Protein molecules have similar linear chain nature as regular polymer molecules. The excluded volume effects play small role for regular polymers, and therefore they can be modeled as random walks or random flights with the Gaussian distribution of the end-to-end distance. In "Statistical Mechanics of Chain Molecules" Flory developed more sophisticated approach by assumption of the discrete values of dihedral angles for polymers (depending on the chemical structure of the monomer) and by introducing transformation matrices depending on dihedral angles. The transformation matrix enables to express coordinates of a vector in a frame defined the backbone conformation of the node i n the frame of the next i+1 node. For a given set of all dihedral angles along the backbone the matrix multiplication of all subsequent rotational matrices enables the computation of the end-to end vector and the calculation of various physical properties of the polymer.<br/>We have applied similar algebraic ideas to model proteins. For simplification we used simple square and cubic lattices and assumed that proteins are Hamiltonian walks or Hamiltonian circults (walks without two ends) within regular shapes of rectangles (in 2D) or parallelpipeds (in 3D). For a given cross-section of the rectangle (2D) or a parallelpiped (3D) we developed transfer matrices that enabled us to exactly enumerate all Hamiltonian walks and Hamiltonian paths on the square and cubic lattices. The transfer matrix method of generating and enumerating compact conformations is extremely efficient. The main advantage is that the piece-wise generation of conformations is attrition-free. Once the transfer matrix for a given cross-section is defined, the more complicated geometrical problem of conformation generation (or calculation of averages such as average energy becomes a simple problem of matrix algebra.

Applications of basic ideas of statistical mechanics of chain molecules to proteins

2019 Fall National Meeting

The official archive and repository for all ACS Meetings & Expositions abstracts and select posters or presentations.