Anisotropic Network Model: Difference between revisions

Revision as of 07:49, 25 September 2017

Revision as of 19:03, 5 September 2018

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<math>{\partial^2 V_{ij}\over\partial{x_i}^2} = {\partial^2 V_{ij}\over\partial{x_j}^2} = {\gamma\over {s_{ij}}^2} {(x_j - x_i)}^2 </math>

<math>{\partial^2 V_{ij}\over\partial x_i\partial y_j} = {-\gamma\over {s_{ij}}^2} {(x_j - x_i)}{(y_j-y_i)} </math> <br />

The above is a direct outcome of one of the key underlying assumptions of ANM - that a given crystal structure is an energetic minimum and does not require energy minimization.

The force constant of the system can be described by the [[Hessian Matrix]] – (second partial derivative of potential V):

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<math>H_{ij} = {-\gamma\over {s_{ij}}^2} \begin{bmatrix} x_j - x_i\\y_j - y_i\\z_j-z_i \end{bmatrix} \begin{bmatrix} x_j - x_i & y_j - y_i & z_j-z_i \end{bmatrix}</math>

Here, the force constant matrix, or the hessian matrix H holds information about the orientation of the nodes, but not about the type of the interaction (such isas whether the interaction is covalent or non-covalent, hydrophobic or non-hydrophobic, etc.). In addition, the distance between the interacting nodes is not considered directly. To account for the distance between the interactions we can weight each interaction between nodes i, j by the distance, sp. The new off-diagonal elements of the Hessian matrix take the below form, where p is an empirical parameter:<br />

<math>H_{ij} = {- 1 \over {s_{ij}}^{p+2}} \begin{bmatrix} {(X_j - X_i)(X_j - X_i)} & {(X_j - X_i)(Y_j - Y_i)} & {(X_j - X_i)(Z_j - Z_i)}\\{(Y_j - Y_i)(X_j - X_i)} & {(Y_j - Y_i)(Y_j - Y_i)} & {(Y_j - Y_i)(Z_j - Z_i)}\\{(Z_j - Z_i)(X_j - X_i)} & {(Z_j - Z_i)(Y_j - Y_i)} & {(Z_j - Z_i)(Z_j - Z_i)} \end{bmatrix}</math>

The counterpart of the [[Kirchhoff matrix]] Γ of the GNM is simply (1/γ) Η in the ANM. Its decomposition yields 3N - 6 non-zero [[eigenvalues]], and 3N - 6 eigenvectors that reflect the respective frequencies and shapes of the individual modes. The inverse of Η, which holds the desired information about fluctuations is composed of N x N super-elements, each of which scales with the 3 x 3 matrix of correlations between the components of pairs of fluctuation vectors. The Hessian, however is not invertible, as its rank is 3N-6 (6 variables responsible to a rigid body motion). In other words, the eigen values corresponding to the rigid motion are 0, resulting in the determinant being 0, making the matrix not invertible. To obtain a pseudo inverse, a solution to the eigenvalue problem is obtained:

<math>H = U\Lambda{U^T}</math>