\n",
"\n",
"The potential energy of a Gaussian Network Model is defined as the summation of harmonic potentials over all unique $(i,j)$-pairs and is a function of only the square of **inter-residue distance vector**, $\\Delta \\mathbf{R}_{ij} = \\mathbf{R}_i - \\mathbf{R}_j^0$. It can be given in terms of the **Kirchhoff matrix for inter-residue contacts**, $\\mathbf{\\Gamma}$,\n",
"\n",
"$$\n",
"V^{GNM} = \\frac{\\gamma}{2} \\Delta \\mathbf{R}^T \\left( \\mathbf{\\Gamma} \\otimes \\mathbf{I}_{3\\times 3} \\right) \\Delta \\mathbf{R} \\,,\n",
"$$\n",
"\n",
"where $\\mathbf{\\Gamma}$ is an $(N \\times N)$-matrix and is defined as:\n",
"\n",
"$$\n",
"\\mathbf{\\Gamma}_{ij} = \\left\\{\\begin{matrix} \n",
"-1 & \\text{if } i \\ne j \\text{ and }R_{ij} \\le R_{cut} \\,, \\\\ \n",
"0 & \\text{if } i \\ne j \\text{ and }R_{ij} > R_{cut} \\,, \\\\\n",
"-\\sum_{j,j \\ne i}^{N} \\mathbf{\\Gamma}_{ij} & \\text{if } i = j \\,. \\end{matrix}\\right.\n",
"$$"
]
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"##