equation:

m i d 2 x i ( t ) d t 2 = F total = − ∇ x i G ( X ( t ) ; E U , U ) − η i d x i ( t ) d t + F i ( t ) ,   i = 1, ⋯ , n . (3)

Here η i is the solvent friction. The random force F i ( t ) is caused by occasionally bumping into another non-solvent molecule. Because of it, we do not have a completely deterministic folding path. Again, mathematical theorems guarantee that such folding path X ( t ; X 0 ) exists. Moreover, mathematics also tells us that it is highly depending on its initial conformation X 0 , so an important issue is to know the protein’s initial conformation as it is out of ribosome.

If the two predictions are positively verified, then we can say that theoretically the protein folding problem is resolved, at least for globular proteins.

To explain our CGFE function, we start with a conformation X . A protein U consists of n atoms ( a 1 , ⋯ , a n ) , a conformation of U can be expressed as a point of 3n-dimensional space ℝ 3 n , X = ( x 1 , ⋯ , x n ) , where x i ∈ ℝ 3 is the nuclear position of a i in the 3-dimensional space ℝ 3 . Not all points in ℝ 3 n are conformations of U , bond lengths, bond angles, and van der Waals distances in general, are natural constraints. Denote X U as the set of all possible conformations of U . A conformational function of U is a function f : X U → ℝ . For example, all force fields used in molecular dynamics simulations are conformational functions.

The 3-dimensional conformation P X of U is P X = ∪ i = 1 n C i ( x i ) ⊂ ℝ 3 , where C i ⊂ ℝ 3 is the shape of the atom a i in U , C i ( x i ) indicates it has been congruently moved to the nuclear position x i . C i exists, will change with X , see [6]. A very good approximation to C i ( x i ) is a solid ball B ( x i , r i ) , centred at x i ∈ ℝ 3 with radius r i that is the a i ’s van der Waals radius. For simplicity, we adopt that C i ( x i ) = B ( x i , r i ) .

In natural, and even in most of artificial environments of protein folding, the immediate environment of a globular protein U is just one layer of water molecules surrounding the conformation P X = ∪ i = 1 n B ( x i , r i ) . This is true even the protein molecule is inside a crystal [7].

P X plus the one layer water molecules consists of a tiny thermodynamic system S X , tailor made for P X . As an open thermodynamics system, S X has a Gibbs free energy G ( S X ) . The CGFE function G ( ⋅   ; E U , U ) : X U → ℝ then is defined as G ( X ; E U , U ) = G ( S X ) .

Between P X and the layer of water molecules, is an interface M X , for example, the solvent accessible surface ∂ P X . The expression of G ( S X ) is via global geometric features of M X and its surface chemical potentials. A protein molecule has many moieties or atom groups, some are charged, some are polar, others are non-polar. They can be classified into hydrophobicity classes H i , 1 ≤ i ≤ H , H > 1 , from most hydrophobic (non-polar) to the most hydrophilic (polar or charged). An atom a i ∈ H j if it belongs to a moiety of class H j . Define P X , i = ∪ a j ∈ H i B ( x j , r j ) , then P X = ∪ i = 1 n P X , i . The space containing water molecules in S X , the ring ℜ X = S X \ P X ¯ , is decomposed into H parts (not necessarily connected) via the distance function d i s t ( x , P X , i ) = min y ∈ P X , i | x − y | , see Figure 1.

ℜ X , i = { x ∈ ℜ X :   d i s t ( x , P X , i ) ≤ d i s t ( x , P X , j ) , forany j ≠ i } , i = 1, ⋯ , H . (4)

The interface M X then is decomposed accordingly

M X = ∪ i = 1 H M X , i ,   M X , i = M X ∩ ℜ X , i . (5)

A water molecule in ℜ X , i will touch M X , i , so it will be attracted ( H i is charged or polar) or repulsed ( H i is non-polar) by P X . So the same water molecule in different ℜ X , i has different chemical potentials μ i . For non-polar H i , μ i > 0 , for charged or polar H i , μ i < 0 . Thus there is a 1 < k < H , such that

μ 1 > μ 2 > ⋯ > μ k > 0 > μ k + 1 > ⋯ > μ H . (6)

is called hydrophobic surface, is called hydrophilic surface.

Since water molecules and electrons can enter or leave, is an open thermodynamic system. Hence, the ensemble used here is the grand canonical ensemble. Applying quantum statistics to the open system, the number of water molecules in turns to be a Hermitian operator with mean value. The same is true to the number of electrons. The Gibbs free energy of then is

Since is fixed, here the kinetic energy operator vanishes.

Since every water molecule in has contact with the surface, is proportional to the area. Therefore, there are, such that

Let be the coordinates of one electron in, the coordinates of water molecules in. Then the electronic density distribution function ((1.3) of ([6], page 6)) gives

The is obtained by mean value theorem of integrals.

Let be the domain enclosed by (), we have roughly the volume, where is the diameter of a water molecule and is the area of. Then taking the mean in (9) we have

Let and. Substitute (8) and (10) into (7), the conformational Gibbs free energy function of a globular protein in its physiological environment is

The function is smooth, i.e., the first and second derivatives exist in except at points such that for some. But such case cannot happen for a conformation, because of the van der Waals distances must be positive. Therefore, on, is smooth.

A scientific hypothesis has to be able to explain natural and artificial phenomena. We will explain several phenomena of protein folding, unfolding, and docking, and suggest an application to drug design, according to the SMTH.

The author declares no conflicts of interest regarding the publication of this paper.

Yi, F. (2019) Single Molecule Thermodynamics Hypothesis of Protein Folding and Drug Design. Journal of Biosciences and Medicines, 7, 164-172. https://doi.org/10.4236/jbm.2019.711015