Structural Fluctuation, Relaxation, and Folding of Protein: An Approach Based on the Combined Generalized Langevin and RISM/3D-RISM Theories
In 2012, Kim and Hirata derived two generalized Langevin equations (GLEs) for a biomolecule in water, one for the structural fluctuation of the biomolecule and the other for the density fluctuation of water, by projecting all the mechanical variables in phase space onto the two dynamic variables: th...
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MDPI AG
2023-10-01
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| Series: | Molecules |
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| Online Access: | https://www.mdpi.com/1420-3049/28/21/7351 |
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| author | Fumio Hirata |
| author_facet | Fumio Hirata |
| author_sort | Fumio Hirata |
| collection | DOAJ |
| description | In 2012, Kim and Hirata derived two generalized Langevin equations (GLEs) for a biomolecule in water, one for the structural fluctuation of the biomolecule and the other for the density fluctuation of water, by projecting all the mechanical variables in phase space onto the two dynamic variables: the structural fluctuation defined by the displacement of atoms from their equilibrium positions, and the solvent density fluctuation. The equation has an expression similar to the classical Langevin equation (CLE) for a harmonic oscillator, possessing terms corresponding to the restoring force proportional to the structural fluctuation, as well as the frictional and random forces. However, there is a distinct difference between the two expressions that touches on the essential physics of the structural fluctuation, that is, the <i>force constant, or Hessian,</i> in the restoring force. In the CLE, this is given by the second derivative of the potential energy among atoms in a protein. So, the quadratic nature or the harmonicity is only valid at the <i>minimum</i> of the potential surface. On the contrary, the linearity of the restoring force in the GLE originates from the <i>projection of the water’s degrees of freedom onto the protein’s degrees of freedom</i>. Taking this into consideration, Kim and Hirata proposed an <i>ansatz</i> for the <i>Hessian matrix</i>. The ansatz is used to equate the Hessian matrix with the second derivative of the free-energy surface or the potential of the mean force of a protein in water, defined by the sum of the potential energy among atoms in a protein and the solvation free energy. Since the free energy can be calculated from the molecular mechanics and the RISM/3D-RISM theory, one can perform an analysis similar to the normal mode analysis (NMA) just by diagonalizing the Hessian matrix of the free energy. This method is referred to as the Generalized Langevin Mode Analysis (GLMA). This theory may be realized to explore a variety of biophysical processes, including protein folding, spectroscopy, and chemical reactions. The present article is devoted to reviewing the development of this theory, and to providing perspective in exploring life phenomena. |
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| language | English |
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| series | Molecules |
| spelling | doaj.art-98bd30460dd040999dbb5561dc73ec212023-11-10T15:08:42ZengMDPI AGMolecules1420-30492023-10-012821735110.3390/molecules28217351Structural Fluctuation, Relaxation, and Folding of Protein: An Approach Based on the Combined Generalized Langevin and RISM/3D-RISM TheoriesFumio Hirata0Institute for Molecular Science, National Institute of Natural Sciences, Okazaki 444-8585, JapanIn 2012, Kim and Hirata derived two generalized Langevin equations (GLEs) for a biomolecule in water, one for the structural fluctuation of the biomolecule and the other for the density fluctuation of water, by projecting all the mechanical variables in phase space onto the two dynamic variables: the structural fluctuation defined by the displacement of atoms from their equilibrium positions, and the solvent density fluctuation. The equation has an expression similar to the classical Langevin equation (CLE) for a harmonic oscillator, possessing terms corresponding to the restoring force proportional to the structural fluctuation, as well as the frictional and random forces. However, there is a distinct difference between the two expressions that touches on the essential physics of the structural fluctuation, that is, the <i>force constant, or Hessian,</i> in the restoring force. In the CLE, this is given by the second derivative of the potential energy among atoms in a protein. So, the quadratic nature or the harmonicity is only valid at the <i>minimum</i> of the potential surface. On the contrary, the linearity of the restoring force in the GLE originates from the <i>projection of the water’s degrees of freedom onto the protein’s degrees of freedom</i>. Taking this into consideration, Kim and Hirata proposed an <i>ansatz</i> for the <i>Hessian matrix</i>. The ansatz is used to equate the Hessian matrix with the second derivative of the free-energy surface or the potential of the mean force of a protein in water, defined by the sum of the potential energy among atoms in a protein and the solvation free energy. Since the free energy can be calculated from the molecular mechanics and the RISM/3D-RISM theory, one can perform an analysis similar to the normal mode analysis (NMA) just by diagonalizing the Hessian matrix of the free energy. This method is referred to as the Generalized Langevin Mode Analysis (GLMA). This theory may be realized to explore a variety of biophysical processes, including protein folding, spectroscopy, and chemical reactions. The present article is devoted to reviewing the development of this theory, and to providing perspective in exploring life phenomena.https://www.mdpi.com/1420-3049/28/21/7351generalized langevin theoryRISM/3D-RISMstructural fluctuationisomerizationprotein foldingGaussian fluctuation |
| spellingShingle | Fumio Hirata Structural Fluctuation, Relaxation, and Folding of Protein: An Approach Based on the Combined Generalized Langevin and RISM/3D-RISM Theories Molecules generalized langevin theory RISM/3D-RISM structural fluctuation isomerization protein folding Gaussian fluctuation |
| title | Structural Fluctuation, Relaxation, and Folding of Protein: An Approach Based on the Combined Generalized Langevin and RISM/3D-RISM Theories |
| title_full | Structural Fluctuation, Relaxation, and Folding of Protein: An Approach Based on the Combined Generalized Langevin and RISM/3D-RISM Theories |
| title_fullStr | Structural Fluctuation, Relaxation, and Folding of Protein: An Approach Based on the Combined Generalized Langevin and RISM/3D-RISM Theories |
| title_full_unstemmed | Structural Fluctuation, Relaxation, and Folding of Protein: An Approach Based on the Combined Generalized Langevin and RISM/3D-RISM Theories |
| title_short | Structural Fluctuation, Relaxation, and Folding of Protein: An Approach Based on the Combined Generalized Langevin and RISM/3D-RISM Theories |
| title_sort | structural fluctuation relaxation and folding of protein an approach based on the combined generalized langevin and rism 3d rism theories |
| topic | generalized langevin theory RISM/3D-RISM structural fluctuation isomerization protein folding Gaussian fluctuation |
| url | https://www.mdpi.com/1420-3049/28/21/7351 |
| work_keys_str_mv | AT fumiohirata structuralfluctuationrelaxationandfoldingofproteinanapproachbasedonthecombinedgeneralizedlangevinandrism3drismtheories |