The Calculation of Enthalpy and Entropy Differences??? (Housekeeping Details for the Calculation of Free Energy Differences) first edition: p second edition: p

The Calculation of Enthalpy and Entropy Differences Free energies can now be calculated with errors of less than 1 kcal/mol in favorable cases. Enthalpy and entropy differences for solvation could be calculated by simulating the two systems separately and taking the differences in the total. –Leads to much larger errors than for the free energy since the free energy reduces to interaction terms only involving the solute. For example: The solvent-solvent interaction term which contributes the so-called cavity (solvent reorganization) term to the energy is said to be canceled exactly by a corresponding term in the entropy. –Yu, H.-A.; Karplus, M. J. Chem. Phys. 1988, 89,

Partitioning the Free Energy If the thermodynamic integration method is used the overall free energy can be partitioned into individual contributions. However, while the total free energy is a state function the individual contributions are not.

Partitioning the Free Energy Calculation of the free energy differences by thermodynamic integration: –When performing this procedure on individual contributions, energy is transferred between the contributors. For example relieving strain in a bond angle may increase the potential energy in certain bond distances.

Partitioning the Free Energy A common practice is to partition the free energy into contributions from the van der Waals and electrostatic interactions. The biotin/streptavidin complex has an extremely strong association constant (-18.3 kcal/mol). The favorable electrostatic interaction, from H-bonding, was canceled by the free energy of interaction of biotin with water. However there was a very large van der Waals interaction. biotin

Partitioning the Free Energy Strong van der Waals interaction.

Potential Pitfalls with Free Energy Calculations Two major sources of error: 1) Hamiltonian. 2) insufficient sampling of phase space. Inadequate sampling: –errors may be identified by running the simulation for longer periods of time (molecular dynamics (MD)) or for more iterations (Monte Carlo (MC)). –The perturbation may be run in the forward and reverse directions. The difference in the calculated energy values, hysteresis, gives a lower bounds estimate of the error. Note, very short simulations may give almost 0 hysteresis while the errors may still be large.

Implementation Aspects Simulation Method: –Molecular dynamics is almost always used for systems with significant degree of conformational flexibility. –Monte Carlo gives good results for small rigid molecules. –Thermodynamic perturbation or integration preferred over the slow growth methods. –Slow growth suffers from “Hamiltonian lag” and adding additional values of requires rerunning the simulation from scratch.

Implementation Aspects Coupling Parameter ( ) –  does not have to be a constant value. Could use small values when the free energy is changing quickly and large values when the free energy is changing slowly. (Dynamically modified windows) Choice of Pathway –A change that involves high energy barriers will require much smaller increments in to insure reversibility than a pathway that proceeds via a lower barrier.

Implementation Aspects Single-topology: Dual Topology: –Both molecular topologies are maintained, but do not interact with each other. –The simplest Hamiltonian that describes the interaction between these groups and the environment is a linear relationship: –The molecular topology at all stages is a union of the initial and final states, using dummy atoms where necessary.

Implementation Aspects Dual Topology: –Can result in a singularity in the function for which an ensemble average is to be formed. –Problem with thermodynamic integration where the derivative of the parameterized Hamiltonian with respect to is the observable. –One solution when performing MC is to change the scaling factors: When n  4 the free energy difference is always finite and can be integrated numerically. –However this results in difficulties in calculating the first and second derivatives of the potential energy function required for MD. Solution: Soft Core Potentials.

Implementation Aspects Soft Core Potential: –The traditional Lennard-Jones interaction can be replaced: –Similar expressions can be developed for for electrostatic interactions. Atom-atom separation Potential energy = 1 = 0 Where  determines the softness of the interaction, removing the singularity.

Potentials of Mean Force May wish to examine the Free Energy as a function of some inter- or intramolecular coordinate. (ie. Distance, torsion angle etc.) The free energy along the chosen coordinate is known as the Potential of Mean Force (PMF). Calculated for physically achievable processes so the point of highest energy corresponds to a TS. Simplest type of PMF is the free energy change as the separation (r) between two particles is varied. PME can be calculated from the radial distribution function (g(r)) using: –Recall: g(r) is the probability of finding an atom at a distance r from another atom.

Potentials of Mean Force Problem: The logarithmic relationship between the PMF and g(r) means a relatively small change in the free energy (small multiple of k B T may correspond to g(r) changing by an order of magnitude. –MC and MD methods do not adequately sample regions where the radical distribution function differs drastically from the most likely value. Solution: Umbrella Sampling. –The coordinates of interest are allowed to vary over their range of values throughout the simulation. (Subject to a potential modified using a forcing function.)

Umbrella Sampling The Potential Function can be written as a perturbation: –Where W(r N ) is a weighting function which often takes a quadratic form: –Result: For configurations far from the equilibrium state, r 0 N, the weighting function will be large so the simulation will be biased along some relevant reaction coordinate. –The Boltzmann averages can be extracted from the non-Boltzmann distribution using: Subscript W indicates that the average is based on the probability P W (r N ), determined from the modified energy function V ‘ (r N ).

Umbrella Sampling This free energy perturbation method can be used with MD and MC. Calculation can be broken into a series of steps characterized by a coupling parameter with holonomic constraint methods used to fix the desired coordinates. (ie. SHAKE)