of fluctuations and correlations The expectation values of residue fluctuations, <ΔRi2> (also called mean-square fluctuations, MSFs), and their cross-correlations, <ΔRi · ΔRj> can be organized as the diagonal and off-diagonal terms, respectively, of a covariance matrix. Based on statistical mechanics, the covariance matrix for ΔX is given by The last equality is obtained by inserting the above p(ΔX) and taking the (generalized Gaussian) integral. Since, <ΔRi2> and <ΔRi · ΔRj> follows Mode decomposition The GNM normal modes are found by diagonalization of the Kirchhoff matrix, Γ = UΛUT. Here, U is a unitary matrix, UT = U−1, of the eigenvectors ui of Γ and Λ is the diagonal matrix of eigenvalues λi. The frequency and shape of a mode is represented by its eigenvalue and eigenvector, respectively. Since the Kirchhoff matrix is positive semi-definite, the first eigenvalue, λ1, is zero and the corresponding eigenvector have all its elements equal to 1/. This shows that the network model translationally invariant. Cross-correlations between residue fluctuations can be written as a sum over the N-1 nonzero modes as It follows that, [ΔRi · ΔRj], the contribution of an individual mode is expressed as where [uk]i is the ith element of uk. Influence of local packing density By definition, a diagonal element of the Kirchhoff matrix, Γii, is equal to the degree of a node in GNM that represents the corresponding residue’s coordination number. This number is a measure of the local packing density around a given residue. The influence of local packing density can be assessed by series expansion of Γ−1 matrix. Γ can be written as a sum of two matrices, Γ = D + O, containing diagonal elements and off-diagonal elements of Γ. Γ−1 = (D + O)−1 = [ D (I + D−1O) ]−1 = (I + D−1O)−1D−1 = (I - D−1O + ...)−1D−1 = D−1 - D−1O D−1 + ... This expression shows that local packing density makes a significant contribution to expected fluctuations of residues. The terms that follow inverse of the diagonal matrix, are contributions of positional correlations to expected fluctuations. GNM applications Equilibrium fluctuations Equilibrium fluctuations of biological molecules can be experimentally measured. In X-ray crystallography the B-factor (also called Debye-Waller or temperature factor) of each atom is a measure of its mean-square fluctuation near its equilibrium position in the native structure. In NMR experiments, this measure can be obtained by calculating root-mean-square differences between different