Hybrid free energy

Fig. 6 Dependence of the DNA hybridization free energy on ionic strength. Base-pairing (a) is small and salt-independent, while base-stacking (b) is the main part of the interaction. Adapted with permission from [9]...

Fig. 32 Unbound particle fraction as a function of temperature for colloids interacting through DNA sticky-end pairing. Experimental data for various sticky end fraction (symbols) are compared with the melting curve for the same sticky end sequence in solution blue line), much broader. The red line shows the hybridization free energy of DNA in solution right axis), while the dashed red line shows the effect of an entropy correction (see discussion in the text). Reproduced with permission from [136]...

The differential change in the hybrid free energy in box i and and the reservoirs is... 

The differential hybrid free energy for an arbitrary state with X molecules is... 

Use of a Monte Carlo or a cluster (Hybrid) algorithm to calculate ionization constants of the titratable groups, net average charges, and electrostatic free energies as functions of pH. 

Finally, an alchemical free energy simulation is needed to obtain the free energy difference between any one substate of system A and any one substate of system B, e.g., Ai- In practice, one chooses two substates that resemble each other as much as possible. In the alchemical simulation, it is necessary to restrain appropriate parts of the system to remain in the chosen substate. Thus, for the present hybrid Asp/Asn molecule, the Asp side chain should be confined to the Asp substate I and the Asn side chain confined to its substate I. Flat-bottomed dihedral restraints can achieve this very conveniently [38], in such a way that the most populated configurations (near the energy minimum) are hardly perturbed by the restraints. Note that if the substates AI and BI differ substantially, the transfomnation will be difficult to perform with a single-topology approach. 

A final important area is the calculation of free energies with quantum mechanical models [72] or hybrid quanmm mechanics/molecular mechanics models (QM/MM) [9]. Such models are being used to simulate enzymatic reactions and calculate activation free energies, providing unique insights into the catalytic efficiency of enzymes. They are reviewed elsewhere in this volume (see Chapter 11). 

Thomas A, Jourand D, Bret C, Amara P, Field MJ (1999) Is there a covalent intermediate in the viral neuraminidase reaction A hybrid potential free-energy study. 1 Am Chem Soc 121 9693-9702... 

H/D exchange of H and Hg protons of sulfone 86 and estimated the difference in the free energies of activation for 79a and 79b to be < 1.2 kcal mol , based on the kjk value of 3 0.5. In the base-catalyzed H/D exchange of 87, kjk = 1.6, where k and k are the rate constants of H/D exchange of H, and H, respectively. Based on the small kjk value. Brown and colleagues suggested that if the carbanion is pyramidal, the steric stabilities of 79a and 79b are almost identical. Meanwhile, based on their C-NMR study Chassaing and Marquet proposed that the hybridization of the carbon atom of the sulfonyl carbanion, PhSOjCHj , would be between sp and sp . 

Simonson, T., Electrostatic free energy calculations for macromolecules a hybrid molecular dynamics/continuum electrostatics approach, J. Phys. Chem. B 2000, 104, 6509-6513. 

It was noted above that a continuum treatment of the solvent can be helpful, although representing certain solvent molecules explicitly may be necessary. The expressions for handling the free energy contributions in such hybrid models have been derived by Gilson et al.11... 

An alternative approach to free energy calculations is the thermodynamic integration (TI) method,18 20 which considers the ensemble average of the first derivative of the hybrid potential with respect to A at various values of A... 

The incorporation of the generalized Bom model into free energy calculation methods using FEP/TI and A-dynamics was carried out by Banba and Brooks.81 They define the electrostatic solvation energy for the hybrid system as follows... 

The basic formalism of the X-dynamics method has taken various forms in its application to problems of interest. In an early prototype calculation to assess umbrella sampling in chemical coordinates, the X-dynamics method was used to evaluate the relative free energy of hydration for a set of small molecules which included both nonpolar (C2H6,) and polar (CH3OH, CH3SH, and CH3CN) solutes.1 By assigning a separate X variable to the Lennard-Jones and Coulomb interactions, a linear partition of the potential part of the hybrid Hamiltonian was constructed... 

The X coupling parameter defines a set of intermediate hybrid states between the unperturbed and the fully perturbed states. Then, the total change in the free energy is... 

---