chemical review_ free energy calculations applications to chemical and biochemical.pdf

7/24/2019 Chemical review_ Free Energy Calculations Applications to Chemical and Biochemical.pdf

1/23

Chem. Rev.

1993,

93,

2395-2417

2395

Free Energy Calculations: Applications to Chemic al and Bioch emical

Phenomena

Peter

Kollman

Depattment of pharmaceutical Chemlstty, University of California, San Francisco, California 9 4 143

Received

May

5

1993 (Revlsed Manusc ript Received August

24

1993)

Contents

E. Combining Quantum Mechanical 2412

I.

Abstract

2395 V. Summary 2413

Calculations with Free Energy Calculations

11.

Introduction

2395

I I I. Methodological Issues 2396

A. Basic Formulation of Free Energy 2396 I . Absfracf

Calculations

B. A Sample Application: The Relative

Solvation Free Energy of Methanol and

Ethane

C. Why is the Calculation

of

AG More

Accurate Than the Calculation of

AH

and

AS?

D. Historical Perspective of Free Energy

Calculations Applied to Chemistry/

Biochemistry

E. Challenges in Free Energy Calculations on

the Solvation

of

Ionic, Polar, and Nonpolar

Molecules

F. Single and Dual Topologies in Free Energy

Calculations

G. Limitations n the Implementationof Free

Energy Methodology in AMBER3 and the

Removal of These Limitations in AMBER4

Theory, Thermodynamic Integration, and

Slow Growth

I . Free Energies Can Be Calculated for

Coordinate as Well as Topology Changes

J.

Dependence of Calculated Free Energies

on Molecular Mechanical Model

K. The Sampling Issue

L. Combining Quantum and Molecular

Mechanical Methods

H. Comparison of Statistical Perturbation

IV . Applications

A. Solvation

1. Aqueous Solvation

2.

Nonaqueous Solvents and Partition

Coefficients

3 . Free Energy as a Function of

Conformation

4. Solvent Effects on Tautomerism,

Reduction/Oxidation, AcMlty/Basicity,

Excited States, and Reactions in

Solution

5. Protein Solvation

1. Small Organic Hosts

2.

Absolute Free Energies of Association

3 .

Protein Hosts

C. Sequence Dependence on Ligand Binding

and Catalysis

D. Sequence Dependent Stabilities

6. Molecular Association

2396

2397

2398

2398

2399

2399

2400

2400

240 1

2402

2402

2402

2402

2403

2403

2404

2404

2404

2405

2405

2407

2407

2410

241 1

I

will review the applications of free energy calcu-

lations employing molecular dynamics or M onte C arlo

methods to a variety of chemical and biochemical

phenomena. T he focus is on the applications of such

calculations t o m olecular solvation, mo lecular associ-

ation, macrom olecular stability, and enzyme catalysis.

T he m olecules discussed range from m onovalent ions

and small molecules to proteins and nucleic acids.

I . Infroductlon

Free energy is arguably the most im portant general

concept in physical chemistry. T he free energies of

molecular systems describe their tendencies to associate

and react. Thu s, being able to predict this quantity

using molecular theory in general would be an enor-

mously important advance and is a seductive goal.

Progress toward this goal has been made in recent years,

and this review attemp ts t o describe this progress as it

applies to the use of molecular dynamics and Monte

Carlo methods to carry o ut free energy calculations in

the following areas:

(1)

solvation of small molecules,

(2)

ligand binding to organic hosts and t o proteins and

nucleic acids, (3) sequence-dependent stabilities of

proteins and nucleic acids, and (4) environmental effects

on reactions in solutions and in enzymes.

I will review of the methodologies used in such free

energy calculations. After presenting som e of th e basic

equations, I present a detailed discussion of th e first

application of the methodology to the calculation of

the relative solvation free energy of the organic mol-

ecules methanol and ethane. Th e agreement between

the calculated and experimental free energy is impres-

sive, as is the inher ent s tatistical error of


2/23

2996 Chamfcai Reviews,

1993,

Vol.

93,

No. 7

K o l h a n

problem an d describe the H amiltonian

H A) as

in eq 2

(2)

where A can vary from 0

H = HA )

to

1 H = HE) .

One

can

then generalize eq

1

as follows:

H A) = MI,+ 1 - A)HA

I

Peter Koliman received his

B.A. in chemlsby

from Grhlnell College

in

1966 and

his

Ph.D.

in chemistry from Princeton

University

in

1970. After

a

NATO

feibwshlpat Cambridge Unhrersity in 1970-

1971, he joined the

faculty

of the Department of Pharmaceutical

Chemistry. School of

Pharmacy,

VCSF. where, since

1980,hehas

been

professor of chemistry

and

pharmaceutical chemistry.

I

will disc us the main barriers ha t hinder the broader

applic ation of free energy calculations. Th ese can be

succintlysummarized

as(1)

rrors in therepresentation

of the energy of the system and

(2)

limitations in

sampling enough of the relevant (low free energy)

conform ations of th e system.

A

number of reviews have appeared which have

concentrated exclusively or in part on free energy

calcuIations.l-l6 Our review has a nu mb er of unique

aspects in ita em phasis on th e practical and greater

detail on applications han many of the previous reviews.

I I I . Methodological Issues

A. Basic Formulation of Free Energy

Calculations

Th e statistical mechanical definition

of

free energy

is in terms of the partition function, a sum of the

Bolzmann w eightsofall

heenergylevelsofthesystems.

However, only for the simplest model system can this

free energy be represented by an analytical function.

One can write a classical analog of the quantum

mechanical partition function where the energy is

viewe$ as a continuo us func tion, rath er th an discrete.

This is likelytobeagood approximationinmosts y s t e m

involving noncova lent interac tions near room tem per-

ature. [

Unfortunately, the free energy represented in this

way requires an integration over

all

3N degrees of

freedoq, where N = number of atoms in the system.

Thus, this is imprac tical in most cases. How ever, if

one focuses on free energy differences betwee n relate d

systems A and B

(AG

= GB

- GA)

represented by

Hamiltonie,

H A

and

HB ,

his free energy difference can

be represented in eq

1

(1)

where

AH = HE- H A

and

) A

refers

to

an ensemble

average over a system represented by Ha miltonian HA.

Equation

1

the fundam ental equation of free energy

more than a trivial way, then eq

1

will not lead to a

sensible free energy. One can, however, generalize he

AHIRT

G B - G A

=

AG = -RTln (e-

) A

perturbatio calculations.

If

systems

A

and

B

differ in

1

AG = G,- G , = -RTln (e-AHRT)A 3 )

A=O

where AH =H A + ~ AHA. One breaks up the free energy

calculation into w indows, each one involving a sm all

enough interval in A to allow the free energy to be

calculated accurately.

An alternative to free energy perturbation calcula-

tions is thermodynamic integration, where the free

energy difference between two systems (one charac-

terized by H = H Aor A = 0 in eq

2

and th e other by H

=

H B

or A = 1 n eq 2

(4)

T he application of eq 4 requires one to evaluate the

ensemble average of t he derivative of th e ham iltonian

with respect

to A, dH/dA)A

a t v arious values of A. One

can then use numerical integrationmethods

to

calculate

AG by eq 4.

The third commonly used method for free energy

calculations is called slow grow th in which th e Ham il-

tonian is changed

an

infinitesimal amou nt over each

step of the sim ulation (eq

5)

A G =

2 H * + l - H J (5)

110. atop. A 4

where

H .

is the Ham iltonian for a given A and

Hn+,

s

the H amiltonian for the next larger A. This equation

can be derived from eqs 1 or

4,

using the assumption

in eq

1

ha t

AG

is small and in eq

4, dH/dA

= AH/AA.

If

evaluated accurately enough, AG should be in de

pend ent of path or simulation protocol, but there are

often a number of practical reasons for using one of

these three approaches.

As noted above in reference

to

eq

1,

he realism of

free energy calculations depe nds on the realism of the

Hamiltonia

H A

and

HE.

To our knowledge, virtually

all

applicationsof this m ethodology make the assum p-

tion tha t the k inetic energy term in the H amiltonian

can be ignored. How realistic is this assum ption?

To

proceed furth er on this point, let us focus on th e first

applic ation of free energy calculations

to

the solvation

of organic molecules:

the study of the relative free

energy of solvationo fmethan ol and eth ane by Jorgensen

and Ravimohan? (JR).

One begins by con sidering a

free en ergy cycle

(6 ) .

B. A Samp le Application: The Relative Solvation

Free Energy of Methanol and Ethane

&GI

C W ) HsCHdQl

do, cyc~) (6)

A

CHaWaQ) H %w)

G e ( c W )

Since free energy is a state function, the difference

in free energ ies of solv ation

of

methanol and ethane,


3/23

Free Energy Calculations

AG1, where AG2 and AG1 are the free energies of

mu tating methanol into ethane in solution and in the

gas phase, respectively. J R used M onte Carlo calcu-

lations and eq

3

(free energy pertur bation ) t o calculate

AGz. Th e O PLS solute model used by J R involves a

united atom CH3 group,

so

CH30H is a triatomic

molecule and CH3 - CH3 a diatomic. J R assumed th at

differences in AG due to kinetic energy differences

would be identical in calculating

AG1

and

AGz,

so these

were not included in either calculation. Van Gu nsteren

has validated this approximation in calculations of

simple systems.18 Given tha t, it is reasonable to mak e

the assumption t h at H -

V.

W hat is a typical potential

energy function

V?

Weiner et a1.19~museq

7;

he OPLS

model uses this form of the eq uation without th e explicit

H-bond term.

AAG

=

AGsolv (CH3CH3) - AG,lv (CH30H)

=

AGz -

Chem ical Reviews,

1993,

Vol.

93,

No. 7

2307

v =

Kr r- req)2+ KB e e e q 2+

bonds angles

-11 +

cos(n4 -

711

+

Jorgensen and Tirado-Rives21 ave adopted the O PLS

model for molecular mechanics and dynamics using the

Weiner et al. parameters for the first three terms in eq

7:

bond stretching, bond bending, and torsional rota-

tion, while employing nonbonded terms they have

derived by carrying out Monte Carlo calculations on

requisite liquids for the nonbonded par t of the potential.

In their M onte C arlo evaluation of AGz (eq

3

and

6) ,

JR assumed rigid bond lengths and angles; thus no

intramolecular contribution to the free energy was

calculated for the m utation of a triatom ic molecule

(CH3OH ) to a diatomic (CHs-CH3). As in the case

of the kinetic energy term, one is assuming that any

such contributions to AG2 are identical to those th at

would appear in AG1. Recently, Cieplak and V eenstra

of UCSF (unpublished) have examined this approxi-

mation using molecular dynamics calculations and have

found it to be valid for mutation s of me thane and e thane

to methanol and dimethyl ether to propane.

Typically, Monte Carlo calculations on complex

molecules assume rigid bond leng ths and angles, as did

the J R calculation of AG2. As we have noted above , in

most prac tical applications of free energy pertur bation ,

one must use eq

3,

in which one creates a num ber of

hybrid systems interm ediate between CH3OH an d CH3-

CH3.

For

example, the C-0 bond length in methanol

is 1.43 A ; the C-C bond in ethane is 1.53 A. A hybrid

state

X =

0.5)would involve a bond distance between

the CH3 group and th e changing atom

(0-

H3) of

1.48A

In th e O PLS m odel of m ethanol, the charge on

hydrog en is 0.435, on oxygen,

-0.700,

and on the methyl

group,

0.265.

In the

X =

0 (ethane), the charges are

zero. T he van der Waals parameters a re similarly

interpolated between methanol and ethan e for the

X

=

0.5 state.

J R began the sim ulation by inserting the m ethanol

molecule in a box of

125

T IP 4P water molecules and

carrying out Monte Carlo calculations to equilibrate

this system in an isobaric ensemble (co nstan t num ber

of particles, temp erature, and pressure). They then

evaluated the free energy for mutating m ethanol to the

X = 0.125 stat e, which is 718 methanol and 118 ethane.

They used double wide samp ling, which they found

to be a useful test for convergence of th e free energy.

Th is involves calculating th e free energy difference for

both the

X

-

and

A

-

intervals.

If

one applies eq

3 and evaluates the ensemble at state

X

and evaluates

the free energy to m utate this into A, this should be the

negative of the free energy determined by using the

ensemble characteristic of

A

and calculating the free

energy to m utate this to state

A.

J R fo und they needed more (closely spaced) values

of X near the methanol state than the ethane state,

because the free energy of interaction with the sur-

rounding w aters is changing more rapidly in this range.

When they had evaluated AGOby mutating methanol

to ethane, they found a calculated AG = 6.75 0.2

kcalfmol, n excellent agreement with the experimental

AG

= 6.93

kcal/mol. Th is was a most exciting result,

since the molecular mechanical models for water and

methanol, derived from reproducing t he enthalpies a nd

densities of the respective liquids, could be used withou t

modification in a binary system involving both mole-

cules and the experimental free energies remained

excellent. Now adays, one can achieve such agreeme nt

with much simpler models, bu t a t the time this was a

most exciting result t o this reviewer. It ed to th e general

incorporation of the free energy approach into the

simulation program AM BER by Singh.22

C. Why Is the Calculation of b G More Ac curate

Than the Calculation of A H and AS?

T o put this result into context, one must appreciate

th at to directly calculate the difference in e nthalp y of

solvation, AH of methanol and ethane would have

required se parate sim ulations of th e two solutions and

taking the difference in the total energy of the two

systems

(125

waters and

1

solute). T he tota l energy of

these system s from M onte C arlo calculations is of the

order of

-1250 10

kcalfmol. Thu s, any directly

calculated enthalpy

AH

would have an inherent error

of f 1 0 kcal/mol, not


4/23

2308 Chemical Reviews, 1993,Voi. 93,No. 7

Kollman

plished with convergence to

-

kcal/mol o ut of

-

00

kcal/mol in

80

pa of molecular dynamics. Polar

dominated mutations such as methanol - thane17

convergence rapidly also. T he m ain difficulty in the

electrostatic dom inated pertu rbatio ns comes when one

changes the net charge of the system.

For

example,

th e free energy calculated for the Ne - a+ m utation

will depend strongly on the nonbonded cutoff in the

simulation. T he solvation free energy estimated using

a continuum model for creating a monovalent ion33

suggests that , using the

8-A

nonbon ded cutoff typical

of simulations will result in a

-20

kcal/mol underes-

tim ate of th e abso lute value of th e solvation free energy.

Using the simple Born formula to correct such errors

is not rigorously correct when one uses nonspherical

bounda ry conditions, although using the correction is

certainly better th an not using it.

Where dealing with this problem becomes very

im portan t is in the calculation of pKas for ionizable

groups in proteins, where the presence of numerous

charge groups complicates m atters a nd it is important

to calculate the solvation free energies of the ionized

groups to f l-2 kcal/mol or better. Th is large challenge

has been undertaken by Lee and WarsheP with

continuing improved success, including improved mod-

els to accurately represent long-range effects.

For

example, the local reaction field method they propose

is significantly more efficient th an no-cutoff m ethods

using spherical boundary conditions at a fraction of

the comp utational expense. Warshel recomm ends a

hybrid m odel, with explicit representation of solvent

to a given distance, further waters with the PDLD

(Langevin dipole) model, and a continuum electrostatic

model beyond that.

In

free energy calculations, one

can alsouse a hybrid approach , where, for the molecule

or fragment which is being mutated, no nonbonded

cutoff is used, with a n 8-A cutoff used for th e rest of

the system. Provided th at the system is net neutral or

close to it, this approach

also

offers a significant

improvement over the standard

8-A

cutoff a t only a

modest additional comp utational expense. The re have

also

been other recent, exciting new approaches to

efficiently incorporate long-range electrostatic effects

into simulations in general.35136

T o accurately simulate sm all, nonpolar m utation s is

a particular challenge because the AG is very sm all and

can be a small difference between the positive exchange

repulsion and the negative dispersion attraction (eq 7).

For example, the following relative experim ental free

energies of solva tion37 in w ater (in kcal/m ol, 1 M

standard state) illustrate this

(8).

+1.94

0.16 0.21

nothing

-

CH,

-

- ,H,

8)

Sune t a1. ave shown how one can simulate m ethane

- thane and ethane - ropane rather accurately,

using Spellmeyers new all atom van der Waals pa-

ram eters a nd Pearlmans b ond pmf correction39 and

new protocol for the representation of th e van der Waals

par t of V(X) for disappearing groups.qo Insuring con-

verged free energies requires

-

00 ps of simula tion in

each direction; ethan e -propane is calculated to within

-0.1

kcal/mol of experiment; w hereas the m ethan e-

ethan e calculation is more dependent on partial charge

model and is overestim ated by -0.1-0.3 kcal/mol.

D. Historical Perspective of Free Energy

Calculations Applied

to

Chemlstry/Blochemistry

Le t me now give some historical perspective t o the

development of free energy calculations for use in

systems of interest in organic and biochemistry. T he

basic equations for free energy perturbation and

thermo dynam ic integration were developed by Zwan-

~ i g , ~ ~

irkwood,25 and Valleau and T omie,26 ut

i t

was

in the early 1980s th at th ey were used in analysis and

simulation of biophysical systems.

Postma

e t

al.27

studied noble gas solvation, W ar sh eP presented pre-

liminary results on th e solvation free energy contri-

bution to an electron transfe r reaction coordinate using

two spheres for donor and acceptor an d a dipolar model

of water, and McCammon showed the usefulness of

free energy perturbation calculations on a model

systemzeprior to JRs study on the relative solvation

free energy of m ethanol and ethane.17 T he a dven t of

a major enhancement in computational power of the

vector CRAY X-MP enabled a major advance in the

generality of application of free energy me thod s by Bash

et ~ 1 . 3 ~ 1 ~ ~

n a pair of pap ers in

Science

they studied

th e relative so lvation free en ergy of a w ide variety of

amino acid side chains, nucleic acid bases, and other

organic molecules, as well as th e relative b inding free

energy of a pair of ligands to the protein therm olysin

undergoing experimental studies as well. Those studies

clearly dem onstrated the pow er and generality of free

energy calculations and th e feasibility of studyin g large

mutations such as alanine -tryptop han and methan e

-

-CHs-guanine.

Th e largest mutation attem pted

prior to Bash

et

d s tudy involved addition or mutation

of one or two atoms. Although the calculations were

of rather limited duration by todays standards, they

clearly showed feasibility of study ing a wide var iety of

chem istry with these ap proaches an d achieving results

in reasonable agreement with experim ent with modest

( f l kcal/mol) error bars.

In the process of their studies, Bash

e t

al. found it

to be u seful to employ electrostatic decoupling, i.e.

changing only the electrostatic p art of the molecular

mechanical potential function first and then the re-

maining. Th is was motivated by the results of some

simulations (e.g. histidine - lanine) where the pres-

ence of hydrogens w ith some remaining charge b ut very

small van der W aals repulsion experienced an artifac-

tually large free energy change upon approach of a

solvent water. One could also imagine solving this

problem with a different X dependen ce in the electro-

static and van der Waals parts of the molecular

mechanics potential function.

E. Challenges In Free Energy Calculations on

the Solvation of Ion ic , Polar, and Nonpolar

Molecules

W hat were the major limitations found by Bash e t

al. in simulating the solvation free energy charges

accurately? One can consider three type s of mole-

cules: nonpolar, polar, and ionic.

The simulation of

polar and ionic solvation effects is dominated by the

electrostatic energies. These can be simulated rea-

sonably accurately and reproduceably with rather

limited simulation lengths. Even the simulation of Ne

-

a+ by S tr aa ts m a a nd B e r e n d ~ e n ~ ~an be accom-


5/23


Why do van der Waals perturbations, where one is

disappearing atom s, require long simulations for con-

vergence th an th e electrostatic dom inated ones, whose

AG are so much larger in magn itude? In our view it is

because electrostatic d ominated changes require mainly

dipolar reorientation of th e solvent, whereas the van

der Waals charges due to growing and disappearing

atoms, involve more slow repacking and translatio nal

diffusion of the solvent.

Hermans41 has show n how one can estim ate noise

and hy steresis for the slow growth method; these two

quantities are related to the relaxation time of the

system and the width of an assumed Gaussian distri-

bution in configuration space. Herm ans e t al. have

also provided de tailed examples of protocols to estimate

errors in different typ es of free energy calculations.42

Both Pea rlma n an d Kollman43 and W ooda have eval-

uated t he H amiltonial lag n slow growth calculations

and th e implications of thi s lag in accurate represen-

tation of the calculated free energies.

Chemical

Reviews, 1993,

Voi. 93,

No.

7 2309

limitations:

(1)

First, intra perturbed group contri-

butions to the energy were calculated in determining

the ensem ble bu t were not included in th e free energy.

Th is was a deliberate choice because it

was

reasoned

that the large intramolecular energy change involved

in, for example, removing an imidizole ring would add

too much noise into th e free energy and obscure the

more important differences in the intermolecular

interactions. For example, th e inclusion of such term s

would require that both

AG1

and

AG2

be calculated in

cycle 6, whereas their neglect would allow the calculation

of only AG2. By using a rigid geometry model, th at is

essentially wh at was done by JR, with excellent results.

(2) Second, because it was not clear how to best carry

ou t th e transfo rma tion of a 10-12 t o a 6-12 nonbon ded

parameter, only the latter was included in the descrip-

tion of th e pertu rbed group. As shown by Ferguson

e t

aL4 one can replace 10-12 parameters with appro-

priate 6-12 parameters so this should not be a large

issue; nonetheless, one could not fully correctly im-

plement the Weiner e t ~ Z . l 9 * ~ 0orce field for free energy

calculations because of this limitation. Such an issue

is not relevant for the O PLS/A MB ER force field, which

conta ins only 6-12 p aram eters.

(3)

T he calculations of

Bash

e t

al.30 and Merz

e t

a1.* showed a significant

sensitivity to th e calculated free energy for disappearing

groups depending on whether th e bonds were shrunk

while th e group disappeared. For example, if one

mutates m ethane to nothing

to

calculate the a bsolute

free energy of solvation of me thane , shou ld one reduce

the C-H bond lengths as th e molecule disappears and,

if

so,

by how much? Since th e free energy mu st be the

same irrespective of the final C-H bond length in the

dumm y m ethane, it was puzzling th at the calculated

free energies were so dependent on whether the bonds

were shrunk in t he process.

Pearlmans implementation of the GIBBS module

in AMBER

440

has removed the three limitations:

(1)

one now has the option to include or not the intragroup

energies, as well as a selected set of them;

(2)

atom

pairs interacting through 10-12 parameters can be

mu tated t o those interacting with 6-12 parame ters with

fully correct representation of t he energy; and

(3)

P MF

correction allows the correct free energy

to

be calcu-

lated, irrespective of bond length changes. Th ere are

many other implementation im provements included

in AMBER 4 GIBB S, including the new combining rules

for nonbonded parameters involving disappearing a t-

oms, which were critical in the accurate representation

of hydrocarbon solvation free energies by Sun e t a1.

T he only remaining limitation in the methodology is

a consequence of the use of a single automatically

generated topology in free energy calculations. Th is

limitation can be illustrated by considering he m utation

of th e thym ine nucleoside to adenosine (only th e C1

of the sugar is represented below).

QU ,QU H e \ ,H

HS \

/H i b

DU? ,QUI y

C p f c 0 D U l

T

NC2

\ c4*N/

CH2

p\,/ \ NH

DU,-H \ \ 5

-

-C,

k..,.

02

7 9 . . ..

...

**s .

*.,

. * X I t C,;

F.

Single and Dual Topologies in Free Energy

Calculations

The re are two distinct approach es in how to represent

the molecular mechanical topologies in free energy

calculations. In the approach used by Jorgensen and

independently incorporated into the molecular dy-

namics programs GRO MOS and AM BER, one uses a

single molecular topology, and the term s in eq 7 hange

its shape and properties as X changes. In the molecular

mechanics program CHARM M40 one keeps two inde-

pendent

topologies,

one for methanol and the other for

ethane. For example, in the methanol to ethane

perturbation, the methanol OH and ethane CH3 both

exist a t the sam e time in th e calculations, bu t they do

not interact with each other. Th e (only about 0.1

A

apa rt) interaction of these groups with their environ-

me nt is calculated using eq

2

or a variant of i t, eq

9.

(9)

T he use of different exponen ts in (9)allows difference

pathway s and corresponding integrands

dHIaX

in eq

4

to be used in determining the free energies. On the

other han d, ther e are a variety of ways within the single

topology method to include th e X dependence more

directly inside the term s in the po tential energy (eq 7).

In th e single topology method, pearl ma^^^^has shown

th at in molecular dynamics one must determine

a

bond

pmf correction when bond length change, in order to

determine a rigorously correct free energy for the

mu tation. Such a correction is not required in the dual

topology method, but in the latter method, the best

way to overlap

or

constrain the topologies when

mu tating, e.g. 1-CH3 hymine to l-C H3 cytosine, is not

obvious, and the resulting free energy may be very

sensitive to the protocol chosen. A t this po int, it is fair

to say th at interesting and useful free energy calcula-

tions have been carried out with both approaches.

H X)

=

XnHB

+ (1

-

X ) n H A

G.

Llmltatlons In the Implemen tation

of

Free

Energy Methodology In AMBER3 and th e

Removal

of

These L lmltatlons In AMBER4

In the Singh

e t

~ 1 . ~mplementation of free energy

approaches with in AMBER 3.0, there were three


6/23

2400

Chemical

Reviews,

1993,

Vol. 93,

No.

7

In the single topology perturb ation, the thymin e He is

mapped into guanine

N9,

Hs into

Ca,

C5, into

N7

DU5

into Ha, and H s ~nd H5 become dummy atoms DU7

an d DU70 DU a nd DU4, become th e Hg and He,

O4

becomes Ng, H3 become DU1, and 0 2 becomes Ha, etc.

Because of the pertu rbation , bond

Cl-Nl

disappears,

to be replaced by Cl-H6

(N9) . A

new bond is formed

between H6 and H5 (N9-Ca). These bonds can have a

force cons tant of zero in the sta te where they d o not

exist, but their presence leads to an inconsistent

implementation of the molecular mechanical model.

For example, the bond between H5 an d H6 m eans

th at no van der W aals param eters are evaluated between

H5 and H6, even at X =

0

(thymine), since the model

does not include nonbonded interactions between atom s

separated by one or two bonds.

The nonbonded

interactions between atom s separated by three bonds

(1-4

interactions) are also handled different than longer

range non-bonded interactions in the W einer

et

al. force

field.19*20n t he above topology, however, one would

trea t H6--H5 and H5 interactions as

1-4

interactions

rather th an regular nonbonded interactions.

These limitations may not be critical in semiquan-

titative studies of sequence dependent perturbations

in DNA helices (e.g.

A T

-

A),

since one is always

taking the difference between two perturbations and

the inconsistent intram olecular energies could largely

cancel. For example, Ferguson has examined the

molecular mechanica l energy dependence of the

x

(el-

N )

angle in thymine n ucleoside using the perturb ation

and regular topologies and found reasonable, if not

quantitative agreement.46 Situations like the T-

pertu rbation are analogous to th e neglect of intragroup

perturbations required in AMBERS. However, th e new

combining rules implemented in A MBE R

440

lso allow

the dual topology approach because atom s th at exist

a t X

=

0 and not at X

= 1

(or vice versa) do no t experience

nonbonded interactions with each other, even in

intermediate

A 0, 1

states).

One would have an

extra angle term

(Nl-Cl-NS),

bu t this could be given

a force con stant of zero. W hether eith er or both single

or du al topologies will allow the more effective simu-

lation of free energy perturbation within nucleotide

double helices remains to be seen, but preliminary

results with th e single topology metho d a re promising.

Recently, Pearlman

J. m. Chem. SOC.,

ubmitted

for publication) h as com pared th e convergence in single

and dual topology approaches for an ethane

-

thane

mu tation, where the true answer mu st be zero and found

convergence more rapid for th e single topology method.

Kollman

H.

Comparlson

of

Statistical Perturbation Theory,

Thermodynamic Integration, and

Slow

Growth

Wh at are the advantages and disadvantages of the

three approaches used in free energy calculations,

statistical perturbation (SP), hermodynamic integra-

tion (TI),and slow growth (SG )? SP does not require

an analytical derivative of H with respect to A; this

derivative is trivial in the d ual topology approach, b ut

can be complex in single topology.

SP can also give

more problems th an SG for van der W aals dominated

changes as atoms are appe aring or d isappearing, if

AX

is too large. It is hard to know in advance w hat size

AX

to choose although Pearlm ans dynamically modified

has alleviated this problem. Th ere are no

fundam ental limitations in SPprovided computer time

is no object and the

AX

are chosen sufficiently sm all.

T he main funda mental weakness in

SP

is tha t the total

free energy cannot be separated into the sum of the

com ponent free energies, because the logarithm of an

exponential with different energy components is not

equal to the sum of the logarithms of the individual

components. T hu s free energy com ponent analysis

cann ot be rigorously carried ou t with th at appr0ach.m

Thermodynam ic integration, unlike SP , requires a

numerical integration of the values of the integrand

dH / dX ) . This is not a fundam ental limitations1 and,

provided th at enough X values are chosen, accurate free

energies can be calculated. An advan tage of TI (or SP)

over slow growth (SG) is th at after one has evaluated

a number of

aH/dX)

nd carried out the numerical

integration, one concludes th at one needsmore sampling

a t already sampled AS or more As in between, this can

be easily done witho ut losing any of the information

previously derived. In SG , one needs to rerun the entire

trajectory ag ain, if one decides to double th e sampling

time. Van Helden and van Gunsteren have shown that ,

if th e system fluctu ates over multiple conformations,

one is more likely to sample th is correctly and efficiently

in TI tha n in SG SS2 G also suffers from the Ham il-

tonian lag, since the Ham iltonian changes a t every

~ t e p . ~ g

All in all, TI seems to be th e best compromise way

to carry ou t free energy calculations a nd do component

a n a l y s i ~ ; ~ ~ ~ ~owever, provided sufficient sam pling is

done and multiple trajectories examined, all these

metho ds can give useful and insightful results. Far

more critical tha n th e choice of which of these m ethods

to use are the two key issues in free energy calculations:

(1)

accuracy of the Hamiltonian (potential energy)

function and

(2)

the sampling problem.

As

noted above, intragro up free energies can be large,

and it is not clear how important they are in specific

cases. We have examined

(P.

Cieplak and D. V eenstra,

unpublished ) their im portance in the calculation of the

relative free energy of solvation of meth ano l and eth ane

or methane an d dimethyl ether an d propane, finding

a


7/23


the neat Aqueous solvation appears to increase

the gauche population by

- lo ,

a noticeable bu t not

exceptionally large effect. Th is calculation required

th e determ ination of th e free energy as a function of

torsion al angle. Analogously the st ud y of CH4 dim er-

ization required th e determ ination of the free energy

as a function of bond distanc e, a so-called pot enti al of

m e a n fo rc e (P MF ) fo r a s so ~ ia t io n . ~ ~he (CH4)2,

Na+-Cl- an d Cl--Cl- PM Fs have revealed con tact

minima, and, in the case of (CH & and Na+ -Cl-, also

solvent separated minima.6m

Tobias and Brooksel have suggested how to imple-

ment free energy as a function of coordinates in

molecular dynamics and have used th is protocol in the

sim ula tion of confo rma tional chan ges in peptides.62-65

Pearlm an ha s carried ou t free energy profiles of th e x

and y dihedr al angles in nucleosides both in vacuo and

in solution.6e Dang has stu die d nucleic acid base67and

K+/

18-crown-6 association in solution using P M F

approaches.B8

Recently Boczko and Brooks69 suggested that a

variant of the multiple histogram method70was more

efficient tha n um brella sampling for calculating free

energy as a function of coordinate. Pearlm an has also

compa red method ologies for calculating free energy as

a function of c0ordinate.7~

Herman s ha s used a combination of conformational

and mutation free energy approaches to characterize

th e relativ e helical propensity of variou s amino acids.7%T4

Not only were these calculations in good agreement

with available (and subseq uent experiments) bu t gave

nice insight into why certain residues had g reater helical

propensity tha n others. For example, th e crucial role

of greater entropy in the den atured state leads Gly to

be less stable in a helix relative to Ala, whereas

a-am inobu tyric acid (AIB) is more stable tha n Ala in

a helix because its g reater rigidity and lowest energy in

th e helical area of th e

,4

map.

Chemical

Reviews, 1993,

Vol.

93,

No.

7 2401

models to reasonably reproduce th e molecular multipole

moments. Th e 6-31G* model inherently overestimates

molecular dipole mom ents by -10-20% and thus

contains some of t he im plicit polarization t ha t the

OPL S achieves by fitting to liquids. Often, however,

there are cancellation of err ors in fre e energy calcula-

tions so rather different charge models can lead to rath er

similar results. For example, Miyamoto has m utated

a 6-31G* charge model of biotin t o a n STO -3G model

both in solution and in th e binding site of steptavidin.76

Both AG values were -15 kcal/m ol, reflective of the

significantly smaller polarity of th e STO-3G model, but

their difference was -1 kcal/mol, small compared to

th e free energy of bindin g of -20 kcal/mol. Deriving

charge models th at accurately represent intramolecular

as well as intermolecular properties is a significant

challenge, as is even reproducing the fact th at cis-and

trans-N-methylacetam ide have a nearly identical sol-

vation free energy.

Th e standard O PLS model finds

kcal/mol fo r th e solvation free energy difference,77

6-31G* electrostatic potential derived charges

for

trans

NMA finds - kcal/mol for the solvation free energy

difference, and using the 6-31G* electrostatic potential

charges for cis-NMA to represent cis-NMA and 6-31G*

electrostatic potential charges for trans-NM A to rep-

resent trans NMA reproduces the nearly identical

solvation free energy.78

Reynolds et a1.79t80ave shown how one can use

multiple conformation fitting t o improve electrostatic

potential derived charge models and Bayly et ~ 1 . 8 ~

Cornel1 et

U Z . ~ ~

nd Cieplak et U Z ~ ave used multiple

conformations, multiple molecules, and restrained

electrostatic potentials

to

provide further improvements

in th e methodology in deriving atomic partial charges

for molecular mechanics/free energy calculations.

One might expect tha t a simple equation such as (7)

would break down in treating ionic systems, which

would be expected t o include significant ionic effects.

However, both Urban and Damewood and Aq vists

have shown th at one can derive effective ion para-

mete rs th at reproduce free energies of solvation even

with additive models. Aqvist and m ore recently Mar-

lone and Mer@ have derived the parameters by

carrying out solvation free energy calculations and

adjusting the van der Waals R* nd to reproduce the

expe rimen tal free energies of solvation an d, as well as

possible, the radial distribution functions. These

models would be expected to be less accurate for small,

gas-phase ion clusters; these can be treated with

nonadditive effects, as a number of studies have

sho~n.87-89Warshel et a1 W have often included explicit

polarization effects in their studies, including the

calculation of relative pK, v alues of protein functio nal

groups.

Cieplak has carried ou t the first free energy calcu-

lation using nonadditive effects on small ion-water

clusters, employing Monte Carlo calculations to derive

free energies for these c1uste1 -s.~~ traatsma and

McCam mon app lied free energy approaches including

nonadditive effects in m olecular dynamics, studying

th e free energy of solvation of a small solute and water

in ~ a t e r . ~ ~ ~ ~ ~ecently, Sun e t

al.

have shown how free

energy pertur bation including nonadditive effects could

improve the calculated Li+ /Na + electivity of an iono-

p h ~ r e . ~ ~

J. Dependence of Calcu lated Free Energies on

Molecular M echanical Model

There are two issues in assessing the accuracy of

potential energy functions such as

(7):

th e accuracy of

the parameters in th e equation and th e inherent ability

of th e functional form to correctly represent th e system.

As

noted above, the OPLS modelzl has proven to be

accurate in its calculation of solvation free energies.

Thi s is because the m odel is inherently w ell-balanced,

having had bo th solute and solvent paramete rs derived

using M onte Carlo calculations to reproduce th e den-

sities and entha lpie s of vaporization of liquids. T he

Bash et

aLso

study used mainly th e Weiner et

al.

force

field,19120which had been derived in a different way,

together with the T IP 3P water m odel; however, they

modified the charge model t o use 6-31G* electrostatic

potential derived charges, which are much more O PLS

like tha n the W einer e t al. charges. Kuyper et ~ 1 . ~ ~

have examined this issue in more de tail with th e relative

solvation energies of m ethoxy an d trimethoxyb enzene

and benzene and have found that 6-31G* derived

electrostatic potential charges led t o excellent solvation

free energies, STO-3G electrostatic po tential charges

gave reasonable ones, but 4-31G electrostatic p otential

derived charges greatly exaggerate the solvation free

energy. This can be related to he ability of these charge


8/23

2402 Chemical Reviews, 1993, Vol. 93, No.

7

Kollman

K.

The Sampling Issue

Thus, there are clearly a wide variety of chemical

phenomena which can be treated quantitatively with

free energy methods using molecular m echanical models

such as (7)

or

variants th at can include some nonadditive

effects. T he major roadblock in broader applications

of free energy approaches is most often no t th e accuracy

of the potential energy function b ut th e sampling issue.

Th e fact tha t existing methods have been so successful

in reproducing free energies of solvation and binding

in simple,well-defined systems supports this. However,

for those systems, even implicit solvation models such

as GBSA ,94 an d AMSOL% can often do a

rather good job as well; but the full free energy

calculations are in any case excellent reference points.

Furthermore, new methodologies for more efficient

sampling in free energy calculations are continuing to

be developed. T he multiple histogram method70and

locally enhanced sampling97 methodologies are recent,

exciting developments.

For sm all well-defined system s the determ ination of

solvation free energies or binding free energies in

solution, adequate sampling is, we feel, no longer an

issue. If need be, such systems can be simulated for

times approaching

1

ns with periodic boundary con-

ditions, and this appe ars adequate to describe, with a

statistical error of


9/23


potential charge models of p-nitro benz ene and ph enol.

This model would not include electronic interactions

of OH and N O2 groups. They the n determined the

6-31G* electrostatic potential charge model of p-ni-

trophenol directly and m utated one charge model into

th e other using free energy calculations. T he calculated

AG was in excellent agreem ent with the ex perimen tal

nonadditivity.

A

second set of m olecules studie d by

Bash

e t

aL3 were acetamide, N-methylacetamide a nd

NJV -dimethylacetamide. Interestingly this is

also

an

example of a nonadditive effect, since the singly

methylated N -methylacetamide is more soluble in water

tha n either acetamide or NJV-dimethylacetamide. Th e

calculations qualitatively reproduced this.

By m utating a m olecule to "dumm y atoms", one can

determ ine the absolute free energy of solvation. For

example, Bash

e t

mu tated a ll of the N -methylated

nucleic acid bases - H4 and then , by m utating CH4

-nothing, were able to predict the absolute free energy

of solvation of the bases, none of which had been

or

have been mea sured directly. More recently, Ferguson

e t

al 47

ave related these calculated free energies to

experimental sublimation energy da ta from which one

can reasonably accurately estimate the solvation free

energies. T he agreement between calculation and

experiment was reasonable.

1. Aqueous

Solvation

The solvation free energy of a wide variety of

molecules has been calculated by Jorgensen and co-

workers using the BOSS programlo2and M onte Carlo

methodologies. Recen t studies include th e stud y of

the solvation free energy of benzene and substituted

benzeneslo3 nd th e free energy of arom atie-aromatic

association in water."

Lee

et ul lo6

have compared simulation approaches

to calcu late solvation free energies of a wide variety of

functional groups, including protein side chains, in

water. They showed th at PD LD (protein dipoles-

langevin dipoles) methodologies provided a useful,

inexpensive alternative t o full free energy calculations

and calculated free energies in impressive agreement

with experiment for a wide variety of molecules and

phenom ena ranging from ionic stre ng th effects on ion

pairing, pKa of protein side chains and molecular

association. Th ey make some useful comparisons with

the results obtained by purely macroscopic electrostatic

models and conclude ha t the semimicroscopic PDL D/S

model is an efficient alternative t o fully macroscopic

or

fully microscopic calculations.

T he W arshell06 approach focuses on the e lectrostatic

energy and uses a n em pirical correction for hydrophobic

effects. They are able to show that , with appropriate

trea tm ent of long-range electrostatics,%very accurate

solvation free energies for polar and ionic molecules

can be calculated. On the other hand , a very accurate

(


10/23

2404

Chemical Reviews, 1993, Vol.

93,

No. 7

Kollman

3.

Free Energy as

a

Function of Conformation

The relative energy of different conformations of

molecules in the gas phase often changes significantly

in solution and free energy calculations can be used to

study this , as described in refs

57

an d 63. One of th e

prototypal systems for study is n-butane an d how its

relative free energy for gauche an d tra ns con formations

changes going from the gas phase t o solution. Tob ias

and Brooks113 have s tudied the conform ational equi-

librium of n -butan e in th e gas phase, in water, an d in

cc l4 and have shown tha t only an all-atom and not a

united atom m odel shifts the equilibrium toward the

gauche conformation and does so more in H2O than

C c4 . Other theoretical studies using united atom

models114 foun d a shi ft toward favoring a gauche

conformation in H2O; the reason for this discrepancy

between the united atom results in refs 113 and 114 is

not clear. Pe ttit an d co-workers have shown the

usefulness of a nalytical theories to reproduce confor-

mational dependent free energies in simple peptide

systems.l15J16

Ha et al.l17 illustrate the subtle balance between

intramolecular electrostatics and intermolecular sol-

vation free energy terms in their study of the a F?

equ ilibrium in D-glucose. T he calcu lated value of

-0.3

0.43 kcal/mol (favoring

a)

s small, consistent with

the magnitude of the experimental value but of the

wrong sign (t he expe rim ental free energy difference is

0.33 kcal/mol).

Nonetheless, this is a deceptively

difficult system to accurately sim ulate, given the large

number of hydroxyl group conformers and potential

for even small force field inaccuracies upsetting the

free energy balance.

Su n an d Kollman118 have shown t ha t one can use

Cartesian coordinate map ping to ca lculate solvation free

energy differences for conformations that differ sig-

nificantly in man y torsional degrees of freedom. One

separately evaluates the intramolecular free energy with

gas-phase minim ization/normal mode analysis. Th ey

validated this ap proac h on 18-crown-6, showing the

significant solvation stabilization of the D 3 d confor-

mation relative to o ther low-energy conformations.

4. Solvent Effects on Tautomerism, Reduction/

Oxidation, Acidity/Basicity, Excited States, and

Reactions in Solution

In order to study more general phenomena which

involve electronic stru ctur e changes, one can combine

free energy/solvation calculations with quantum me-

chanics. Cieplak et ~ 1 ~ ~ ~howed that high level ab

initio calculations could reproduce tautomerism of

simple aromatic systems (e.g. 2-hydroxypyridine

in

equilibrium with its keto tautomer) and then, by

mutating one tautomer into another w ith free energy

calculations, one could determine th e tautom eric equi-

librium in solution.

The re are often dramatic differ-

ences in these tautom eric equilibrium in the gas phase

and solution, and these could be of relative in DNA

base mispairing.

Such a combination of quantum

mechanics and free energy calculations have been

successfully applied to rationalize other tautomeric

equilibria as we11.120J21

As in tautom eric equilibria, basicity/acidity involves

proton movement and a large electronic structure

change; thus quantum mechanical calculations are

necessary to describe such a process. B ut again, one

can com bine such calculations with free energy calcu-

lations to determine relative basicities/acidities in

solution.122-12s h is is particu larly useful in estimatin g

difficult to measure pKa values, such as that of

ethane.122T he ac curat e calcu lation of pKa)s of enzyme

groups is important for interpretation of enzyme

mechanisms. Even negative esu lts are considerable

importance, as in Merzs demonstration126 hat Glu-

106 should not be considered as a general acid/base in

the mechanism of carbonic anhydrase. Along these

same lines, Aqvist has shown how the PKa of H2O is

perturbed by metal ions, which also has implications

for enzyme catalysis,127 iscussed further below.

Richards an d co-workers128J2Bave nicely combined

qua ntu m m echanical calculations for redox processes

with solvation free energy contrib utions to reproduce

an d make predictions of aqueous redox prop erties of

quinones.

Duffy et al.130 has combined ab initio calculations to

determine th e amide rotational barrier with solvation

free energy calculations to estimate the barrier of

isomerization in different solvents. Interestingly, here

is an increase in

AG*

f

2

kcal/mol in aqueous solution

compared to the gas phase, which the calculations

reproduce an d clarify. D ebolt and Kollmanl3l have

used a com bination of ab in itio calculations an d free

energy calculations to sim ulate the relative solvation

free energies of ground and excited states of

C=O

groups of formalde hyde and acetone in H20 , CH30H ,

and C Cb. Th e calculations are able to rationalize and

reproduce the blue shift of the n

- r*

transition in

CH30H and H2 0, but not the red shift in CC4. To

reproduce the latter likely requires tha t t he diffuseness

of th e excited-state charge distribution an d its greater

dispersion interaction (th an the ground state) w ith the

solvent be represented; the model used treats both

ground an d excited states as simple point charge models,

derived from fitting to the respective electrostatic

poten tials surrounding the molecule. A more general

model for including solvation in the study of excited

state phenom ena has been presented by Luzhkov and

Warshel.

132

Combining quantum mechanical calculations with

explicit solvation models has been implem ented in the

Warshel group both using ab i nit io p s e u d ~ p o t e n t i a l s l ~~

and semiempirical quantu m mechanical meth0ds.13~

Particularly, the wide range of applicability of the

methods of ref 134 is very impressive. It would be

interesting to compare th at approach on a similar set

of molecules, with AMSO L,% which uses a m ore

macroscopic solvation model.

5.

Protein Solvation

One can consider protein groups as a solvent; h us,

one has suggested that charges in protein are often

stabilized by helix dipoles. Aqvist et ~ 1 . l ~ ~sing free

energy calculations, have shown th at t he stab ilization

of charges in barnase a nd sulfate binding protein come

mainly from the groups in the first turn of the helix,

not from a macrodipole. They were also able to

sim ulate the actua l perturbation of th e PKa of H is-18

and sulfate binding constant in sulfate binding protein

is impressive agreem ent with experiment. Earlie r,

Dagg ett et al. qualitatively simulated th e electrostatic


11/23


effect of helices on charged groups using simpler m odels

to represent s01vent.l~~

The question whether protein cavities provide a

suitably attractive environment for water molecules has

been studied by Wade

e t

a1.137-138By mu tating water

+nothing in water

(AA =

6.3 kcal/mol), the calculations

have been able to nicely rationalize the presenc e of water

in one cavity

(AA

or disappearanc e - 6 kcal/mol) and

its absence in anoth er

(AA

or disappearan ce -6 kcal/

mol).

T he solvation of the [Fe4S4SCysI2-/%edox couple in

four different environments has been simulated by

Langen et

al.

and the available experimental data

reproduced.139Although som e of th e differences could

be rationalized by the presence of amide groups

stabilizing these a nions, th e crucial role of w ater

penetration in rationalizing the relative redox potential

of two struc turally similar protein systems was noted;

this is a beautiful example of sim ulations pointing to

something tha t is hard to analyze from the experimental

structure alone.

Chemical Reviews, 1993,

Vol.

93,No.

7

2405

B. Molecular Association

Free energy calculation have been applied in many

exciting examples to molecular association in solution.

T he basic equation describing this is given in scheme

10.

- AB

AGl

A + B

A

+

B B

AG2

Consider

A

as a host and B and B two guests for this

host. One can measure the free energy of association

of

B

and

B

to A using experimental methods and

m utate B into B free in solution and when bound t o

A

using theoretical methods. Since free energy is a

stat e function, the difference between the experimen-

tally measured a nd calculated free energies should be

equal (eq

11).

AAG

=

AG2 - AG,

=

AGbind- AGsolu (11)

exper imental) computat ional)

Molecular association is a balance between solute an d

solvent interactions; this is what h appens in molecular

association described by (10) and (11);by bringing

molecules together, we replace solvent-solute interac-

tions by solute-solute interactions. We calculate the

solvation difference between B and B (AG3) and the

host interaction free energy difference (in solution)

between B and B (AG4).

1

Small Organic

Hosts

Th e first application of eqs 10 and 11 to a complex

molecular association was the stu dy by L ybrand e t ~ 1 . l ~

on the h ost SC 24/4H+ with the two guests C1- and Br-.

T he calculations were successful in reproducing th e -

kcal/mol preference of the host for C1-, which was

impressive given the large charges involved. T he key

issue, which was appreciated before free energy cal-

culations, but which could not be calculated in a

qua ntitative way, was the balance between AGbbd and

AGsolvin determining the free energy of association of

guests to hosts. In the case of the Lybran d

e t al.

study,

AGbind was -7 kcal/mol and AG80~v as -4 kcal/mol,

so

the

AG,l,

modulated

AAG.

A more general picture, in qualitative agreement with

available experiments, emerged from the studies of

Grootenhuis,141who studied both dibenzo-18-crown-6

(DB186) and dibenzo-30-crown-10 (DB3010) with var-

ious cations Na+,

K+,

Rb +, Cs+. As is the case with

most ionophores, binding free energy is lowest at an

interm ediate p oint in th e alkali ion series, i.e. DB186

binds

K+

most tightly and DB3010 binds Rb+ most

tightly. Th is comes abou t because AGbind and AGsolv

are both in the order Li+

chemical review_ free energy calculations applications to chemical and biochemical.pdf

Documents