The Calculation of Enthalpy and Entropy Differences???

(Housekeeping Details for the Calculation of Free Energy Differences)

first edition: p. 493-502second edition: p. 574-585

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, 2366-2379.

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.

( )

=

= ƒ

ƒ=Δ

1

0

,λ

λλ

λλ

dANN rpH

( ) ( )

L

L

+Δ+Δ=

+ƒ

ƒ+

ƒƒ

=Δ =

=

=

=

anglesbonds

1

0

angles1

0

bonds

AA

ddAλ

λλ

λ

λλ

λλ

λλ

λ

λ HH

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 hasan extremely strong associationconstant (-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.

N

NS

H

H

COOH

HH

H

Obiotin

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.

==

==+=Δ

1;

0;11 )(

λ

λ

stepNi

iiiA H-H

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:

H

HHH

O X H

HXH

O H

CH3H

OCH2

H

O H – The molecular topology at all stages is a union of the initial and final states, using dummy atoms where necessary.

( ) ( ) XY 1 HHH λλλ −+=

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.

( ) ( ) XY 1 HHH nλλλ −+= n

Implementation Aspects• Soft Core Potential:

– The traditional Lennard-Jones interaction can be replaced:

– Similar expressions can be developed for for electrostatic interactions.

( ) [ ] [ ]√√√

↵

+−

+=

66LJ

6

266LJ

12

4ijij

ij

ijij

ijij

LJij rr

vσα

σ

σα

σελ

Atom-atom separation

Pot

enti

al e

nerg

y

= 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.

constant)(ln)( B +−= rgTkrA

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 kBT 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(rN) is a weighting function which often takes a quadratic form:

– Result: For configurations far from the equilibrium state, r0N, 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 PW(rN), determined from the modified energy function V ‘(rN).

)()()( NNN W rrr +=′ VV

20 )()( NN

WN kW rrr −=

[ ][ ]

W

N

W

NN

TkW

TkWAA

B

B

/)(exp

/)(exp)(

r

rr

+

+=

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)