On the Use of Electrostatic Potential Derived Charges in
Molecular Mechanics Force Fields. The Relative Solvation
Free Energy of Cis- and Trans-N-Methyl-Acetamide
Piotr Cieplak* and Peter Kollmani
Department of Pharmaceutical Chemistry, University of California, Sun Francisco, San Francisco, CA 94143
Received 8 February 1991; accepted 1 July 1991
We have carried out free energy perturbation calculations on the relative solvation free energy of cis- and
trans-N-methyl-acetamide(NMA). Experimentally,the solvation free energy difference has been found to be
near zero. Using 6-31G* ab initio electrostatic potential derived charges for both the cis and trans
conformations, we calculate a solvation free energy difference of 0.1 & 0.1 kcal/mol. Using the 6-31G* charges
derived for the trans conformation for both the cis and trans models leads to a solvation free energy difference
of 0.9 0.1 kcal/mol, compared to the value of 2.2 kcal/mol determined for the OPLS model for trans-NMA.
*
INTRODUCTION
One of the most important issues in molecular mechanical, molecular dynamical, and Monte Carlo simulations of polar and ionic molecules is how to
represent the electrostatic energies of the system.
For full microscopic models, this often reduces to
what to use for the partial charges on all the atoms
of the molecule(s). There have been a number of
methods to derive such charges and these fall into
two classes-empirical, with the possible use of
electronegativity ' and by using quantum mechanical
calculations-through the use of Mulliken populations," electrostatic potential derived charges; and
distributed multipole analysis derived charges!
Jorgensen and co-workers, with their OPLS
model: have developed the most extensive and validated set of empirical charges. They have used both
Monte Carlo simulations of liquids and solutions and
a b initio calculations on intermolecular complexes
to derive and validate these models.
One of the critical assumptions of most molecular
mechanical models is that the partial atomic charges
are "two-body additive," that is, they are independent
of environment or conformation. Jorgensen and Gao6
sought to test the validity of this assumption by using
the charges derived for trans-N-methyl-acetaide
( N U ) to calculate the solvation free energy difference between trans- and cis-NMA. They calculated
a solvation free energy difference of 2.2 kcal/mol,
significantly above the inference from experiments
that this solvation free energy should be ca. 0. By
*Permanent address: Department of Chemistry, University of
Warsaw, Warsaw, Poland.
tAuthor to whom all correspondence should be addressed.
Journal of Computational Chemistry, Vol. 12, No. 10, 1232-1236 (1991)
0 1991 by John Wiley & Sons, Inc.
comparing the hydrogen-bond energies for water *
cis-NMA complexes with those found by a b initio
calculations, they concluded that the N-H
* OH2
hydrogen bond was relatively too strong in
trans-NMA and adjusted the charges to reproduce
the relative ab initio energy difference between
C=O
HOH and N-H ... OHzhydrogen bonds (ca.
2 kcal/mol).
We have long advocated the use of electrostatic
potential derived charges as being simple to derive,
transferable, and not subject to bias. We have further
validated their effectiveness in nucleic acid base
interactions7 and simple associations between
crown ethers and polar molecules.8 We thus wished
to assess how well electrostatic potential derived
charges could reproduce the relative solvation free
energy of trans- and cis-NMA. Our calculated relative solvation free energy when trans-NMA charges
were used for both cis and trans structures was 0.9
kcal/mol, significantly closer to experiment than
found with the original OPLS model (2.2 kcal/mol).
Furthermore, when using the trans electrostatic potential based charges for trans-NMA and cis charges
for the cis-NMA, we calculate a solvation free energy
difference of ca. 0.1 kcal/mol, in excellent agreement
with experiment?
METHOD
In our studies we performed a series of molecular
dynamics/ free energy perturbation simulations for
NMA in water solution using AMBER version 3A.l'
The atomic charges for cis and trans form of NMA
were obtained by fitting them to the molecular electrostatic potential." The structure of cis- and
trans-NMA were found by carrying out molecular
CCC 0192-8651/ 91/ 101232-05$04.OO
1233
SOLVATION FREE ENERGY OF CIS- AND TRANS-NMA
Table I. ESPOT derived
cis-N-methyl-acetamide conformation.
Atom
for trans- and
Table 11. Parameters used for nonbonded interaction
energy calculations."
cis
trans
Atomh
A
C
c1
- 0.4919
H 10,Hll,H 12
c2
03
0.1301
0.7562
- 0.6366
- 0.4603
0.3152
- 0.2922
0.1397
-0.3610
0.1102
0.7583
- 0.6655
- 0.5944
0.3421
0.0446
0.0484
O(SP3
CT(SP?
C(SP'1
N(SP')
OW
HW'
710,751.2
1,044,297.7
1,044,297.7
1,802,925.0
582,100.0
0.0
0.0
7532.0
653.03
676.01
676.01
968.25
595.0
0.0
0.0
21.75
N8
H15
c9
H16,H17,H18
"6-31G*basis set; using algorithm described in reference
11.
"See Figure 2 for atom names.
mechanics minimization using the parameters of the
Weiner et al. all atom force field.12 The electrostatic
potential around each molecule was calculated using
the Gaussian 80-UCSF program13 and a 6-31G* basis
set.14The charges found in our simulations are listed
in Table 1. For the nonbonded LennardJones intermolecular interaction between NMA and water molecules we used slightly modified OPLS parameters.
The modifications of A and C were based on requiring appropriate hydrogen-bond distances for
NMA-water proton-acceptor and proton-donor
type dimer interactions. The nonbonded parameters
used in our simulations are presented in Table I1 and
the results for two different NMA-water complexes
after energy minimization are presented in Table 111.
All the remaining molecular mechanical parameters
come from reference 11. We note that this nonbonded set is not necessarily our final set in the
derivation of a complete set of parameters for proteins and nucleic acids using 6-31G* electrostatic
potential derived m0de1s.l~
All the molecular dynamical simulations were carried out at T = 300 K and 1 atm of pressure in a
cubic box of 261 TIP3P16 water molecules. In all
simulations we used an 8 A cutoff for calculating
interaction energies and periodic boundary conditions. The time step of At = 0.0015 ps was used with
temperature coupling parameters 0.4 ps-' and pres-
Hd
HC'
"A = kcal/mol * Ale; C = kcal/mol . &.
"All other parameters for NMA from Weiner et al."; intramolecular 1-4 nonbonded (electrostatic and van der
Waals) energies are not scaled by 2, but left at their full
weight. TIP3P water parameters for OW and HW from
Jorgensen et al.16
'From TIP3P water model.Iti
dAmide hydrogen.
"Hydrogen attached to sp3 carbon.
sure coupling constant 0.2 ps-'. A 10 ps equilibration
was done for each simulation before mutating the
system. The SHAKE procedure was applied to constrain all bond lengths to their appropriate equilibrium values. For the 1-4 van der Waals and 1-4
electrostatic intramolecular interaction energies in
NMA, we use a scale factor equal to 1.
In order to determine the solvation free energy
differences between the trans and cis forms of NMA,
we gradually transformed one into the other during
the course of molecular dynamics simulations. We
mutated the H in the N-H group and its attached
dummy atoms into a full methyl group. At the same
time the methyl group was transformed into a hydrogen and three noninteracting dummy atoms. In
some simulations we changed only the charges (4)
on a given molecule from those which are characteristic of the trans isomer into those characteristic
of the cis isomer. To calculate free energy differences we applied windowing as well as the slow
growth a p p r 0 a ~ h .In
l ~ the first case the 21 windows
were chosen with 500 MD steps of equilibration followed by 500 steps of data collection. In some sim-
Table 111. H-bond energies and distances for trans-NMA-water complexes?
Complex
trans charges
NMA
water proton donor
> = 0 ... H,O
NMA + water proton acceptor
>N-H
..* OH,
cis charges
NMA + water proton donor
+
NMA
+ water proton acceptor
"Energies in kcal/mol.
"See Figure 2.
'Results from quantum mechanical calculations."
AE,,,
H-bond distances
- 9.1h
( - 7.3)'
- 6.5b
( - 5.4)'
ro
ro
rh-,,
r,
- 9.5b
( - 7.3)'
- 7.0h
( - 5.4)'
ro
rO
Th-1,
r,
1.81A
2.77 A
1.92 A
2.94 A
(,
=
=
=
=
()
= 1.78
= 2.75
(,
=
,)
,)
A
1.91 A
= 2.93 8,
1234
CIEPLAK AND KOLLMAN
Table IV. Calculated free energies for different NMA simulations:"
Reactant -+ product
Method"
Total time'
AQ
21 windows
trans (qtrans)d-+ cis (qtrans)
21 windows
cis (qtruns) -+ trans (qtrans)
21 windows
trans (qtruns) + cis (qtrans)
slow growth
trans (qtrans) -+ cis (qtrans)
slow growth
5.
cis (qtrans) + trans (qtrans)
11 windows
trans (qtrans) + trans (qcis)
6.
21 windows
cis (qtmns) -+ cis (qcis)
7.
11 windows
8.
cis (qtruns) + cis (qcis)
"Free energies in kcal/mol.
bSee Singh et al.I7for description of the difference between windows and slow growth.
Total simulation time, in the case of windows, and equal time was spent in equilibration and data collection.
dtruns (qtruns) means trans-NMA with charges derived from trans-NMA.
31.5 ps
31.5 ps
63.0 ps
63.0 ps
63.0 ps
16.5 ps
63.0 ps
16.5 ps
1.
2.
3.
4.
0.92 k 0.001
- 1.22 2 0.32
1.06 +- 0.06
0.81 ? 0.01
-0.77 ? 0.01
-1.17 t 0.01
-0.80 t 0.01
-0.83 ? 0.01
~~~
ulations we doubled the number of MD steps for
each window. In the case of slow growth, 42,000 MD
steps were performed to make the total perturbation
of one form of molecule into another. The results
and conditions of different simulations are summarized in Table IV.
RESULTS AND DISCUSSION
Table IV and Figure 1 summarize the key results. We
consider the last three simulations the most reliable
for the process N M A trans (qtrans) --.f NMA cis
(qtrans) so we have reported the AG for that in the
figure. This leads to a AG = 0.88 2 0.12 kcal/mol.
Including the first two simulations as well leads to
a AG = 0.96 t 0.15. Because such a mutation involves both electrostatic and van der Waals charges,
it requires longer and more independent simulations.
The two vertical processes in Figure 1 involve only
electrostatic perturbations and are expected to be
much less sensitive to simulation time/protocol,ls so
they were only calculated once. The free energy for
the bottom horizontal process was estimated by difference.
A reason that the calculations can be carried out
in the way described is the large (ca. 20 kcal/mol)
barrier between the cis and trans conformations.
Thus, the molecule does not undergo rotation around
the amide bond during the 10-100 ps of the simulation. The final structures were analyzed to insure
that no rotation has occurred.
In the previous study by Jorgensen and Gao,G they
noted that ab initio calculations suggested a significantly larger difference between C=O
HO and
N-H -.-OH2hydrogen-bond energies than found in
the original model. This was their basis for charge
adjustment. In Table I11 we report the N-H
OH2
and C=O
HO hydrogen bond energies and distances for trans-NMA and show stereoviews of the
molecular mechanics optimized structures in Figure
1. As one can see, the molecular mechanical results
are consistent with a significant preference for C=O
HO over N-H ... OH2hydrogen bonding, as found
at the ab initio level. We did not study cis-NMA in
- 1 -
1 . -
0.88 It 0.12
NMA - trans (Sums)
b
NMA - cis (qtrans)
-0.80f 0.01
-1.17 fO.O1
7
V
NMA - trans (qcis)
b
NMA - cis (qcis)
( 1.25)
Figure 1. Thermodynamic cycle for cis- -+ trans-NMA with cis and trans
charges. The top horizontal process was the average over runs 3, 4, and 5
(Table IV), with the average deviation noted. The two vertical processes were
for runs 6 and 7, respectively.The bottom horizontal process was the amount
required to make the sum of free energies around the cycle sum to zero.
See Table IV, footnote d for notation.
1235
SOLVATION FREE ENERGY OF CIS- AND TRANS-NMA
r2
r2
i
,,
i
”
‘
8
8‘
8
815
(b)
Figure 2. Stereoviews of (a) N-H
bonded structure.
... 0 hydrogen bonded structure and (b) 0
that way because the lowest energy water complex
for cis-NMA involves a single water in both C = O
HO and N-H
OH2 hydrogen bonds.l9 Our H-bond
energies are larger than found in the QM calculations
partially because TIP3P16 involves a “polarized’
water, with enhanced attraction even over the 6-31G
( d ) ab initio quantum mechanical model.
The calculations presented here demonstrate the
usefulness of electrostatic potential derived charges
in molecular mechanics/ dynamicdfree energy studies. Without any adjusting of charges, the difference
in the trans
cis solvation free energy of NMA is
calculated to be ca. 0.1 kcal/mol, in excellent agreement with experiment. The use of a 6-31G*ab hzitio
basis set for these calculations is an appropriate
choice because such a basis set overestimates polarity (dipole moments) in molecules by ca. 10-20%,
close to the enhancement of the TIP3P water dipole
moment over the gas phase value. As noted before,
it is most important for simulations that the charges
of solute and solvent be balanced and the TIPSP
solvent/6-31G* electrostatic potential derived
charges for the solute meets this criterion.
One of the fundamental assumptions in most molecular mechanical models is the conformational independence of the electrostatic charges. This
assumption cannot be correct, but it is important to
-
s . 1
H-O
hydrogen
assess the magnitude of the error made by this assumption. Comparing the difference in free energy
of solvation of trans N M A and cis-NMA using
trans-NMA charges lets one assess the errors inherent in assuming a single set of charges and that error
turns out to be ca. 0.9 kcal/mol. This is comparable
to the “best case” error (* 1 kcal/mol) for free energy calculations on macromolecular systems. Since
electronic structure differences between trans- and
cis-NMA are probably larger than found between
different single bond rotamers, a single set of electrostatic charges should be at least adequate for
many sets of free energy calculations. This has been
nicely demonstrated by Williarns,2O who showed that
a single set of charges for 13 different conformations
of alanyl dipeptide led to only an average 7.4%error
in the electrostatic potential values. This level of
percent difference is reflected in the difference in
hydrogen bond energies for trans-NMA using cis and
trans charges (Table 111), the values differing by ca.
0.4-0.5 kcal/mol, or 5-7% of the total hydrogen bond
energy. Obvious exceptions are free energy changes
between ground and excited states21or different rotamers involving charges in multiple bond conformations (e.g., trans or cis-NMA vs. N M A with
Cp(CNC0) = 90’).
This study demonstrates some significant advan-
C I E P M AND KOLLMAN
1236
tages of electrostatic potential (ESPOT) derived
charges for molecular mechanics over empirical
ones. What is the relation of ESPOT derived charges
to other quantum mechanical charges? The superiority of ESPOT charges over Mulliken population
charges is quite clear.“J1:22Since ESPOT charges
seem to reproduce the multipole moments of the
molecule well, ESPOT derived charges should be
very similar to those derived by “Multipole Con~ t r a i n t . What
” ~ ~ is the relationship of ESPOT charges
to distributed multipole analysis (DMA) charges?3
The method of deriving ESPOT charges is to ensure
a model that reproduces the electrostatic properties
outside the van der Waals envelope of the molecule,
the region most relevant for molecular interactions.
On the other hand, the DMA analysis focuses on the
electron distribution and derives point multipoles
(charges, dipoles, and quadrupoles) at various centers. The DMA analysis is in principle, more “transferable,” but requires a significantly more complex
model. Is the extra complexity worth it for molecular
mechanics/dynamics, given that the intermolecular
interactions are also sensitive to the dispersion and
repulsion parameters of the model, not to mention
the charge transfer and polarization effects usually
left out of such models? In our opinion, ESPOT
charges at the monopole level are an excellent current compromise between computational efficiency,
“balance of errors” and accuracy. The recent results
of William@’ and those presented here delineate its
accuracy and limitations. It seems to us that the
burden of proof for the use of the DMA approach is
on its proponents to show, for typical H-bonded complexes such as in Table 111, that such a model leads
to sufficiently more accurate intermolecular energies to be worth the computational cost.
Although 0.9 kcal/mol is a small error for many
cases, it does correspond to a factor of ca. 5 in equilibrium constant. Thus, one seeks to improve the
charge models for molecular simulations. The accuracy with which the AGsOlvis calculated using cis
charges for cis-NMA and trans charges for
trans-NMA suggests that, if necessary, these can be
used directly, albeit the book-keeping in simulation
programs would be much increased. An alternative,
more elegant approach will be to see if nonadditive
polarization models can represent conformational
dependent electrostatic potentials as well as different intermolecular electrostatic fields. Such an analysis is underway in our lab?4
We are glad to acknowledge research support of the
NSF (CHE-85-10066) and NIH (GM-29072). P.C. thanks the
Polish Academy of Science for partial support through
project CPBP 01.12. Some of these calculations were carried out at the San Diego and Pittsburgh Supercomputer
Centers through supercomputer support provided to
P.A.K. We also acknowledge the facilities of the UCSF
Computer Graphics Laboratory (supported by the NIH,
RR01081 to R. Langridge). Useful comments by Allison
Howard are acknowledged.
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