1549380323-Statistical Mechanics Theory and Molecular Simulation

362 Free energy calculations

fromx=−atox= 0, i.e., the value ofyremains fixed in each ensemble

member during this process. The work performed on each system inthe

ensemble is related to the change in potential energy in this processby

W=U(0,y)−U(−a,y)

(see eqn. (1.4.2)). By performing the average ofW over the initial en-

semble, that is, an ensemble in whichx=−afor each member of the

ensemble, show that〈W〉>∆A.

c. Now perform the average of exp(−βW) for the work in part b using the

same initial ensemble and show that the Jarzynski equality〈exp(−βW)〉=

exp(−β∆A) holds.

8.7. Calculate the unbiasing (Z(r)) and curvature (G(r)) factors (see eqns. (8.7.20)

and (8.7.31)) in the blue moon ensemble method for the following constraints:

a. a distance between two positionsr 1 andr 2 ;

b. the difference of distances betweenr 1 andr 2 andr 1 andr 3 , i.e.,σ=

|r 1 −r 2 |−|r 1 −r 3 |;

c. the bend angle between the three positionsr 1 ,r 2 , andr 3. Treatr 1 as the

central position;

∗d. the dihedral angle involving the four positionsr 1 ,r 2 ,r 3 , andr 4.

8.8. For the enzyme–inhibitor binding free energy calculation illustrated in Fig. 8.1,

describe, in detail, the algorithm that would be needed to perform the calcu-

lation along the indirect path. What are the potential energy functions that

would be needed to describe each endpoint?

∗ 8 .9. a. Write a program to perform an adiabatic free energy dynamicscalcu-

lation of the free energy profileA(x) corresponding to the potential in

problem 8.6. Using the following values in your program:a= 1,U 0 = 5,

kTy= 1,kTx= 5,my= 1,mx= 1000,λ= 2.878. Use separate Nos ́e–

Hoover chains to control thexandytemperatures.

b. Use your program to perform the histogram test of Section 8.12. Does

your histogram peak atp= 1/2?

8.10. Write adiabatic dynamics and thermodynamic integration codesto generate

theλfree energy profile of Fig. 8.2 using the switchesf(λ) = (λ^2 −1)^4 and

g(λ) = ((λ−1)^2 −1)^4. In your adiabatic dynamics code, usekTλ= 0.3,

kT = 1,mλ= 250,m= 1. For the remaining parameters, takeωx= 1,

ωy= 2, andκ= 1.

∗8.11. Derive eqns. (8.10.13).