Separating the role of protein restraints and local metal-site interaction chemistry in the thermodynamics of a zinc finger protein

We express the effective Hamiltonian of an ion-binding site in a protein as a combination of the Hamiltonian of the ion-bound site in vacuum and the restraints of the protein on the site. The protein restraints are described by the quadratic elastic network model. The Hamiltonian of the ion-bound site in vacuum is approximated as a generalized Hessian around the minimum energy configuration. The resultant of the two quadratic Hamiltonians is cast into a pure quadratic form. In the canonical ensemble, the quadratic nature of the resultant Hamiltonian allows us to express analytically the excess free energy, enthalpy, and entropy of ion binding to the protein. The analytical expressions allow us to separate the roles of the dynamic restraints imposed by the protein on the binding site and the temperature-independent chemical effects in metal-ligand coordination. For the consensus zinc-finger peptide, relative to the aqueous phase, the calculated free energy of exchanging Zn ²⁺ with Fe ²⁺, Co ²⁺, Ni ²⁺, and Cd ²⁺ are in agreement with experiments. The predicted excess enthalpy of ion exchange between Zn ²⁺ and Co ²⁺ also agrees with the available experimental estimate. The free energy of applying the protein restraints reveals that relative to Zn ²⁺, the Co ²⁺, and Cd ²⁺-site clusters are more destabilized by the protein restraints. This leads to an experimentally testable hypothesis that a tetrahedral metal binding site with minimal protein restraints will be less selective for Zn ²⁺ over Co ²⁺ and Cd ²⁺ compared to a zinc finger peptide. No appreciable change is expected for Fe ²⁺ and Ni ²⁺. The framework presented here may prove useful in protein engineering to tune metal selectivity.