The fundamental law for protein folding is the %DIFDELCMD < {\bf %%% Thermodynamic Principle : the amino acid sequence of a protein determines its native structure and the native structure has the minimum Gibbs free energy among all possible conformations of the protein. The very essential of the thermodynamic principle, a Gibbs free energy formula G( %DIFDELCMD < \bX%%%  ) for every possible conformation %DIFDELCMD < \bX %%%  of the protein , has never been theoretically formulated from the fundamental physical laws. We will apply quantum statistics to derive such a formula. The formula has two versions, the chemical balance version is: equation*%DIFDELCMD < } %%% G(%DIFDELCMD < \bX%%% ) = \mu_eN_e(%DIFDELCMD < \bX%%% ) + \sum_{i=1^H \mu_i N_i(}%DIFDELCMD < \bX%%% ), equation*%DIFDELCMD < } %%% where N_e(%DIFDELCMD < \bX%%% ) is the mean number of electrons in the space included by the first hydration shell of%DIFDELCMD < \bX%%% , \mu_e is its chemical potential; the index i, 1%DIFDELCMD < \le %%% i%DIFDELCMD < \le %%% H, is the hydrophobicity classification of atoms. N_i(%DIFDELCMD < \bX%%% ) is the mean number of water molecules in the first hydration layer that directly contact to the atoms with the hydrophobicity degree i; \mu_i is the chemical potential. Label all atoms of hydrophobicity degree i by H_i and let M_\bX be the molecular surface for the conformation%DIFDELCMD < \bX%%% , defining M_{\bX_i as the set of points in M_\bX that are closer to atoms in H_i than any atoms in H_j, j }%DIFDELCMD < \ne %%% i. Then the geometric version of G(%DIFDELCMD < \bX%%% ) resembles to well-known surface area models , plus a volume term: equation*%DIFDELCMD < } %%% G(%DIFDELCMD < \bX%%%  ) = a\mu_e V(\Omega_\bX)+ ad_w\mu_eA(M_\bX) + \sum_{i=1^H\nu_i \mu_i A(M_{\bX_i}), }%DIFDELCMD < \quad %%% a , \nu_i > 0, equation*%DIFDELCMD < } %%% where V(\Omega_\bX) is the volume of the domain \Omega_\bX enclosed by M_\bX, d_w is the diameter of a water molecule, and A(M_\bX) and A(M_{\bX_i) are the areas of M_\bX and M_{\bX_i}} \nablaxi  .
The fundamental law for protein folding is the %DIFDELCMD < {\bf %%% Thermodynamic Principle : the amino acid sequence of a protein determines its native structure and the native structure has the minimum Gibbs free energy . If all chemical problems can be answered by quantum mechanics, there should be a quantum mechanics derivation of Gibbs free energy formula G( %DIFDELCMD < \bX%%% X ) for every possible conformation %DIFDELCMD < \bX %%% X of the protein . We apply quantum statistics to derive such a formula. %DIFDELCMD < } %%% %DIFDELCMD < \bX%%% %DIFDELCMD < \bX%%% ^H \mu_i N_i(}%DIFDELCMD < \bX%%% %DIFDELCMD < } %%% %DIFDELCMD < \bX%%% %DIFDELCMD < \bX%%% %DIFDELCMD < \le %%% %DIFDELCMD < \le %%% %DIFDELCMD < \bX%%% %DIFDELCMD < \bX%%% as the set of points in M_\bX that are closer to atoms in H_i than any atoms in H_j, j }%DIFDELCMD < \ne %%% %DIFDELCMD < \bX%%% For simplicity, only monomeric self folding globular proteins are covered. We point out some immediate applications of the formula. We show that the formula explains the observed phenomena very well. It gives a unified explanation to both folding and denaturation; it explains why hydrophobic effect is the driving force of protein folding and clarifies the role played by hydrogen bonding; it explains the successes and deficients of various surface area models %DIFDELCMD < } %%% %DIFDELCMD < \bX%%% . The formula also gives a clear kinetic force of the folding: Fi(X ) = ^H\nu_i \mu_i A(M_{\bX_i}), }%DIFDELCMD < \quad %%% %DIFDELCMD < } %%% ) are the areas of M_\bX and M_{\bX_i}} -\nablaxi G(X). This also gives a natural way to perform the ab initio prediction of protein structure, minimizing G(X) by Newton's fastest desciending method .
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The formula has two versions, the chemical balance version is:
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where N_e(
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H, is the hydrophobicity classification of atoms. N_i(
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) is the mean number of water molecules in the first hydration layer that directly contact to the atoms with the hydrophobicity degree i; \mu_i is the chemical potential. Label all atoms of hydrophobicity degree i by H_i and let M_\bX be the molecular surface for the conformation
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, defining M_{\bX_i
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i. Then the geometric version of G(
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) resembles to well-known
For simplicity, only monomeric self folding globular proteins are covered. We point out some immediate applications of the formula. We show that the formula explains the observed phenomena very well. It gives a unified explanation to both folding and denaturation; it explains why hydrophobic effect is the driving force of protein folding and clarifies the role played by hydrogen bonding; it explains the successes and deficients of various
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, plus a volume term:
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equation*
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G(
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. The formula also gives a clear kinetic force of the folding: Fi(X
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where V(\Omega_\bX) is the volume of the domain \Omega_\bX enclosed by M_\bX, d_w is the diameter of a water molecule, and A(M_\bX) and A(M_{\bX_i
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G(X). This also gives a natural way to perform the ab initio prediction of protein structure, minimizing G(X) by Newton's fastest desciending method