Why is it inappropriate to calculate free energy change from end points alone?

In molecular dynamics, free energy changes are estimated using a variety of protocols to establish a path between the starting and ending states. The classic example is umbrella sampling in which a sequence of intermediate states between the two endpoints is simulated.

The complication and computational cost incurred by these procedures strongly suggests that it would be incorrect to simply simulate the two end points, calculate a free energy for each, and then subtract. This feels vaguely incorrect to me, somehow the entropic contribution to free energy is being mistreated, but I cannot articulate how.

What is the reason that calculating a free energy change from the end points alone would be incorrect?

As an example, suppose you wanted to calculate the free energy change of a red block and blue block binding. If you ran a simulation for the unbound state and another simulation for the bound state and then tried to directly calculate the free energy change you would be trying to calculate a free energy according to the following diagram. The "All atoms infinitely separated" state would be implicit in the parameters that were used in the simulation. For example, if the simulation consisted of a Coulombic force between each pair of atoms then we usually would express energies of the system as relative to all atoms being infinitely separated and arbitrarily assign zero energy to that state. As a result, ΔG3 can be calculated from ΔG1 and ΔG2. The diagram obviously feels wrong but I can't put it into words.

Update: After reading into this a bit more it looks like end-point only methods are in use for binding free energy estimations via MD. Their weak-points are thought to be subtracting two large numbers to get a small number (two absolute free energies to get a change in free energy) and the handling of entropy. Since non-endpoint methods seem preferred the question still seems worth an answer but the question is more appropriately phrased as "Why are non-end point methods preferred?". At the moment I feel shaky on the details but eventually I may write an answer.

• I'm not versed on molecular dynamics simulations, but it seems to me that they want to minimize the fluctuation contributions on the calculation. Since free energy is an average value on an canonical ensamble, there are fluctuations, and if you have only to states to compute it, probably the errors are going to be large. – Hydro Guy Jan 26 '15 at 3:02