network (residues of a protein) and springs represent the connections between the nodes (covalent and non-covalent interactions between residues). For nodes i and j, equilibrium position vectors, R0i and R0j, equilibrium distance vector, R0ij, instantaneous fluctuation vectors, ΔRi and ΔRj, and instantaneous distance vector, Rij, are shown in Figure 2. Instantaneous position vectors of these nodes are defined by Ri and Rj. The difference between equilibrium position vector and instantaneous position vector of residue i gives the instantaneous fluctuation vector, ΔRi = Ri - R0i. Hence, the instantaneous fluctuation vector between nodes i and j is expressed as ΔRij = ΔRj - ΔRi = Rij - R0ij. Potential of the Gaussian network The potential energy of the network in terms of ΔRi is where γ is a force constant uniform for all springs and Γij is the ijth element of the Kirchhoff (or connectivity) matrix of inter-residue contacts, Γ, defined by rc is a cutoff distance for spatial interactions and taken to be 7 Å for amino acid pairs (represented by their α-carbons). Expressing the X, Y and Z components of the fluctuation vectors ΔRi as ΔXT = [ΔX1 ΔX2 ..... ΔXN], ΔYT = [ΔY1 ΔY2 ..... ΔYN], and ΔZT = [ΔZ1 ΔZ2 ..... ΔZN], above equation simplifies to Statistical mechanics foundations In the GNM, the probability distribution of all fluctuations, P(ΔR) is isotropic and Gaussian where kB is the Boltzmann constant and T is the absolute temperature. p(ΔY) and p(ΔZ) are expressed similarly. N-dimensional Gaussian probability density function with random variable vector x, mean vector μ and covariance matrix Σ is normalizes the distribution and |Σ| is the determinant of the covariance matrix. Similar to Gaussian distribution, normalized distribution for ΔXT = [ΔX1 ΔX2 ..... ΔXN] around the equilibrium positions can be expressed as The normalization constant, also the partition function ZX, is given by where is the covariance matrix in this case. ZY and ZZ are expressed similarly. This formulation requires inversion of the Kirchhoff matrix. In the GNM, the determinant of the Kirchhoff matrix is zero, hence calculation of its inverse requires eigenvalue decomposition. Γ−1 is constructed using the N-1 non-zero eigenvalues and associated eigenvectors. Expressions for p(ΔY) and p(ΔZ) are similar to that of p(ΔX). The probability distribution of all fluctuations in GNM becomes For this mass and spring system, the normalization constant in the preceding expression is the overall GNM partition function, ZGNM, Expectation values