The single-parameter Gamma matrix of force constants proposed by the Gaussian Network Model (GNM) is iteratively modified to yield native state fluctuations that agree exactly with experimentally observed values. The resulting optimized Gamma matrix contains residue-specific force constants that may be used for an accurate analysis of ligand binding to single or multiple sites on proteins. Bovine Pancreatic Trypsin Inhibitor (BPTI) is used as an example. The calculated off-diagonal elements of the Gamma matrix, i.e., the optimized spring constants, obey a Lorentzian distribution. The mean value of the spring constants is approximately -0.1, a value much weaker than -1 of the GNM. Few of the spring constants are positive, indicating repulsion between residues. Residue pairs with large number of neighbors have spring constants around the mean, -0.1. Large negative spring constants are between highly correlated pairs of residues. The fluctuations of the distance between anticorrelated pairs of residues are subject to smaller spring constants. The importance of the number of neighbors of residue pairs in determining the elements of the Gamma matrix is pointed out. Allosteric effects of binding on a single or multiple residues of BPTI are illustrated and discussed. Comparison of the predictions of the present model with those of the standard GNM shows that the two models agree at lower modes, i.e., those relating to global motions, but they disagree at higher modes. In the higher modes, the present model points to the important contributions from specific residues whereas the standard GNM fails to do so.