The Role of Entropy in Biomolecular Modelling

Three Examples

1. Force Field Development

How to parametrise non-bonded interaction terms?

Include Entropy

=

Hydration

Hyd Hyd

G

H T S

of variety of solutes

Simulation at finite T

The Role of Entropy in Biomolecular Modelling

3. Protein-Ligand Complexation:

Ligand binding to the Estrogen Receptor:A variety of configurations (ensemble) contributes to binding, both in the protein and in waterContinuum representation of the solvent is unable to mimic binding subtleties of

individual solvent or co-solvent molecules

2. Partitioning of Solutes between various Solvent Mixtures

Solvation of small molecules:

HS Enthalpy co-act or may depending on mixture and solute

SS Entropy counteract

Continium methods will not catch these entropic effects

Four Ways to Compute Entropy Differences

( , ) ( , , ) state

( , ) ( , , ) state

a a

b b

H p r H p r

H

a

p br H p r

33333333333333333333333333333333333333333333333333333333

33333333333333333333333333333333333333333333333333333333

3( ) ln ! exp ( , , ) /NNVT b bA k T h N H p r k T dpdr

3333333333333333333333333333

Coupling Parameters approach

Hamiltonian is made function of :

Free energy depends on :

1. Entropy Difference via Thermodynamic Intergration (TI)

Free Energy Difference and End States Energy Difference

( ) ( )

( ) ( )

TI

end

end TI

T

ba

b a

ba

ba ba

ba

I

b b

b a

a a

b

b

a

a H

dA HA A A d d

d

U

U U

H

U

U AS

T

accurate

not so accurate

Four Ways to Compute Entropy Differences

,N V

AS

T

, ,N V T

S

2

1TIb

bab a

S dk T

H HH H

using

and

2. Entropy Difference directly via TI

correlation between and not so accurate

only -dependent terms all terms

H

H

3. Entropy Difference via finite Temperature Difference

using

,N T

AS

T

)

2

( ( )TI TIT ba ba

ba

T TA AS

T T

T

difference between almost equal accurate values

Four Ways to Compute Entropy Differences

4 . Solvation Entropy Difference via Solute-Solvent Entropy Difference

(using TI) and

End States Solvent-Solvent Energy Difference

,2

,,

1 1b

b a

a

ba

ba

all uv uvuv u

TI

endTI

Bv vv vv

vvu

bv

a

H HS H H d H H

k T T

US

T

accurate

only solute-solvent terms

not so accurate

all solvent terms

solvent: vsolute: u

Comparison of1. Excess Free Energy, Entropy of Water

2. Hydration Free Energy, Entropy of Water

Using four different Methods

Three Models or Hamiltonians:

1. SPC Model: Coulomb plus van der Waals interaction

2. SPCnc Model: no Coulomb interaction

3. SPCnn Model: no (non-bonded) interaction

Christine Peter

Thermodynamic Cycle System

1000 H2O moleculesperiodic boundary conditionsT = 280K, 300K, 320Ksimulations = 100-600psNVT NPT

Change:1 H2O hydration

all H2O excess more accurate

SPC (liquid)

SPCnc (liquid, no Coulomb)

SPCnn (ideal gas)

G, S, H = 0

Free Energy and Entropy of Water

Reference: J.Chem.Phys. (2004)

method2 1 4 4

Free Energy and Entropy of Water

Reference: J. Chem. Phys. (2004)

method 3

63

close

All 1000 H2O Molecules Changed

TI TI

A Single H2O Molecule Changed

A via TI S via TI Suv via TI

NVT

NPT

same patternas for 1000 H2O changed

erraticnot converged

same patternas for 1000 H2O changed