The Second Law, Gibbs FreeEnergy, Geometry, and Protein Folding

The fundamental physical law of protein folding is the second law of thermodynamics. The key to solve protein folding problem is to derive an analytic formula of the Gibbs free energy. It has been overdue for too long. Let U be a monomeric globular protein whose M atoms

Decades of experiments by many researchers proved that once the peptide chain of a natural protein is put in correct environments it will spontaneously fold to its native structure.Therefore, the guiding fundamental physical law must be the second law of thermodynamics.Anfinsen summarized this as the thermodynamic principle (he modestly called it hypothesis) of protein folding, that the native structure has the minimum Gibbs free energy and only depends on the peptide chain of the protein in physiological environment [1].Thus, a cross section of complicated life phenomena, the protein folding, is reduced to a physical problem and should and can be solved accordingly.
Theoretically, all problems such as structure prediction and mechanics of folding process will be answered once we know the Gibbs free energy of protein folding, p H, ionic strength, presence of other components such as metal ions or prosthetic groups, temperature, and other", [1].We think that pressure belongs to the other.In fact, because constant pressure and variant volume, the second law takes the version of minimum of Gibbs free energy.We will derive for a monomeric globular protein U via quantum statistical mechanics.
We start with the observable physical quantity, the electron density distribution function [2], is the wave function of the Born-Oppenheimer approximation to the Hamiltonian of one molecule of U , and N is the number of electrons in U .
Since in natural, nascent peptide chains already have their peptide bonds and covalent bonds in residues formed, we will not discuss the bond lengths and angles.Instead, we assume that the values of those covalent bond lengths and angles in X are very close to the standard bond lengths and angles.
To apply statistical mechanics, we have to create a thermodynamic system X T tailor made for and its immediate environment, leaving everything in Because of X T is tailor made for X , requirment 1 is automatically satisfied.For reuqirement 3, only monomeric globular proteins can be assumed that in the immediate environment of X P there are no other large objects except water molecules, hence here we consider only this class of proteins for the simplicity of environemnet.Note that the method itself is general, only that for complicated environment the derived formula will also more complicated than that obtained here.For the requirment 2, the dependence on the peptide chain of U is via the electronic density distribution function X p that indicates how the X P will interact with the immediate environment, in our case, water.For which we need to discretize X p with general knowledge of amino acids.It was well-known as early as the 1920's that proteins are multi-polar or "bristling with charges" as described in [3], resulting in different atomic groups have different hydrophobicity levels, say, there are The atomic space distribution of these hydrophobicity classes are highly depending on X and the peptide chain of U .
Exploiting these space distributions gives a way of applying the X p while not being able to calculate it.J a n u a r y 0 3 , 2 0 1 4 is connected, then it is a closed surface.Thus, we always have For any compact (closed and bounded) set The X R is the first hydration shell surrounding X P , and X T is the thermodynamic system tailor made for the conformation X .Define compact sets ) , ( = , 1, =  .We will allow water and electrons enter or leave X T , so

X
T is an open system.

Interaction of a water molecule with an atom of
where A q is the electronic charge in the nucleus of Proof of Theorem 1:Since water molecules are very small comparing to U , we can apply the Born-Oppenheimer approximation, fix  A water molecule is treated as a single particle centered at the oxygen nuclear position Let be the nuclear centers of water molecules in X R and   be electronic positions of all electrons in X T .Then the Hamiltonian for the system , and where E .The Born-Oppenheimer approximation has the Hamiltonian (here we use the definition in [6]): ). , ( The eigenfunctions Since N varies, we can adopt the grand canonic ensemble.The grand canonic density operator is given as ( [7] and [8]) According to [7], the entropy is the grand canonic potential  in [8] and the macroscopic potential in [7], it is a state function with variables , and e  .By the general thermodynamic equations [8]: that is exactly (2).

Since every water molecule in
the protein structure prediction to a pure mathematical problem of finding minimizers of an analytic function atomic (nuclear) center of i a .According to Anfinsen, the physiological environment En N consists of elements such as "solvent, 3. the peptide chain of the protein U .To be realistic, these are general requirments for any attempting of creating a Gibbs free energy functon of protein folding.

H
atomic group.For example, we may assume that the classification is as in[4] where there are 5 = H classes, C, O/N, O  , N  , S. Unlike in [4], we also classify every hydrogen atom into one of the H hydrophobicity classes.

Figure 1 : 1 SXM are contained in 1 S
Figure 1: Molecular surface.To describe the formula, we have to do some preparations.Rolling a probe sphere of radius r on the boundary

x
and the subset U .Define .

iHTheorem 1
will gain a Gibbs energy i  , the chemical potential.Let i  be the average number of water molecules that can simultaneously touch i M X in a unit area, theni i i    =is the J a n u a r y 0 3 , 2 0 1 4 chemical potential per unit area of i M X .Moreover, since the curvature of X M is uniformly bounded for all conformations of U , i  does not depend on X .Similarly we define e  and e  to be the chemical potentials of per electron and per unit volume.Let U be a monomeric globular protein with M atoms .Then in the physiological environment

XP
and let all water molecules and electrons in X T move.Then we will apply the grand canonic ensemble of statistical mechanics to the open system X T .

Figure 2
Figure 2: covalent bonds in it are fixed.In the Born-Oppenheimer approximation, only the conformation X is fixed, all particles, water molecules and electrons in X T , are moving.
masses of a water molecule and an electron; assume that a water molecule cannot occupy spaces in X P , thus by the design of X

.
Then taking the mean we have