67. 1.7.3. Force-field parametrization using tnostly expérimental data. 70. 1.7.4.
..... C'est pourquoi l'application d'une force motrice artificielle extrêmement forte ...
Research Collection
Doctoral Thesis
Molecular dynamics simulations using empirical force fields principles and applications to selected systems of chemical and biochemical interest Author(s): Hünenberger, Philippe Henry Publication Date: 1997 Permanent Link: https://doi.org/10.3929/ethz-a-001730163
Rights / License: In Copyright - Non-Commercial Use Permitted
This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use.
ETH Library
Diss. ETH No. 12052
Molecular
dynamics simulations using empirical Principles and applications to
force fields:
selected Systems of chemical and biochemical interest
A dissertation submitted to the SWISS FEDERAL INSTITUTE OF TECHNOLOGY
for the
degree
of
Doctor of Natural Sciences
presented by Philippe Henry Hiinenberger Dipl. Chem. UNIL boni 23.5.1970 citizen of Basel BS and Zurich ZH
accepted
on
Prof. Dr. W.F.
the recommendation of van
Gunsteren, examiner
Prof. Dr. R.R. Emst, co-examiner
1997
To my
parents,
my two
grandmothers,
and to Valérie
CONTENTS
Résumé
1
Summary
5
Préface
8
CHAPTER 1:
Empirical
classical force fields
for molecular Systems 1.1.
12
Summary
12
1.2. Introduction
12
1.3. Choice of the 1.3.1. 1.3.2. 1.3.3. 1.3.4. 1.3.5.
1.4.
explicit degrees of freedom Gas-phase force fields Condensed-phase force fields Mean-solvent force fields Low-resolution force fields Hybrid force fields
of the model
17 19 20 21 21
21
Assumptions underlying empirical classical interaction functions 1.4.1. Implicit degrees offreedom and the assumption ofweak corrélation
22
1.4.2. Energy tenus and the assumption
27
oftransferability 1.4.3. Coordinate redundancy and assumption oftransferability 1.4.4. Choices mode in the averaging processes 1.5. General characteristics of the
empirical interaction
1.5.1. Interaction function parameters and molecular
function
topology
22
30 31
32 32
1.5.2. Atom types and combination rules
33
1.5.3. Expression for the classical Hamiltonian
34
1.6. Interaction function ternis used in current force fields 1.6.1.
Bond-stretching term 1.6.1.1. Functionalforms
37 37
1.6.1.2. Combination rules
1.6.2.
Bond-angle bending
35
40
term
40
1.6.2.1. Functional forms
40
1.6.2.2. Combination rules
41
Contents
n
1.6.3. Torsional dihedral
1.6.4.
angle
41
term
1.6.3.1. Functional forms
41
1.6.3.2. Combination rules
44
Out-of-plane
44
coordinate distortion term
1.6.4.1. Functional forms
44
1.6.4.2. Combination rules
45
7.6.5. Valence coordinates
45
cross terms
1.6.5.1. Functional forms 1.6.6.
45
der Waals interaction
48
1.6.6.1. Functional forms
48
1.6.6.2. Combination rules
53
van
1.6.7. Electrostatic interaction
57
1.6.7.1. Functional forms 1.6.7.2. Combination rules
57 63
1.6.8. Coupling between covalent coordinates and electrostatic interactions 63 1.6.9.
Hydrogen-bonding
63
term
1.7. Force fleld
parametrization procédures basicproblem Source of data for force-field parametrization or validation
1.7.1. The 1.7.2.
1.7.3. Force-field parametrization using tnostly expérimental data 1.7.4. Systematic parametrization for simple condensed-phase Systems 1.7.4.1. By trial and 1.7.4.2.
Using sensitivity analysis
Systematic parametrization using resultsfrom calculations in
65
67 70 71
71
error
71
1.7.4.3. Using the weak-coupling method /. 7.4.4. Using a search method in parameter space 1.7.4.5. Using the perturbationformula 1.7.5.
65
72 72
73
ab initio
73
vacuum
7.7.6. Technical
difficultés in the calibration offorce fields 1.7.6.1. Parameter interdependence 1.7.6.2. Parameter dependence on degrees offreedom (D), functional form (F), combination rules (C) and approximations (A) 1.7.6.3. Parameter dependence on the molécule training set and
75 75
76
calibration observables
77
1.7.6.4.
77
1.7.6.5. 1.7.6.6. 1.7.6.7.
Non-convergence of important observables Existence of conflicting requirements Force-fieldmixingproblems Validation ofaforcefield and comparisonofforcefields
78 78 79
1.8. Conclusion
79
1.9. Références
80
Contents
m
Alternative schemes for the inclusion of a reaction-field
CHAPTER 2:
correction into molecular Influence
on
and dielectric 2.1.
dynamics simulations: energetic, structural
the simulated
87
properties of liquid water
87
Summary
88
2.2. Introduction
90
23. Theory
straight cutofftruncation application ofa reaction-field correction
2.3.1.
Schemesfor the Coulomb interaction with
2.3.2.
Schemesfor
2.3.3.
Comparison ofthe différent schemesfor dipoles
to
the
90
93
the truncation at
a
distance close
cutoff
101
2.3.4. Ewald summation scheme
103
to
the
2.4. Molecular model and
computational procédures
104
2.4.1. Simulation setup
104
Analysis ofthe energetic properties 2.4.3. Analysis ofthe diffusion properties 2.4.4. Analysis ofthe structural properties 2.4.5. Analysis ofthe dielectric properties
105
2.4.2.
2.5.2.
107 108
110
2.5. Resuite 2.5.1.
106
Thermodynamic and energetic properties Diffusion properties
110 114
2.5.3. Structural properties
116
2.5.4. Dielectric properties
121
2.6. Conclusion
126
2.7. Références
128
potential and field in a spherical cavity permittivity tensor
APPENDK A:
Reaction
APPENDK B:
Calcuiation of the dielectric
based
CHAPTER 3:
on
the
dipole
moment fluctuations
132
137
Hydrogen-bonded diastereotopic interactions in a model complex
Contents
145
rv
3.1. Summary
145
3.2. Introduction
146
3.3. Molecular model and
computational procédure
148
3.4. Résulte 3.4.1.
149 149
Population analysis
Thermodynamic properties 3.4.3. Sensitivity ofthe thermodynamic properties
152
3.4.2.
3.4.4. 3.4.5.
to
selected
model parameters
157
Lifeûmes ofthe complex and hydrogen bonds Hydrogen-bonding pattems
160 163
3.5. Discussion
165
3.6. Conclusion
167
3.7. Références
168
APPENDK C:
APPENDK E:
Thermodynamic extrapolations using the perturbation formula Thermodynamic extrapolations using a Taylor expansion The homogeneity function
CHAPTER 4:
Fluctuation and cross-corrélation
APPENDK D:
protein
analysis
170 172 174
of
motions observed in nanosecond
molecular
dynamics
simulations
175
4.1. Summary
175
4.2. Introduction
175
43. Methods
178
4.3.1 Molecular model and computational procédure 4.3.2. Analysis procédure
4.4. Résulte and discussion 4.4.1. 4.4.2. 4.4.3. 4.4.4.
178 179 180
Stability ofthe simulations Simulated B-factors calculated using averaging Windows of différent lengths Corrélation between simulated and expérimental B-factors over différent averaging Windows Build-up time offluctuations and cross-correlations Contents
181
183 187 188
V
4.4.5. Cross-correlotions
190
4.4.6. Influence
192
ofthe fitting procédure
4.5. Conclusion
194
4.6. Références
195
CHAPTER 5:
Computational approaches to study protein unfolding: lysozyme as a case study
Hen egg white
5.1.
197
197
Summary
197
5.2. Introduction
Study of protein folding and unfolding by computer simulation 5.2.2. Expérimental data on the folding and unfolding of hen egg white lysozyme 5.2.7.
5.3. Methods
197 199
200
5.3.1. Molecular model and
200
5.3.2.
200
5.3.3. 5.3.4.
computational procédures Température induced unfolding: T-run Pressure induced unfolding: P-run Constant radial force induced unfolding: F-run
5.3.5. Kinetic energy
gradient driven unfolding:
K-run
5.4. Resuite 5.4.1.
5.5. Discussion
5.5.3. 5.5.4. 5.5.5.
202
202
T-run
unfolding: P-run 5.4.3. Constant radial force induced unfolding: F-run 5.4.4. Kinetic energy gradient driven unfolding: K-run 5.4.5 Peptide amide hydrogen bonding to water
5.5.2.
201
202
Température induced unfolding:
5.4.2. Pressure induced
5.5.1.
201
209 210 214
215 217
Methodology: température induced unfolding Methodology: pressure induced unfolding Methodology: artificial driving forces Peptide amide proton protection Models forfolding and unfolding
217
217 218 219 220
5.6. Conclusion
221
5.7. Références
222
APPENDK F:
Algorithm for driven
the kinetic energy
unfolding
gradient 225
Contents
VI
APPENDK G:
Définition of the uniform déformation rate vectors
APPENDK H:
Conservation of the overall kinetic energy in the
kinetic energy
gradient driven unfolding
229
231
Outlook
232
Acknowledgements
234
Curriculum vitae
235
Contents
1
Résumé Ces dernières décennies ont vu,
ordinateurs, méthodes
une
avec
l'accroissement continuel de la
augmentation rapide du nombre,
informatiques utilisées
en
chimie
des
performances
théorique.
Dans
et
puissance
des
précision
des
de la
méthodes, les champs de
ces
classiques empiriques utilisés en modélisation moléculaire occupent une place unique, car ils permettent actuellement de générer jusqu'à 10*-107 configurations (en termes force
de
dizaines de
dynamique, quelques
106
nanosecondes) de systèmes comprenant jusqu'à 105-
thermodynamiques
atomes. Le calcul d'un certain nombre d'observables
mécanique statistique est ainsi
biologiques
de macromolécules
L'application
de
ces
champs
méthode permettant de traiter extrêmement solution
en
portée,
grand
et
de membranes
ou
hors
problèmes chimiques
de macromolécules
équilibre (Chapitre 5). aux
biologiques
Dans les
perdre
de
vue
point
de
rapport
Le
aux
possibilités
et
Chapitre
méthodes basées Sont
1 est
également discutées
qui déterminent
introduction
champs
rapport
celles des
en
l'équilibre
solution à
une
attention
ou
qui
précision
de la
de la fonction
l'espace des conformations,
aux
employées.
aux
champs
de force
empiriques.
à d'autres modèles
Les
théoriques.
du système et des observables étudiées,
devrait être choisi pour résoudre
champ
nous
résultats des simulations et par
comparées
caractéristiques
l'utilisation d'un
l'omission de certains atomes
générale
de force sont
le type de modèle
spécifique. Quand
par
limitations des méthodes
une
sur ces
critique
vue
biochimiques
d'équilibres
que la fiabilité d'études utilisant des
d'interaction utilisée et du domaine échantillonné dans un
en
et
et
exemples présentés,
champs de force classiques dépend essentiellement
adopté
liquides, exemple.
développements méthodologiques ainsi qu'à une analyse
détaillée des résultats simulés. Sans
avons
par
constitue à l'heure actuelle la seule
empiriques nombre de
lipidiques,
importants: description de liquides polaires (Chapitre 2)
particulière est portée
à l'aide de la
de même que que la simulation de
solution
de force un
(Chapitre 3), dynamique
(Chapitre 4) toute
à notre
de force
classique
un
semble le choix
problème
approprié,
groupes d'atomes dans la fonction d'interaction
(p.ex.
solvant, chaînes latérales de protéines...) permet de réduire le temps de calcul. L'adjonction de certains
degrés
de liberté
permettre l'étude des
d'interpréter élémentaires, l'interaction
quantiques (p.ex. protons, électrons...) peut quant
propriétés
acide-base
les fonctions d'interaction nous
ou
classiques
pouvons identifier les
comme une
somme
des réactions
conditions qui doivent être satisfaites pour
terme de
en
hypothèses qui
de termes
analytiques.
qu'un champ
Résumé
(bio-)chimiques.
principes physiques
mènent à Sont
à elle
En essayant
une
description
également
de force donne
de
discutées les
une
description
2
thermodynamique
et
dynamique
énuméré les différents choix
correcte
possibles
systèmes
des
moléculaires. Enfin,
après
avoir
dans le nombre, le type et la forme fonctionnelle des
utilisés, ainsi que les règles de combinaison employées dans les champs de force
termes
d'usage
courant,
nous avons
abordé le difficile
problème de l'optimisation des paramètres
partir d'informations expérimentales.
à
problème
Le
du traitement des interactions
abordé dans le Chapitre 2. La précision
est
globale
l'approximation la plus grossière qui entre dans (typiquement l'eau) interactions et
la
des macromolécules
ou
électrostatiques
précision
des
courant
sa
souvent un
basée
une troncature
périodiques
aux
rôle crucial dans la fiabilité
limites de l'échantillon
sur
des groupes de
sur
propriétés
étudie, il
charges,
ou
d'usage
(iii)
liquide
et
comparées
sur
en
la théorie du
sommes
sont
appliquées
détail. Les résultats montrent que
champ de réaction,
interatomiques.
Dans le second cas, les
celles calculées
en
à des forces
point
de
utilisant des
propriétés calculées
sommes
de réseau. De
plus,
électrostatiques qui s'annulent à la distance vue
une troncature
été étudiées par simulation de
reportés
dans le
complexation
Chapitre
des deux
3.
présentons
deux
systèmes dans
un
Expérimentalement,
paires diastéréomériques
modèle
accord
avec
méthode conduit
sur
les
sommes
de réseau.
de complexes de deux molécules
dynamique moléculaire.
des simulations de
des
sur
des distances
cyclohexanediamine) liées par liaisons hydrogène
à 298K, montrent des différences inattendues Nous
lorsqu'on
de troncature et est moins coûteuse
thermodynamique et la dynamique de formation
organiques simples (cyclopentanediol
sur
dans des
basée
sont en excellent
cette dernière
du temps de calcul que les méthodes basées
et
de réseau,
l'inclusion de corrections
groupes de charges donne de moins bons résultats qu'une troncature basée
ont
est
simulées. Ces artefacts
en utilisant des
champ de réaction. Ces trois méthodes
la théorie du
inclut une correction basée
La
polaires
simulés, le traitement des
de tronquer ces interactions à une distance donnée. Cependant, la troncature directe
simulations d'eau
du
liquides polaires
calculées. Pour restreindre le temps de calcul et permettre
peuvent être réduits p.ex. par (i) le calcul de l'interaction
(ii)
les
de force est limitée par
définition. Lorsque des solvants
donne lieu à de graves artefacts dans beaucoup de
basées
champ
d'un
solution sont
en
longue portée joue
à
propriétés
l'utilisation de conditions
électrostatiques dans
Les résultats de cette étude sont
paramètres thermodynamiques
de
de molécules, mesurés dans le benzène
qui appellent
dynamique
simplifié
les
une
moléculaire de
de solvant. La
justification théorique.
longue durée (0. lus)
longue durée de
ces
des
simulations
permet d'étudier de manière exhaustive les espèces présentes à l'équilibre, rendant possible un
calcul
d'énergie
libre par comptage direct. Les résultats
Résumé
sont en
bon accord
avec
les
3
valeurs
expérimentales
hypothèses
pour l'une des
inhérentes
au
paramètres du modèle
modèle lui-même. Une analyse
sur
hypothèses sous-jacentes au modèle entre résultats
théoriques
modélisation
et
lui-même doivent être mises
inadéquate des liaisons hydrogène
ou
possibilité
a aucune
conséquent,
Par
en
aux
de l'influence des
les
doute. Ce désaccord
être dû à des interactions
d'aggrégats de plus de deux
désaccord pour
en
du modèle soit
systématique
ajustement des paramètres.
expérimentaux peut
à la formation
(benzène),
paramètres
indique qu'il n'y
les résultats simulés
d'améliorer l'énantiosélectivité par
le solvant
paires diastéréomériques, mais
Ceci peut être dû soit à l'inexactitude des
paire.
l'autre
spécifiques
avec
molécules de soluté, à
à des effets de redistribution de
une
charge
électronique. Le
problème
l'équilibre
en
de la convergence d'observables dans des simulations de
solution
est
abordé dans le
Chapitre
moléculaire de deux
protéines, BPTI et HEWL,
position atomique
et de
fluctuations sont
souvent
calculées
car
liaison
avec sur
un
avec un
extrapolation ait un
sens:
substrat). L'hypothèse
sur
le
système
l'examen de l'évolution
clairement que si la condition (i)
(ii)
n'est pas
remplie à l'échelle
facteurs B simulés et de temps et
est
Les
facteurs
se
déroulent
représentatives
produiraient sur des temps plus longs
soit
en
qu'une
équilibre durant la simulation
convergé sur
temporelle
durant la
période
et
ou
telle
(ii)
que
de simulation.
des fenêtres de temps de différentes
des
propriétés mentionnées,
montre
satisfaite pour les deux protéines étudiées, la condition
de temps de la nanoseconde. De
expérimentaux
techniques
se
trajectoires, basée
deux
aux
est faite que les corrélations de
substrat. Deux conditions sont nécessaires pour
les fluctuations et leurs corrélations aient
et
importants qui
la courte échelle de temps des simulations sont
(i) que
L'analyse systématique de
atomiques.
beaucoup plus longues que la nanoseconde (p.ex.
de mouvements de plus grande amplitude qui lors de la liaison
à
dynamique
cristallographiques. Les corrélations croisées sont utilisées
dans les protéines à des échelles de temps
fluctuations observées
protéines
de fluctuations de
en termes
elles peuvent formellement être reliées
indirecte de certains processus fonctionnels
comme mesure
longueurs
étudiées
corrélations croisées entre les mouvements
B déterminés à partir de données
repliement,
sont
4. Des simulations de
est douteuse
en
plus, la comparaison
des
raison des environnements, échelles
de détermination très différents
impliqués
dans
l'expérience
et
la
simulation.
Au
appliquées
Chapitre 5, sur
nous
étudions l'influence de différents types de
le processus de relaxation observé lors de simulations de
solution hors-équilibre. Dans la communauté scientifique,
Résumé
on
observe
un
perturbations
protéines
en
effort considérable
4
pour tenter d'élucider la manière dont une
structure tridimensionnelle se
processus
résolution
produit.
nanoseconde
protéine
nous
déploiement
une
explicite),
se
replie
seul le processus de
chemins de
Simuler le
d'une force motrice artificielle
dépendra alors
protéine
et
de celles de la force motrice choisie. Pour aborder
une
déploiement
et
comparoons des simulations de
de la
méthode basée
dynamique :
de mécanisme de
de la
d'une force radiale
redistribution des vitesses
atomiques.
Les et
repliement est plutôt douteuse. Malgré cela,
les résultats simulés et certaines données
protons amidiques, capacités
ce
moléculaire du
augmentation
pression (lOkbar), application sur une
simultanément
observés sont fortement dépendants de la force motrice choisie
en termes
comparaison entre
une
repliement
le chemin de relaxation
de HEWL utilisant différentes forces motrices
interprétation
au
être
et
présentons
ce
déploiement peut
déploiement d'une protéine en
pourquoi l'application
en
de force de
nécessaire,
de la
à tous les atomes, et
la
pratiques, lorsqu'on utilise champ
repliement.
température (500K), augmentation
leur
donnée d'acides aminés
quel(s) chemin(s) dynamiques)
selon
accélération d'un facteur 106-10' par rapport
in vitro. C'est
caractéristiques
problème,
finales du
représente
extrêmement forte est des
solvant
séquence et
L'hypothèse est alors faite que les étapes initiales du déploiement peuvent
comparées aux étapes d'une
Pour des raisons
atomique (incluant un
être étudié.
une
unique,
expérimentales (échange de
calorifiques, compressibilités) indique
que
ces
simulations
peuvent aider à comprendre la stabilité des protéines ainsi que la stabilité relative des éléments de structure secondaire
qui
les constituent.
Résumé
5
Summary
a
With the continuing increase of the power of computers, the past décades hâve seen rapid increase in the number, performance and accuracy of theoretical computational
mefhods in
chemistry. Empirical
unique place
among thèse since
classical force fields for molecular simulation occupy
they
can
currently
and generate up to lf/-107
configurations of thèse Systems (tenss
of dynamics). This
that
using lipid
means
only available biochemical solution under
example,
methods in
problems,
(Chapter
are
the
as
dynamics
simulated results.
Keeping
dépend crucially
the outcome of the
are
description
of
force fields
can
are
extremely important
be computed
chemical and
equilibrium (Chapter 4)
The main focus in the and
a
or
therefore the
polar liquids (Chapter 2), equilibrium
of solvated biomolecules at
in mind that the
on
Empirical
address many
methodological developments
in or
applications reported
detailed
analysis ofthe
reliability of studies using classical force
fields
the accuracy of the interaction function and the extent of
conformational space that has been to
to
non-equilibrium conditions (Chapter 5).
in the présent work is towards
will
thermodynamic observables
within reach.
position
a
such
3), the
number of
of nanoseconds in terms
and that the simulation of solutions, solvated biomolecules
statistical mechanics,
membranes, for
a
a
handle Systems of up to ÎO'-IO6 atoms
sampled,
a
critical
point of view
adopted with respect
is
simulations, and the power and limitations of the methods employed
discussed. Fn
Chapter 1,
Empirical
gênerai
a
introduction
force-field methods
are
empirical
to
compared
classical force fields is
given.
with other theoretical models, and the
characteristics of the System and observable(s) of interest that détermine which type of model should be chosen in order to solve
classical force fields
seem
the
a
given problem
appropriate choice,
are
discussed. When
solvent, protein sidechains) from the interaction function may lead
computational efficiency. Alternatively,
degrees
of freedom
properties
or
the inclusion of selected
(e.g. protons, électrons)
may allow for
(bio-)chemical reactions. By trying
classical interaction functions in terms of first
analytical description be satisfied for
a
as a sum
force field to
molecular Systems
are
give
form and combination rules
a correct
the
quantum-mechanical
description physical
assumptions
(building blocks)
(e.g.
to an increased
of acid-base basis of the
that lead to
an
and the conditions to
thermodynamic and dynamical description
Finally, possible
defining
a
understand the
principles,
of force field terms
identified.
to
empirical
the omission of selected atoms
of
choices in the number, type, functional
the terms found in
currently
used force fields
are
listed, and the difficult problem of parameter calibration from expérimental data is discussed.
Summary
6
Chapter 2,
In
the
problem of the treatment of electrostatic
is addressed. The overall accuracy of that enters into its définition.
(bio-)molecules
the crudest
sol vents
computational effort to truncate
and allow for the
of
use
thèse interactions at
long-range
(typically water)
or
plays
a
To limit the
properties.
it is
common
convenient (cutoff) distance.
Straight
periodic boundary conditions,
some
polar liquids
approximation
electrostatic interactions
reliability and accuracy of the simulated
crucial rôle in the
practice
simulating polar
When
in solution, the treatment of
interactions in
by
force field is limited
a
truncation, however, results in serious artifacts in many simulated properties. Thèse artifacts may be reduced e.g. by (i) the group truncation scheme,
methods
applied
are
show that when
instead of the
using are
a
a
in simulations of
sum
method and those calculated
complexes
cyclohexanediamine, thermodynamic
of
Systems in
simulations,
equilibrium possible.
are
studied
parameters for
using
an
a
a
simplified
be studied
The results
are
nearly identical results can
organic
binding
properties
lattice
are
molécules,
solvent model of the and
are
reported.
inaccuracy
dynamics a
sum
experiment
with
of hydrogen-
and
no
and
for
one
of the
the
dependence is
problem
or
electronic
length of
more
by
direct
at
counting
(ii)
with the
experiment.
assumptions
of simulation results to
improve
the
the model
spécifie solute-solvent
that two soluté molécules,
charge redistribution effects.
of the convergence of observables in
simulations of solvated proteins is addressed. Nanosecond
Summary
a
the
species présent
assumptions underlying
(benzène) interactions, formation of clusters involving
as
for
diastereomeric pair,
found
itself should be questioned. Possible causes for discrepancies may be
bonds
or
the
simulations ofthe
disagreement
possibility
appealing
Due to the
of the model parameters
enantioselectivity by parameter tuning. Therefore,
the
methods.
of molécules,
are
free energy calculation
systematic analysis
improper modelling of hydrogen
pairs
which
is obtained, the
obtained for the other pair, in
parameters is performed
Chapter 4,
calculated
cyclopentanediol
(0.1 us) molecular-dynamics
timescale
good agreement
be due to either (i) the
model
In
in détails. The results
reaction-field correction scheme
a
of the two diastereomeric
exhaustively,
in
charge-
by molecular dynamics simulations. Experimentally,
well-converged picture
can
of a
atomic truncation scheme
the simulated
using
display unexpected différences,
Long
inhérent to the model itself. A on
compared
and
computationally cheaper than
simple
two
theoretical rationalization.
but
use
addition, the latter scheme leads to electrostatic forces which
measured in benzène at 298K,
This
method, (ii) the
Chapter 3, the equilibrium thermodynamics and dynamics of formation
bonded
is
water
sum
reaction-field correction. Thèse three
(usual) charge-group truncation scheme,
lattice
in excellent agreement. In
two
liquid
a
reaction-field correction is included
vanish at the cutoff distance and is In
of a lattice
use
the inclusion of
(iii)
or
molecular-dynamics
equilibrium simulations
7
proteins, BPTI and HEWL, are
of two
studied in terms of atomic positional fluctuations and
cross-correlarions of atomic motions. The former
formally an
indirect
for many
measure
longer than
timescales much
functionally important
nanoseconds (e.g.
of the
larger amplitude
meaningful
that (i) the System is in
fluctuations and corrélations
are
trajectories using time
development of the above properties
In
Chapter 5,
the
comparison
experiment and
which dynamical
expérimental
be studied. The
pathway(s) this used
of an
pathway
will
selected
driving
unique
For
practical
unfolding
of a factor 106-109 with respect to
extremely strong
artificial
dépend simultaneously on force. To address this
to
elucidate the way
reasons, when
the
driving
of
a
protein
within
protein folding
a
method based
on
velocity scaling.
questionable. data
(peptide
On the other hand,
amide proton
simulations may provide
their
and their
constituting
The
are
presented
protein
and
unfolding pathways
interprétation
comparisons
exchange,
insight
heat
into the
in terms of
and those of the
are
folding
Summary
of
proteins
to
increased ail atoms,
found to be
strongly
mechanism is rather
of the simulation results with
stability
Thus, the
simulations of the
compared:
capacities, compressibilities)
structural éléments.
nanosecond
in vitro.
température (500K), increased pressure (lOkbar) application of a radial force
driving-force dépendent,
may with
force is required, and the relaxation
the characteristics of the
forces
along
compared a
a
force field
a
unfolding process
problem, molecular-dynamics
unfolding of HEWL using différent driving
the type of
three-dimensional structure and
then made that the onset of unfolding may be the
on
solvated protein is addressed.
community in trying
occurs.
process
folding. Simulating
speedup
a
a
(including explicit solvent), solely
assumption is
the latest stages of
and
proteins
B-factors is
the simulation.
simulations of
non-equilibrium
in
at atomic resolution is
application
and by monitoring although condition (i) is
problem of the dependence of the relaxation pathway
the
séquence of amino acids folds into
a
(ii)
period. Systematic
lengths
that
of simulated and
There is considérable interest in the scientific
represents
the simulation and
or
be
to
due to the very différent environments, timescales and détermination
involved in the
perturbation applied given
clearly
are
timescale
extrapolation
an
as on
assumption
not satisfied on the nanosecond timescale for the two
study. Additionally,
techniques
shows
The
longer
within the simulation
Windows of différent
of the two
the time
questionable
occur on a
equilibrium during
converged
analysis
fulfilled, condition (ii) is
binding).
can
used
proteins
in
occur
they
are
the short simulation timescale
Two necessary conditions for such
binding.
under
substrate
on
motions that would
upon substrate are
B-factors. The latter
processes that
folding,
is made there that fluctuation corrélations observed
représentative
often monitored because
are
expérimental X-ray crystallographic
be related to
expérimental
indicate that thèse
and the relative
stability
of
8
Préface The to a
primary goal
of theoretical
approaches
chemical and biochemical problems is
to
rationalize the behaviour of macroscopic Systems in terms of a microscopic model. Once
model has allowed for
one
may
assume
a
successful
that it has
and
description of a class
of properties,
captured the essential underlying physics governing them,
the model may be further used for
possible microscopic
A
understanding
prédiction
of related
model (Model I) for
properties
macroscopic
and
in related Systems.
molecular Systems is
an
infinité statistical-mechanical ensemble of Systems in which nuclei and électrons interact
according électrons thèse
quantum-mechanical
to an exact
can
particles
appear
are
seldomly available
to say that Model I would allow for a correct
and biochemical an exact
Hamiltonian. Of course, both nuclei and
be described in terms of smaller
problems. Unfortunately,
description
of
particles,
in the
but the
description
the
use
macroscopic phenomena
énergies
day-to-day world.
a
fundamental model to obtain
impossible.
First of ail, exact analytical solutions for macroscopic properties based
virtually further
never
cases,
available. Statistical mechanics leads
approximations
mechanics
can
équations
heavily
on
computing
roughly an
are
describe
made
analytically solely
is
on
solutions
particle Systems (H, He+,...).
numerically. Theoretical
a
solve ail
the method that is the most suitable to One option is to insist
computational costs
on a
to any
can
a
are
when
chemistry such
In ail other
study is
the available
currently
possible problems, but that a
by
increases
still far
are
me
be studied. It has thus become clear that there
problem
one
should rather sélect
of interest.
quantum-mechanical description of the Systems. The high
of quantum mechanical models, however,
statistical ensembles of
only
tradeoff has to be found between the accuracy of
and the size of Systems that to
Model I
therefore relies
order of magnitude every 5-7 years, numerical solutions to Model I
universal method able
no
analytical
the power of modem computers
from reach for many Systems and
description
two
computers, and the fundamental limitation
Although
to
(e.g. idéal gases, idéal solutions,...), whereas quantum
hâve to be solved
resources.
before
of most of the relevant chemical
of such
is
to invest
It is thus reasonable
sufficient size
to
prohibits
the
génération
of
compute many thermodynamic observables with
reasonable accuracy, which
implies that the statistical description has to be simplified using crude analytical approximations or simply abandoned. The quantum description is a
most
of the time
simplified by
découplés the motion of nuclei H) an arbitrary
the
use
of the
Born-Oppenheimer approximation
and électrons. This leads
to a
second type of models
which
(Model
collection of Systems in which fixed nuclei interact with their électron cloud
according to an approximate quantum-mechanical Hamiltonian.
Préface
The
only exact numerical
9
solution to this
problem
would be
Hartree-Fock calculation
a
déterminants into account (full CI) and using
using a limited
molecular orbitals. Even
basis set, full CI
Systems and further simplifications in the Hamiltonian of ab initio
méthodologies
hâve been
approximations: plain Hartree Fock, of
use
proposed,
alternatively,
use
of
density
become
the lowest
at
extremely poor.
theory.
quantifies the
(cheapest) levels
constants but as
reasonable
seem
referred to
as
adjustable
of the interaction
nature.
or
theory
be handled. On the
experiment may
first-principle
can
be tuned to
combine
(Hamiltonian)
the
replace
but
derived
improve
correct
(or
the constants
the basis of expérimental
on
quantum-mechanical, thèse methods
semi-empirical version, study
phase, however, the important observables
of
is well-suited to the
gas-phase problems.
In
statistical mechanical in
are
Therefore, their calculation involves the considération of many degrees of freedom
use
study
not
play an
intermolecular forces in the gas
forces in the bulk
phase.
This is
chemical and biochemical To
significant size,
which
of Model II. In other words, quantum mechanical models
where intermolecular forces do to
which
and thus to the
and statistical ensembles of configurations of a the
or
semi-empirical.
single-molecule properties
the condensed
can
to treat
collection of adjustable parameters tuned
Model II, either in its ab initio
calculation of
that
parameters that
called empirical. When the Hamiltonian is
are
small
A wealth
discrepancies with experiment appear to
into it
a
cases.
various types of
by
of theory, the agreement with
entering
by
expand the
to very
Of course, the lower the level of
data
are
characterized
Such methods,
agreement with experiment.
approximated) analytical form
applied
needed in most
larger the System
In thèse cases, and when
bear somewhat systematic trends, it may no more as
are
be
excited
ail thèse with basis sets of différent sizes,
functional
and the smaller the size of the basis set, the other hand,
only
can
ail
to
inclusion of a limited number of excited déterminants,
many-body perturbation theory, the
taking
infinité basis set of functions
an
bypass to
description
a
phase, they
are not
suited for the
and their average effect replaced
by
(ii)
an
the électrons
be used
of thèse
important
analytical
through large (although finite) ensembles
may become
can
the nuclei
move
removedfrom the
représentation,
description
possible. This leads
finite statistical mechanical ensemble
Préface
be
interaction function between
method is used, the statistical mechanical
a
study
of many
Systems
simplified by considering that,
nuclei. Due to the lower computational expense of this
HT),
study
Born-Oppenheimer approximation, (i)
the laws of classical mechanics and
of models (Model
to
although they can
serious limitation for the
this limitation, Model II may be further
adéquate sampling
currently prohibits
restricted
problems.
within the framework of the
according
clearly
dominant rôle, and
are
and when
an
of the System
to
the third type
of atoms
interacting
10
according to a classical mechanical Hamiltonian.
Even if the real
classical, classical and quantum statistical mechanics lead to the
partition
and classical
changes
are
involved
functions become
(no electronically
équivalent, provided
excited states
significantly populated (high enough température, of interest. If thèse conditions function
describing
give
Models I
no
that (i)
electronic
solely physical
rearrangements), (ii)
motions of the nuclei
to
light particles)
satisfied, it is possible
II
by averaging
derived from first
the electronic
out
principles,
most often defined
parametrical
as a sum
functions of
Many choices
find
to
classical interaction
a
of the system with the
are
degrees
they
of freedom, due to the
but rather tailored and calibrated in of
physically-based
(or
one
a
few)
internai
an
microscopic
force field terms. Thèse
coordinate(s)
that
ofthe
empirical parametrization
dynamical properties
properly only if (v)
the omitted
fields) degrees of freedom explicit degrees offreedom ofthe model. resolution force
Although they or
popularity,
(electronic,
are
up to
now
extremely important chemical
the
(tens of nanoseconds in
dynamical properties
a
are
terms
of solids,
within reach.
of
hâve
a
capture the
not sufficient to ensure a
Dynamical properties
study
will
supra-atomic
of processes
are
only available methods
dynamics),
to
much shorter relaxation time than the
and biochemical
liquids,
analytical
of the force-field
but also atomic in
in
involving
rapidly increasing a
position to
in
address
problems. They can currently handle
Systems of up to 105-106 atoms and generate up to 106-107
such
are
ofthe system.
do not offer the best framework for the
they
are
interatomic distances.
or
becomes reliable. Since the real
proton transfers, empirical classical force fields since
generally not
thermodynamical description of macroscopic
world is not classical, thèse four conditions
description
be described
so
computational
empirical way. It is
possible for the number, type and functional définition
physics of the system,
example,
level of
may be used to
Systems is then that (iv) the selected functional form is sensible enough
many
same
the interaction function is
practice,
terms. A further condition for the correct
électron
(iii) statistical
and
it is most often not feasible to dérive the classical interaction function from
or
limitations of thèse models. Therefore in
correct
ail are
allow for the convergence of the observables
if classical molécules do not exist,
even
world is not
results, i.e. quantum
thermodynamical description of macroscopic Systems. Although possible in
a correct
principle,
are
to
thermodynamic properties
Model I. Thus,
as
accuracy
the
generated
are
or
corresponding
discrète quantum mechanical energy levels
ensembles of a sufficient size
microscopic
same
configurations
and thus, the
of thèse Systems
study of thermodynamic and
solvated (bio-)molecules
or
lipid membranes,
for
Although challenging, the possibility of addressing problems of
complexity, but also
of extrême
practical relevance,
should not obscure the inhérent
limitations of empirical classical force fields. Thèse limitations
Préface
are
essentially determined
11
by the five conditions listed in the previous paragraph. Consequently, in
empirical
force fields,
outcome of the
a
critical
point
of view should be
adopted
ail
applications of
with respect to the
simulations, and the power and limitations of the methods employed. As
for any scientific
study,
three criteria
reliability, reproducibility, and
are
essential for the scientific
quality of a given
accuracy, the assessment of which should form
part of any theoretical study.
Préface
an
work:
intégral
12
Chapter
Empirical classical
1.1.
force fields for molecular Systems
Summary When
spécifie
a
molecular system is to be studied
Computing
largely
power,
be consirered will
prohibit the
functional methods. In this case, Unless
a
one
interaction function
are
to further remove the
description
of
(expensive)
may turn to
an
their ability
to
large Systems,
to
of freedom of
use a mean
the
use
of
reproduce and predict
spécifie
solvent
or a
empirical
or
such
a
force-field
the
a vast
density
description will be at best
when many évaluations of the
atoms
one
may
attempt
(solvent, protein sidechains...)
low resolution force field
amount
or
description.
(Section 1.3).
classical force fields résides in
interesting to try to vmderstand their physical basis Classical force fields
molecular orbital
empirical
required to compute the observable(s) of interest,
degrees
and
only justification
together with
chemist and biochemist, the number of
to the
use
hybrid quantum/classical treatment is performed,
at the atomic resolution. For very
from the
of interest will,
détermine which theoretical method is to be used
(Section 1.2). For many Systems of interest atoms to
using a theoretical computational method,
observable(s)
characteristics of the system and
available
The
1
principle
in
of expérimental data. It is nevertheless
in terms of first
principles (Section 1.4).
originate from the averaging out ofthe electronic (Born-Oppenheimer
surface) and possibly also some of the atomic (potential-of-mean-force) degrees of freedom from the
quantum-mechanical
which the
averaged
Hamiltonian of the System. The
interaction function
hâve been removed from the
performed correctly,
any
dépends
description
are
are
called
thermodynamic property
ensemble averages of instantaneous observables
Chapter
degrees
implicit. that
can
If this
be
depending solely 1
of freedom upon
called explicit, whereas the
averaging
expressed in on
the
ones
that
process is terms of
explicit degrees
13
of freedom
can
quantum effect
cornes
play (high enough température,
into
will however be described
properties
of freedom hâve
degrees
exactly by the classical interaction function, provided that
be described
freedom. In
practice,
a
functions of are
one
(or
a
few)
It is most often defined
empirically. coordinate(s)
internai
or
atoms
a
sum
Many
interatomic distances.
term
as
of
principles, of
analytical parametrical
are are
number, type and functional définition of the force field
1.6). Combination rules, which détermine force-field types of
explicit degrees
not derived from first
(Section 1.5). Thèse
force field terms
for die
possible
(electronic, atomic)
if the omitted
generally
the interaction function is
no
light particles). Dynamical
much shorter relaxation time than the
but rather tailored and calibrated
physically-based
properly only
no
parameters
defining the corresponding internai coordinate,
are
as a
terms
choices
(Section
function of the
also
an
important
component of the définition of a force field. Once functional forms and combining rules
corresponding parameters
hâve been selected for ail terms, the
(Section 1.7).
This calibration is
possibly complemented by parameter tuning is the
problem
important
can
a
the results of
difficult task in the
be solved
high
of
an
of
of
functional forms and
force field.
require long simulations
observables in terms of simulated
straightforward. Finally, the
spécifie
Only parts
procédures. Convergence
empirical
or
be unreachable, and
properties
values of each parameter will be
with the values of other parameters, with the choice of the
the
be calibrated
to
level quantum chemical calculations. The
design
somewhat automatic
using
simulated observables may
interprétation of expérimental not be
hâve
generally performed using mainly expérimental data,
explicit degrees
combining rules, and
may sometimes
strongly correlated of
freedom, with
with the choice of calibration
Systems and observables. Thèse corrélations will strongly reduce the domain of validity of a
given
force field, and there is thus
no
"universal" force field (Section 1.8). At last, the
optimal choice of a combination of terms,
their functional forms, the
the parameters should be such that die terms (force-field
transferability
from
one
molecular System
to
the
combining rules
building blocks)
and
hâve the best
other, witiiin the domain of validity of the
force field.
1.2. Introduction With the
rapid
continuing increase
increase in the number,
methods in
chemistry (van
performance
Gunsteren et
distinguish three major classes listed in order of
of the power of computers, the past décades hâve and accuracy of theoretical
computational
al., 1989ff, Lipkowitz & Boyd, 1990ff). One
of methods for the theoretical
decreasing computational
methods (Hehre et al., 1986), (ii)
seen a
expenses:
semi-empirical
Chapter
study
(i)
can
of molecular properties,
ab initio molecular-orbital
molecular-orbital methods (Stewart, 1990,
1
14
Zerner, 1991), and (iii) empirical classical force-field methods. The computational expenses of ab initio methods
are
of the order
0(Nf 4) (Hartree-Fock level)
or
higher
(Configuration Interaction, Many Body Perturbation Theory), Nf being the number of basis functions used. or
Density functional approaches
and
lower. The costs of empirical methods scale
stands for the number of of the
scaling with
usually
much
than any other methods
for the simulation of Systems
Since the available
semi-empirical methods
Computing
an
possible problems,
ail
but that
problem of interest.
observable(s)
(size
of the
typically
As is
one
prefactor ÎO'-IO6
up to
0(Nf3)
no
the
to
scaling)
and
atoms.
often the Crue
resources are most
limiting
factor
to
universal method able to solve
should rather sélect the method that is the most suitable
schematically represented
and system under considération that
in
Figure 1.1,
the
properties ofthe
will, together with the available
Computing power, largely détermine which type of method &
as
or groups of atoms). Independently empirical interaction function remains
numerical calculations, it has become clear that there is
to a
scale
0(Na2) down to nearly 0(N„), where N,
elementary particles (atoms
the system size, évaluation of
cheaper
currently allows
as
can
be used,
(van Gunsteren
are
Berendsen, 1990):
A. the
required system size
B. the
required volume of conformational
terms
C. the
of
dynamics:
particle,
requirements
in conflict wim
hierarchical
or
a more
die
group of atoms, treated
sampled (in
the smallest
explicitly in
entity,
model)
the
A and B,
hybrid models, or
use
of
a
potential
average effect without
case
the observable cannot be
together mosdy determining die computational effort,
only the
by the design of degrees of freedom are treated
most relevant
resolution method. This is often done, for
base-catalysed, organic, of
in which
C and D, this conflict may be resolved
where
expensive, higher
of acid-
incompatible,
currently available computer resources (van Gunsteren et al.,
requirements
(Warshel, 1991,Field, 1993, is the
or
may be
1995b). When requirements
study
or
required energetical accuracy of die interaction function
Thèse
widi
of particles (determined by
in terms
atom,
computed adequately with are
space that has to be searched
required timescale)
required resolution
subatomic D. the
the
or
enzymatic
example,
in the
reactions in fhe bulk
phase
Whitnell & Wilson, 1993,Liuetal., 1996a). Another example
mean
including
force its
représentation
degrees
for the solvent, which includes its
of freedom
explicitly (van
1994). Mean fluctuations in the solvent may also be included through
équations
of motion
as
in Stochastic
Dynamics (Yun-Yu
Berendsen, 1990).
Chapter
1
et
a
Gunsteren et al.,
modification of the
al., 1988,
van
Gunsteren &
15
OBSERVABLE OF INTEREST
Required
Required
resolution
energetical
of
accuracy
Required
Required
System
conformanonal
terms
m
particles
space to be
size
Hybnd
model
PMF solvent
sampled
Structural7
?
Thermodynamic
?
Dynamical
9
Choice of
explicit
Choice of
c
a
sampling method Number of
Number of
H évaluations
explicit degrees offreedom
Choice of
Computational
interaction Hamiltonian
costs
Hqm OrHclass
Figure 1.1: Schematic représentation of the system m order
simulate
to
an
basic choices made while
Molecular-orbital methods
are
well suited for the
clusters of molécules
in
(supermolecule) averaged solvent environment (Ângyân, 1992, Persico, 1994, Muller-Plathe & as
building
a
model of the molecular
observable of interest Thick hne boxes represent the three essential choices
van
vacuum
study of small molécules
(Keith
&
Frisch, 1994),
or
or
small
within
an
Cramer & Truhlar, 1992, 1995, Tomasi &
Gunsteren, 1994), and give
access to
properties
such
equilibrium geometnes, vibrational frequencies, heats of formation, relative énergies of
conformers and isomensation barriers. Thèse
increasing
accuracy
1991, Maple
et
by empirical
problems
methods (Bowen &
al., 1994a,b). Due
to the size of the
conformational space, simulation of organic molécules
phase
is the domain of atom-based
empirical
also addressed with
are
Alhnger, 1991,
an
Hagler,
and volume of accessible
problem or
Dinur &
macromolecules in the condensed
classical force fields (van Gunsteren &
Berendsen, 1990). Long timescale (or long relaxation time) problems involving large Systems, such
as
protein folding
by
residue-based force fields
&
Smith, 1996). Finding
resolution areas
(i.e.
a
sufficient
or
de
novo
protein design,
can
currently be addressed only
(Gerber, 1992, Jones, 1994, Ulrich
an accurate
description
et
al., 1994,1996, Lathrop
of the interaction at this low
energetical resolution) is, however,
a
major difficulty.
of development with respect to treatment of degrees of freedom
in Section 1.3.
Chapter
1
are
particle Current
briefly discussed
16
Choosing
die
explicitly
force field calculation
handled
(Figure 1.1).
sample
the conformational space
1991,
Scheraga, 1992, 1993,
degrees
of freedom is the first step in
The second is the choice of
(Howard & Kollman, 1988, Leach, 1991,
Osawa & Orville-Thomas, 1994,
1995a). This choice will also
dépend
me
on
information
empirical
an
method to search
a
van
Gunsteren et al.,
van
required
to
compute die
namely:
observable(s)
of interest,
A. Structural
information (searching):
The purpose of thèse mediods is to search conformational space for
one or a
number
of relevant low energy conformations. In the latter case, the conformations obtained not
related
by
of choice is the B.
highest
probabilistic
any well defined
me one
that searches the
low-energy
number of
Structural and thermodynamic
to
get
a
or
largest
dynamical relationship,
of conformational space,
extent
returning
information (sampling):
sample
conformational space
collection of conformations which build
a
part of it in order
or
correct statistical ensemble, that is,
ensemble in which the conformations appear with
a
Boltzmann
probability.
The
séquence of the conformations is not relevant and the method of choice is the which achieves die C.
are
and the method
structures.
The purpose of thèse mediods is to
an
or
Gunsteren,
one
highest sampling efficiency.
Structural, thermodynamic and dynamical information (simulating): The purpose of thèse methods is to simulate the motion in conformational space of it, in order to get
a
séquence of conformations which build
ensemble, but are also consécutive in time (dynamics). In this which
explicitly
contain time
are
Lagrange, Hamilton, Langevin
or
The tiiird choice to be made in interaction function
(or, together with
the selected explicit
degrees
constructed
such
Liouville
équations
me
of freedom
using expérimental
empirical
an
as
information
1.1 ). In
a
(possibly complemented
to the
underlying quantum mechanical reality. Empirical on a
generalization
possibly
or
predict
also of individual atoms) to obtain
implicit degrees
an
Chapter
analytical
1
a
an
to
are
with theoretical
large
amount of
empirical description
classical force fields
Born-Oppenheimer approximation, over
of
principle, empirical force fields
based
of die quantum mechanical Hamiltonian
one
Hamiltonian) corresponding
It is however instructive to try to relate the
ofthe
équations of motion
of motion.
results) and their only justification is their ability to reproduce
expérimental observables.
part
Schrôdinger, Newton,
force-field calculation is die
kinetic energy,
(Figure
case,
die Dirac,
required,
or
statistical
a correct
diat is,
of freedom
are
on an
formally
averaging
(electronic
interaction function
and
depending
17
solely on
die
explicit degrees
interaction will be called
Averaging occurs
of freedom of the model. Due to this
potential of
ofthe
process, the
interaction
function.
over
the
différent chemical/topological
depending
terms
environments
on
1.1
internai coordinate
the same coordinate, that is,
possible options
The list of methods is
functional
on me
of force field
The choice of of
by far not exhaustive and
me
description
parametrization
explicit degrees
elementary
an
freedom)
will be
1.5 and
unit (i.e.
elementary
togefher witii die corresponding type This choice will détermine Gunsteren & Mark, 1992,
or
is somewhat biased towards
me
tiiat will hâve
particle
design
unit and
of
an
empirical
explicitly
of interaction function,
Gunsteren et al.,
and tiras die
The extent of conformational space mat
no
explicit internai
classical force field.
treated
are
degrees
of
freedom,
summarized in Table 1.1. & Berendsen, 1990,
van
1995b):
A. The number of degrees of freedom mat will hâve to be handled
spécifie molecular System,
Finally,
briefly discussed (Section 1.7).
strongly influence (van Gunsteren
van
of
1.6).
of freedom of the model
is the first step in the
Possible alternatives for die
a
représentation
(Sections
simulations and simulation of large molécules (biomolecules).
1.3. Choice of the
degrees
différent geometrical
over
interaction function in atom and united-atom based force fields
problem
the other
in the three basic choices outlined in
(thick lined boxes) and mainly concentrate
condensed-phase
over
(Section 1.4.3)
The présent text will discuss
Figure
environments
différent molécules (Section 1.4.2)
Averaging of a force-field term corresponding to an force-field
B.
averaging
effective
1.4.1)
Averaging of a force-field term
C.
the
or
tiiree levels:
at
model (Section
présent in
me
force
mean
Averaging ofthe quantum mechanical interaction over the implicit degrees offreedom
A.
B.
a
computational
explicitly for describing
effort.
(or in
tenus of molecular
dynamics, the reachable timescale). Because available Computing
power is most often
a
limiting factor,
potential
for
a
can
system of a given size,
energy function will
be searched
me
number of
rapidly decrease witil
freedom.
Chapter
1
possible évaluations
the number of
of the
explicit degrees
of
unplicit solvent
adisk
idem
possible
van
Gunsteren, 1994, (e) Brooks
et
form, parameters,
the équations of motion
based interaction fonction
m
the functional
H
mtramolecular
side-chain
solvent
idem
allH
ail H bound to C
aliphaùc
electromc
solvent
none
none
none
(h)
(g)
(0
(f)
(e)
(e)
(e)
(e)
(d)
(c)
(b)
(a)
REF
Keim & Pnsch, 1994, (d) Ângyan, 1992, Cramer & Truhlar, 1992, al, 1983, Gerber & Muller, 1995, (f) van Gunsteren et al, 1994, (g)
average întermolecular interaction fonction
statistics
terms or
m
solvent terms
corrections
încludrng exphcit
by additional
idem
idem
idem
Hierarchy of explidt degrees of freedom incmded in the model see for example (a) Hehre et al, 1986, (b) Stewart, 1990, Zerner, 1991, (c)
1995, Tomasi & Persico, 1994, Muller Plathe & Jones, 1994, (h) Bâtes & Luckhurst, 1996
Références,
Table 1.1:
or
liquid phase
rod
(or crystal)
a
sphère,
molécules
represented by
a
one or a
(or crystal)
represented by
few beads
proteins
in
"bead(s)":
atom groups
as
unplicit solvent
idem
eg amino-acids
explicit solvent
(ail)
idem
united atoms
groups)
idem
united atom (ail CH,
classical empirical interaction funcuon
gas phase
reaction field contribution
idem, additional
mplicit solvent
semi-empincal approximated HamUtoman
surface
allatoms
(alipbatic groups only)
density functional
pnnciple quantum mecbamcal Hamdtoman,
Bom-Oppenheimer
first
ab mitw,
OUT
idem, supennolecule methods
phase
DF AVERAGED
TYPE OF INTERACTION
(OPERATOR/FUNCnON)
exphcit solvent
gas
PHASE
united atom
(united-)atoms:
électrons and nuclei
ELEMENTARY UNIT
19 C.
The maximum resolution, in terms of particles
of atoms,
or
reactions) that
molécules) can
be achieved
D. The type of functions diat
units in E.
adéquate
an
that
to
describe
is, with
me
interaction between
reasonable
a
energetical
The type of observables the force field may be able to describe
which will
necessarily
Current in terms of
developments
degrees
empirical
in
1993, Whitnell & Wilson, 1993,
are
correctly, and those
Allinger, 1991,
mainly
follow five basic Unes
Dinur &
Gunsteren et al., 1994, Jones,
van
Hagler, 1991, Gelin, 1994), which will
described in Sections 1.3.1-1.3.5. Notediat in 1.3.3-1.3.5, die number of
particles.
is to limit the
dimensionality
1.3.1.
Gas-phase force
et
or
replace
or to
more
force fields is
me accurate
description
of molécules
frequencies, heats
vibrational
fields is made amount and
of
ab initio molecular orbital calculations
possible by (i)
and
such
as
to
(Maple
eitiier et
al.,
equilibrium geometries,
formation, relative énergies of conformers and energy
(Hwang
reliability of data
systematic
al., 1994). Thèse force fields may be used
et
expensive
predict expérimental gas-phase properties
barriers for isomerisation
of
e.g.
fields
al., 1988, 1994a,b, Hwang
complète
see
be discussed hère.
(Bowen & Allinger, 1991, Dinur & Hagler, 1991, Hagler & Ewig, 1994, Maple
vacuum
1994a),
not
primary purpose of gas phase
The in
size of the conformational space to be searched
discretize die coordinates (lattice mediods,
or to
Binder, 1992). Thèse mediods will
me
be
explicit degrees
the force-field resolution in terms of
essentially by decreasing
An alternative way to reduce
terms
sufficient.
classical force fields
offreedom (Bowen &
of freedom is reduced
elementary
accuracy.
(B), die force field resolution in
(C), and die force field accuracy (D)
particles
chemical
stay inaccessible. Accessible observables will be tiiose for which
die extent of searchable conformational space
of
changes,
conformational
die force field.
by
likely
are
manner,
(e.g. subatomic particles, atoms, group
(e.g.
and processes
et
al., 1994). Rapid progress in die design of such force
the absence of intermolecular forces,
(ii)
the
from ab initio molecular-orbital calculations and
increasing (iii)
the
use
relatively inexpensive procédures for parameter calibration using both
theoretical and expérimental data (Section 1.7.5). Thèse force fields, sometimes called class II force fields covalent
(Maple
degrees
and terms that
examples
are
et
al., 1994a,b),
of freedom,
couple
(Hagler
involving
usually characterized by
anharmonic
the internai coordinates
a
detailed
description
(non-quadratic) potential energy
(non-diagonal
energy terms).
of
terms
Typical
the force fields CFF (Lifson & Warshel, 1968, Warshel & Lifson, 1970,
Lifson & Stem, 1982) and et
are
a
recently
al., 1979a-c, Lifson
et
modified version
(Engelsen
et
al., 1995a,b), CVFF
al., 1979), EFF93 (Dillen, 1995a,b), MM2 (Allinger,
Chapter
1
20
1977, Bowen & Allinger, 1991), MM3 (Allinger Bowen &
Allinger, 1991)
The term for
gas-phase force
applications
structures
and QMFF/CFF93 field does not
in condensed
phase
is sometimes used in
mean
al., 1989, Lii & Allinger, 1989a,b,
et
(Maple
al., 1994a,b, Hwang
et
Expérimental information
simulations.
parametrization procédure (Warshel
me
al., 1994).
et
that such force fields cannot be extended
crystal
on
Lifson, 1970,
&
Dillen, 1995b, Engelsen, 1995b). For applications in liquid phase problems, however, tiiese force fields will suffer from die
force fields gas
(Section 1.3.2),
same
difficulties in
and whemer the
inclusion of anharmonic and
phase by
parametrization
significantly improved off-diagonal
terms
as
will resuit in
condensed-phase properties
increase of accuracy in the simulated
condensed-phase
accuracy
gained in a
die
significant
is still matter of
discussion.
Condensed-phase force
1.3.2.
fields
The primary purpose of condensed-phase liquids, solutions of organic compounds or
Tildesley, 1987,
McCammon &
force fields is die accurate macromolecules and
description
of
crystals (Allen
&
Brooks ffl et al., 1988,
Harvey, 1987,
van
Gunsteren &
Berendsen, 1990). Progress in die development of such force fields is slow, since (i) die dominant forces in the condensed described and
impossibility
rely mostly
phase,
and
possible (see
not
intermolecular forces which
(iii) die
on
design
a
large
are
small amount of
a
of
however Section
applied)
not
easily
is
limited,
expérimental
data
systematic optimization procédures
1.7.4). One major
is that the estimation of observables to be
generally requires
are
die relevance of data from ab initio molecular
(even when reaction-field corrections
has to
die condensed
gênerai
is in
vacuum
parametrization
concerning
are
parametrized adequately, (ii)
orbital calculations in and the
phase
reason
for tiiis
compared to expérimental
number of évaluations of die
potential
energy
results
function, and is
dierefore computationally expensive. In thèse force fields, die main effort is aimed at die description of non-bonded forces and torsional potential energy terms. Potential energy terms
involving other covalent internai
coordinates
are
often eitiier
called class I force field)
or
the force fields AMBER
(Weiner & Koliman, 1981, Weiner
et
simply
al., 1995), CHARMM (Brooks
1992, MacKerell Jr. DREIDING
1983a,b,
(Mayo
Levitt et
1987, Scott &
et
et
1989),
(Rappé
et
by me
use
quadratic-diagonal (so-
of constraints. et
Typical examples
al., 1983, Nilsson & Karplus, 1986, Smith & Karplus,
al., 1990), ECEPP/3 (Némethy EREF
(Levitt, 1974),
Gunsteren, 1995),
et
GROMOS
al., 1992), ENCAD (Levitt,
(van Gunsteren & Berendsen,
MAB (Gerber & Muller, 1995), MacroModel
al., 1990), OPLS(Jorgensen & Tirado-Rives, 1988), Tripos (Clark et
are
al., 1984,1986, Pearlman
al., 1995), CHARMm/QUANTA (Momany & Rone, 1992),
al., 1995),
van
(Mohamadi UFF
et
zeroed
al., 1992) and YETI (Vedani, 1988).
Chapter
1
et
al.,
21
1.3.3. Mean-solvent force fields The purpose of but widiout
an
a
explicit
1994). Almough
treatment
accurate
an
computational
by
expenses, e.g.
a
an
et
al.,
solvent, the
treatment of me
explicit
degrees of
almost ail solvent
or
of freedom (van Gunsteren
degrees
description of die structure, mobility, dynamics and energetics
generally requires
of molécules in solution omission of ail
of the solvent
of molécules in solution,
description
mean-solvent force field is die
dramatically reduces
freedom
the
factor 10-50 for biomolecules in solution. The explicit
influence of die solvent is
approximated hère by its mean effect, and possibly also me effect
of its
as
fluctuations,
mean
1993).
implicit
The main
dynamics (Yun-Yu
1995, Fraternali &
van
al., 1988,
are
Gunsteren, 1996) and of
Gunsteren,
structural effect,
mimicked
by a modification
équations
me
van
or
additional terms,
(différent functional form,
of me interaction function
et
hydrophobic
drag,
random fluctuations and viscous
screening,
dielectric
in stochastic
influences of solvent, i.e.
see
e.g. Banks et al.,
(Langevin
of motion
équation). 1.3.4. Low-resolution force fields The purpose of low-resolution force fields is
addressing long timescale phenomena, such de
novo
Computing
power, dièse
(HUnenberger
amino-acid residue level
problems et
are
adéquate expression
energetical
treated to
be
1.3.5.
study
récognition
are
van
for
being developed
for
me
of native mean
force term
expected from Hybrid
A whole
Lathrop
peptides
& Smith,
and
1996).
proteins (Gerber, 1992, The main
difficulty is to
provides
a
in
A correct
a
calibrated via
description
of the
a
sufficient
functional
a
are
statistical
normally
dynamics
is not
such models.
variety of models include me combination
high particle resolution
a
usually
are
structures. The effects of solvent
(Section 1.3.3).
available
force fields
instance, die first
explicidy
using
currently
force fields at atomic
interaction between residues tiiat
(and non-native) protein
while
proteins, protein folding,
Witii die
resolution to discriminate correct from incorrect structures. Once
by a
freedom at
large Systems,
of
in
Gunsteren et al., 1995b). Force fields at the
form is selected, die interaction function parameters
analysis
me
diffïcult to address,
al., 1995a,b,
Jones, 1994, Ulrich et al., 1994,1996, an
fold
protein design and protein-protein association.
resolution
find
as
or
first few
and
a
of
treatment of die
hydration shells
of
a
a
treatment of
odiers at
1
a
few
degrees
of
lower resolution. For
macromolecule may be included
simulation, fhe bulk solvent being modelled dirough
Chapter
a
a mean
force
(Section
22
1.3.3). Anodier typical example is die simulation of chemical,
acid-
or
base-catalyzed
or
reactions, in solution or in enzymes (Warshel, 1991, Field, 1993, Whitnell & Wilson, 1993, Liu et al., 1994,1996a,b).
be
applied to
me
resolution in such
a
quantum mechanical
description
of
computational
costs, such
a
due to the
full system under
be treated in diis way.
1.4.
Clearly,
required. However,
die protons is
study,
ability
to
models is hère die main
hybrid
only justification of empirical
reproduce
and
predict
information used in their
a vast
long
a
few relevant
design
degrees
of freedom
useful to try to understand die
reason
can
of particle
difficulty.
classical atomic interaction functions résides in tiieir
expérimental
and calibration
force field is successful at
or
classical interaction functions
amount of
quantum mechanical calculations. Thus, as a
only
électrons
Finding die proper interface between die différent degrees
Assumptions underlying empirical The
as
and
me
treatment cannot
no
cornes
theoretical
reproducing
results.
justification
data from
of die agreement
Usually,
experiment
from
is in
cause
of die
principle required
experiment.
(or the
most
and not from
It is nevertheless
of discrepancies)
by
considering die relationship between die force-field building blocks (energy terms) and die underlying quantum 1.4.1.
mechanical
Implicit degrees
of freedom and die
degrees
Whatever die
reality.
the reality behind remains
of freedom chosen to be treated
quantum-mechanical
and électrons. Since the electronic of
nuclei, do
not
function, but still fundamental
assumption of weak
are
explicitly
and involves
degrees offreedom,
appear in die définition of die
corrélation within
force field,
a
interaction between nuclei
me
and sometimes tiiose of classical
empirical
a
potential
number energy
présent in die underlying reality, they may be called implicit. The
assumption (or approximation)
on
which
classical force fields
empirical
are
based, is diat the corrélation between die fluctuations in thèse implicit degrees of freedom and die fluctuations in those which
assumption, only
their
the fluctuations in die
mean
effect. This
Oppenheimer principle, of freedom based
on
die framework of dlis energy
surface, PES)
implicit
degrees
me
explicitly
can
degrees of freedom is in
essence
a
be
can
neglected.
be
averaged
séparation of die nuclear
principle, can
be
a mean or
effective potential
Under this out,
generalization of
leaving
die Born-
and electronic
large différence between nuclear and electronic
masses.
degrees Witiiin
energy function (or potential
defined, which describes die interaction of the nuclei in die
instantaneously averaged potential electronic
handled
assumption
which allows
die
are
of die électron cloud. More
offreedom and i die nuclear ones, die
Chapter
1
mean
precisely,
potential
if
u
dénotes die
energy
describing
23
die interaction of the
nuclei,Vnuc( {r,} ),
is defined
time-independent Schrodinger équation
A^[? },{?,))
at a
lowest
as me
eigenvalue of die
given configuration
WFJiir,))
of die nuclei
electronic
{r,}
^((r,}) HiM({rM};{r,})
-
(1.4.1.1)
«„({?„},{?,))
with
=
«„({?„},{?,))
-
K,(lr,}) A
A
is
where Di
equal
to
die total Hamiltonian of die system, Jt M, minus die kinetic energy
A
operatorK, corresponding die
the nuclear
to
ground state electronic wave-function,
treatment is valid
which
only
electronically
the nuclei
are
for
an
degrees
dépends on {r,} only parametrically. This
which
isolated System (time
excited states
play
no
\|»|i({ïM};{r,})is
freedom, and
of
rôle. In
independent
total
Hamiltonian), in
Equation (1.4.1.1), die assumption
motionless while solving die electronic problem allows for the
mat
decoupling
A
of die K
nuclear
operator from the Hamiltonian. The nuclear problem is tiien described by
,
a
time-independent Schrodinger équation
«.({F,}) *,({?,})
E„ *,({?,})
=
(1.4.1.2)
«,({?,})
with
where die
eigenvalues Ew
are
Vnuc({r,})
=
*
die allowed values for the total energy of die system in its
$,({?,})
différent vibrational and rotational states, and wavefunctions. die
mean
view, this
Very often,
me
potential Vnuc({ rj)
approximation
ÂT,({r,})
further
can
assumption
be treated
die
corresponding
is made tiiat the motion of
classically.
is normally valid for ail but
From
me
a
me
nuclear nuclei in
thermodynamical point
lightest
atoms
and at
of
high enough
température, that is, when die classical and quantum partition functions become équivalent. When this classical treatment is adéquate,
équivalent formulations
VJLW)
"ap
or
where me
,,
=
be
given
to
using
die
Hellmann-Feynman tiieorem
two
Equation (1.4.1.1)
^
=-FHICJlirl})=M
j.
This
only
and
means
FnilC0 ({r,}) is
diat when die
potential (first définition)
u
and die
24
potential of mean force (second définition) case
in
an
When classical
an
degrees
of
are no more
freedom,
m, are further removed from the interaction
of freedom i will be described resolution model (in
particle
or
of
protein side-chains),
équivalent. Thermodynamic quantities
an
NVT
equivalendy by
an
explicit
the
ail atom model and
if the Boltzmann factors
ensemble)
die two
defined in terms of
microscopic (instantaneous) observable depending on
ensemble average of a
degrees
constant). This is the
energy surface.
function by averaging (e.g. nuclei of solvent molécules types of définition
a
field, where the classical interaction is described by die Born-
ail atom force
Oppenheimer potential
équivalent (witilin
are
lower
a
identical, that
are
isif
e-V„{[r,Wkf where
e-Vm({rJ.{r,))lkBT
is the interaction at die lower
VMF({Ï1})
constant, and T die
ïj
die correct statistical mechanical définition of
—^
=
W„
system, V„p becomes the
as an
areas
its internai
entropie force,
which
of différent Boltzmann
energy surface. This force
(1.4.1.6)
potential
averaging
{î,}.
Equation (1.4.1.4)), and VmMn
actually
leads
is
Vmea„ ( {r,} ),
out from the
particle j,
/. d-4.1.5)
df
atom
energy,
thus be
coordinates
k
a.potential ofmean-force
averaged
(1.4.1.6)
explicit
sizes in die nuclear
as
>m-m
({r },{?})
ôV
m
(1.4.2.1)
321/
ôr, drk
Chapter
1
io
28 ofthe matrix
eigenvectors
The
used to define
system
as
in
a
unique,
a
spectroscopic force
practical application A.
The
containing
second derivatives
me
(Hessian matrix)
basis set for
orthogonal
non-redundant and
mathematically satisfactory,
field. This is
can
description
me
be
of die
but of limited
since
equilibrium conformation
is
known but
usually not
something
one
would like to
predict. A system at
C.
Other conformations neither die
generally characterized by
is
equilibrium
B.
(non-equilibrium)
Taylor expansion
at
{ï,°)
are
nor me
than
more
one
often also of interest
conformation. in which
-
corresponding well-defined
cases
basis set
are
usable. D.
description through
The accurate
Taylor expansion
a
for
one
provide much insight into other parts of me configurational die E.
physics
configuration
space,
so
does not
does not describe
of die system.
The accurate
of
description
molécule is useless for
one
prédictions
about odier
molécules. The second
approach
functions (energy terms)
relies
on
depending
the
use
of
a sum
functionally simple analytical
of
selected internai coordinates, chosen
on
on
die basis
of chemical intuition. This is justified because A. A wealtii of chemical data tells
angles
and non-bonded
corresponding
entities such
us mat
interactions
are
as
bonds, bond-angles, torsional
physically meaningful, and tiius,
internai coordinates and distances in space appear
in which die functional forms of the interactions
are
likely
to
as
die
die natural choice
adopt
die most
simple
forms. B.
may C.
potential
Since such internai coordinate
give
an
appropriate description
Since internai coordinates involve
hope
to
obtain
building
a
of
energy terms
a
larger part
are
of
physically meaningful, they
configurational
limited number of atoms
blocks transférable from
one
(one
to
space.
four), tiiere is
a
molécule to another (assumption
oftransferability). In odier words,
one wants to
physically meaningful (and transférable from are
one
split me interaction function into a sum of functionally simple, tiras
molécule
insight providing)
to
called force-field terms. The
assumption
mentioned Section 1.4. l,i.e. diat for each term environment
can
be
averaged
terms, which would in addition be
another, and tiius, bear prédictive power. Thèse
out
that thèse
terms
(e.g. bond, bond-angle...),
by considering
Chapter
1
an
terms
exist is similar to the me
one
effect ofthe
ensemble of molecular Systems
29
(topologies)
More
conformations (geometries).
and
analytical expression
explicidy,
one
would like to hâve
V^ar,))
E n^^jea})
=
(1422)
terms a
jea indicates
where die notation
Vml({ r,}) a.
This
die
is die
of
description
Boltzmann factors
are
If this
équation
is
me
j
is involved in die force-field term
potential energy surface
force defined in
mean
îj, jeP
,
and its
on
with respect to
dr~, aV^,(tr;j6tt},{rt,tea})
équation
p,
over
the coordinates
{rk,këP}.
force fields where die coordinates
(1.4.2.4) will always be présent in force fields,
and distance
dépendent
means
often act
terms
that when
die energy term
p
me
on
coordinate
by solving consistently Equation (1.4.2.4)
for
P
a
given
?j
différent set for another
only
initio data
by considering
be removed to
an
set
Equation (1.4.2.4)
an exact
topological
set
of terms
a.
This type of corrélation
design
of force fields
using a
ab
proper
sélection of conformations. Unlike in the définition
of die potential of mean force in Section 1.4.1 has
given
of conformations used tiiere is, however, not
arbitrary
The
collection of différent molecular Systems. A
tiiis is followed in die consistent
(Section 1.7.5). The
statistical ensemble, but
a
one.
a
terms.
may be removed
molecular system. The
may be correlated to
for
procédure analogous
because covalent
is also involved in force field
me term
may
Witii the
normal mode
becomes correlated to thèse odier
corrélation arises from die fact that a
are
the Cartesian coordinate of the
géométrie corrélation arising from diis so-called coordinate redundancy
molecular system, and to
(1.4.2.4)
orj
averaging
spectroscopic
vectors, the second term in
one
one
*
of harmonie
terms a odier tiian
which do not appear
gets after rearrangement
where k dénotes ensemble
This
(1A2.3)
{rk, kf p}
the coordinates
H'Pjmx
same atom.
and
Equation (1.4.1.4) will give
négative logarithm differentiated
~
energy terms
a
of terms
of any molecular system if the
«-"--«vr-.W
D
dF,
exception
as a sum
„-v'°.~A(v-'t"i>/*i>7'
n
=
integrated with respect to
of die coordinates
me
tiiermodynamic properties
v»f (i',)vv
given force-field term p,
a
the
of
identical
e
in
tiiat die atom
analytical représentation
analytical représentation and
same
an
of die form
solution. In
(Equation (1.4.1.4)), practice,
Chapter
1
one
it is not
guaranteed that
fixes functional forms for die
30
flexibility
the
on
As
and calibrâtes die parameters. The
V",,^,
energy terms
entity incorporating
parameters corresponding
that die
for which die term
compounds
give
given
calibrated (Section
was
enough
so
(1.4.2.4)
that
picture
a correct
tiiat
guaranteed, however, 1.4.3. Coordinate Force fields
of the
usually
a
dynamical picture of
practice atoms
with
a
higher
valence
redundant, i.e.
internai coordinate
Hagler, 1991).
can
likely
to
A conformation of formamide is
hâve
an
example, the HCO, carbonyl
group,
assign spécifie vary in
cannot
redundancy is the is
case
of molecular Systems. It is not
will be obtained.
non-bonded interactions (even if in
me
systematic counting,
one
influence
HCN and OCN
angles
are
die six
a
set
however, tiiat tiiere
an
properties
independent
handled
differentiy
are
hydrogens (this only (this
nine
12
=
bonds, 6
necessarily redundant.
mese are
dépendent since,
due to die
manner
from
one
to, say,
a
generic
from the odier
planarity
amide OCN
angles.
For
of die
force field to another, can
as
tiiree
not
possibilities: (i)
use a
easily generalizable to
angle,
since it
Valence coordinate
is illustrated hère for
be defined in émane,
only
torsional coordinate which involves
less
symmetrical cases), (ii)
induces asymmetry in the system and
use one
of
requires arbitrary choices)
or
a
force constant
(tiiis may be computationally inefficient). Botii choices (ii) and (iii)
Chapter
1
one
groups with respect
(iii) accept die redundancy and calculate die nine four-body torsions witii
by
5
tiiey must sum up to 2it. This raises me question whetiier it is possible to and transférable
anodier. There
divided
of 3N-6 are
of
(Dinur &
tiiat is, 17 available internai coordinates,
of etiiane. Out of the nine torsional dihedrals
die dihedrals
problem
of formamide
fully specified by sees,
the energy. Five of
on
case
required to describe me relative rigid body motion of the two methyl
to one
molécule includes
a
from the others. The
be illustrated in die
angles, 4 dihedrals and 2 out-of-plane coordinates, ail
are
transferability
linearly dépendent
are
redundancy
internai coordinates. By
averaging
than two, die valence internai coordinates tiiemselves become
of them
some
also
die class of
When die
calculated in Cartesian coordinates). When
are
on
virtual
defined in terms of internai valence coordinates for the
covalent interactions, and atom-atom distances for the forces
dépendent
1.7.6.3).
a
means
die interaction function will be able
good solution,
assumption
and
really
is
selected for die energy terms
tiiermodynamic properties
a correct
redundancy are
has
term
environments. This
may be
term
performed correcûy and me analytical functions
process is
sensible
to a
a
force-field
possible
average effect of various
me
but radier characteristic of
spécifie molécule,
ensemble of molécules and conformations. Thus,
an
dépend
averaging process, the parameters characterizing die
not tiiose found in any
a are
solution will
quality of the
sensibleness of die selected functions.
conséquence of this second
a
force-field term
to
physical
and
are
31
found in current force fields. At last, the définition of relevant internai coordinates some
insight into
pyramidality
the
which choice may lead to die best transférable entities. For around
nitrogen
a
center
energy terms, it will be difficult to get at the
and
frequencies
is maintained same
time the correct
molécule
are
pyramidal
it will be introduced
dépendent
on
bond-angle vibrational
inversion barrier.
by limiting
Even if redundancy in the valence coordinates is avoided
3N-6, ultimately,
by
is about 0.16
a
die valence coordinates, and dius, die non-bonded interactions
interaction should induce strain
lengtii
tiieir number to
die non-bonded interaction. Distances witiùn
example,
will introduce die strain effects into the valence coordinates. For
its
if
bond-angle potential
three
by
requires
example,
on
the central C-C bonds in
[nm], whereas in
most
tri-ierr-butyl
equilibrium
force fields die
non-bonded
médiane
diat
so
length
bond
is
about 0.152-0.153 [nm].
averaging processes
1.4.4. Choices made in die
Choices to be made witii respect
example
of
a
C-C bond. As
influence die effective A. The bond
pointed
lengtii
potential
in
a
to me
out
averaging process
in die two
given conformation (a potential of
energy term
will be discussed
of a
given compound
mean
force
using
die
the factors diat will
previous sections,
over
are
ensemble of
an
molécules) B.
A
possible explicit dependence
when différent C atom types C. A
possible explicit dependence
cross-terms
D. The For
are
on
topology through différent
classes of C-C
on
topology
and geometry
tiirough valence
implicit dependence
example, die
on
topology and geometry tiirough non-bonded strain
DREIDING force field
for biomolecules,
not
(Mayo
very accurate, but makes
option
et
al., 1990) excludes botii factors B and
parametrization easy.
a
a
This is
spécifie calibration of the chemical entities required for
simple
accurate for
In most force fields
B is used and différent classes of C-C bonds
depending on die connectivity and environment of the bonded atoms. allows for
coordinate
(Section 1.6.5)
C, which is perhaps
leads to
interaction function. In class II force fields, which
a
are
defined,
more
accurate,
given
are
purpose and
meant to
complicated by
are
required
for ail but
me
simplest Systems,
the interaction function is
die cross-terms, and parameters hâve to be calibrated ail
togetiier
consistent way. On the other hand, die interaction function is very accurate and atom
two, C and
be very
molécules in vacuum, option C is mostiy used. The inconvenience hère is that
many parameters
since few
bonds,
used.
types hâve
to be
defined (e.g. in
H).
Chapter
1
me
in
a
élégant,
CFF93 force field for alkanes,
only
32 1.5. General characteristics of the
empirical
interaction function
1.5.1. Interaction function parameters and molecular An
empirical
interaction function,
topology
loosely called
or
a
force field, V, is defined
by
its
functional form and die parameters diat enter into its définition, i.e. its interaction function
parameters, {s,}. In order
to
express tiiis latter
V
can
be used, where
=
V(
spécifie System,
to model a
?;{*,})
some
information
This results in very différent interaction at die an
electronic level exist. For
Na+ and
a
Cl
not
will hâve
If me ions
for
me
really required
a
a
From this
example,
exceeds
function like
our
régimes
or even
a
it is clear that
needs, (ii)
correct
a
analytical approximation
can
pair and
one can
a constant
around die
K^R-R^)2 gives (i)
die
degrees
me
more
analytical description
is
more
information about die ions,
the parameters
pairs
between atoms
be calculated
by solving
say
quite safely
diat die
parameters is likely molecular
to
a
again
can
distance
approximation
R^.
is
electronic
the
we
énergies
is sufficient, and
same
required
(iii)
me
to
me
other hand,
it will
require
régime (bonded, non-bonded)
and
It is also clear that die transition between
problem.
die framework of which functional form and
analogy,
be solved
However,
die true energy.
to
computationally cheaper,
Even when
me
interaction between two
functional form, die best choice of function
be différent if die bonds
topology information
potential
die quantum mechanical character of the
(K^.K^R^) spécifie to die pair. by die
die
for two separate ions. However,
equilibrium
of relative
intuitive and
namely, die
of bonded atoms is described
are
of freedom.
Schrodinger équation contains information tiiat
bypass
bonded and non-bonded régime will be
This
die interaction between
interaction, but radier captures die essential physics from its solution. On if
In order
required.
and R die distance between die ions.
reasonable
a
description
does not
is
relationships
molécule, the Schrodinger équation
séparations
System) die
complète.
topology
Hagler, 1991),
phase
useful, since is
if différent
&
électrons of the ion
closer and form
for différent internuclear know diat
not
principles techniques, empirical force fields
example (Dinur
K^/R dependence, where K
come
die molecular
on
ion at 1 [nm] in the gas
Schrodinger équation tilis is
any coordinate
energy function diat averages out die electronic
potential
on a
defining (in
This information is, however,
arises from die fact diat, in contrast to first based
(1.5.1.1)
is die 6N dimensional vector
q"
configuration of die molecular system.
die notation
dependence,
are
not identical. To
summarize, die
décide which interaction is to be treated in
using which
values for die parameters.
By
only proper molecular topology information required for an ab initio molecular Chapter
1
33
orbital calculation
at a
certain level of
molecular
topology
is in most
is the number of protons and électrons for
theory,
each atom. Note diat due to coordinate
redundancy (Section 1.4.3),
die
spécification
of
a
unique.
cases not
1.5.2. Atom types and combination rules In
a
number of
considered
as a
charged
freedom. When
implicitly atoms.
point
mass
heavy
significandy reduces
This
with
constitute about 50% of
me
dynamics point
die number of
problematic
degrees
and enables correct
a
when the donor is treated
as a
(non-polar) hydrogens
négative
effects of the
quadrupole
moments
united atom,
steric effects (me united atoms
approach fails be
even
are
of
hydrogens
spherical). Finally,
non-polar hydrogens,
for
of
removing
me
a
high
larger time-step to intégrale force fields
handled
explicit hydrogens
serious for
not too
a
some
are
atoms
becomes mixed
use a
explicidy, whereas
ail
included into united atoms (see Table 1.2). The
are
suppression
(this is
of
hydrogen
modelling of me hydrogen bond
metiiod, where polar (and possibly aromatic) hydrogens the odier
atoms are
and 30% in DNA. From
advantage
me use
of
often
form so-called united
of freedom, since
proteins
of view, this also offers die
since
to
usually
degrees
internai
no
hydrogen
bearing tiiem,
total atom number in
équations of motion. However,
and
of biomolecules,
atoms mat are
frequency C-H bond stretching motion die
directionality
no
simulating large Systems
included into the
molecular
force fields, the basic unit is the atom, which is
empirical
and
an
there
are
die loss of
linked to are cases
dipole
carbons) and
and
loss of
a
where the united atom
explicit inclusion
of ail
hydrogens may
required for a proper description of me System (Miiller-Platiie et al, 1992,
Kaminski et
al., 1994). Common force fields
atom). Thèse to
tiieir
(e.g.
are atoms
usually define
(or groups)
limited number of atom types
are
physically
(Hwang
et
information for
a
n-body interaction
spécifie system.
interaction term between dièse atoms,
The
n atoms a
irrespective to
chemically
to
united
(i.e. witii respect
force field to anotiier & Tirado-
facilitate the attribution of interaction
terms, while
assumption
one
(possibly
OPLS/proteins (Jorgensen
al., 1994), 65 in
Rives, 1988) ). The purpose of dièse atom types is function parameters to
and
alike. This number varies from
physical environment)
2 in CFF93/alkanes
a
which
generating die
molecular
is that the parameters s, for
of atom type a„ is
solely
determined
by
topology
an
n-body
the types of
tiieir environment, i.e.
s,
=
s,
(K
•
a
=
l-n})
Chapter
1
(1.5.1.2)
34 Such rules
are
called combination rules and
Depending
tiiey can
weakly (i.e. easily overriden)
be
preferred,
is to be
on
possible
Mark, 1992). One ofthe
die form of tables
of die atom type ofthe
of required parameters. For
also include atoms which
atoms and
the
van
of discussion (Section
environment
topology
editing
flexibility (van
offers
more
information
using
Gunsteren &
(and physically based) combination rules
of the interaction between two atoms is to each atom
given by
type. Combination rules for
next
complexity
function of die
as a
diat the
indicating
constituting
same
atom, which reduces
specified
atom
1.6.6.2).
Note diat in
principle
directly participating
example,
in the
covalendy
of the force field and, to
knowledge,
différent atom types
use
of few atom types has the
has
significantiy modified by
usually defined
two
are
our
advantage
to
of
never
still
by
die bonded
rapidly increase
been done. Instead, when
die type of
distinguish
are
interaction, but define die
bond type could be defined
a
amount
die combination rules could
bonded atoms. This would, however, very
is
be used
der Waals parameters, proper combination rules
are not
an atom
to
significantiy me
precisely.
die environment of
types. Thèse
parameter is
For
more
the
a
die type of rule,
strongly implemented. The former possibility
or
most well established
tables may include "wildcards",
matter
on
(bond, bond-angle, torsional dihedral angle and out-of-plane coordinates)
generally given in
irrespective
of the définition of
important part
of the molecular
product of die (point) charges corresponding
valence terms are
manual
law, where the magnitude
is Coulomb's the
génération
since
combination rules and
are an
die structure of die simulation program and
force field.
neighbouring
atoms,
die différent environments. The
simplicity and
ease
of
parametrization.
For
example, if four atom types are defined for carbon (C(sp3), C(sp2), C(sp) and C(aromatic)), only
10 bond types hâve to be
die environment is low.
parametrized,
but the
sensitivity
Twenty types would surely allow
influence of die chemical environment, but this would then as
much
1.5.3.
as
210 bond types, which may be
Expression
particles
imply
on
better for die detailed
the
parametrization
of
hard task.
for die classical Hamiltonian
As in die quantum energy of the
a
of die bond behaviour
to account
description
of
a
molecular system, die classical Hamiltonian (total
system) dépends simultaneously on
in the system. In
a
similar
manner as
electronic Hamiltonian is
approximated by a sum
classical Hamiltonian
be
can
approximated by
the coordinates and die momenta of ail
in Hartree-Fock calculations, where die of
a sum
Chapter
1
one
of
and two électron operators, the
n-body terms
35
3W(^,i)
E iVK&yvvty
-
]
E E 0
phase shift,
and
which
(1.6.3.1.2)
(n)ô,
plays
die
0,7t
=
same
rôle
as
sign
the
is\, in die first formulation. Since die slope of the potential has to vanish at 0 and n, die only possible values of (n)ô, are 0 and 7t. If (n)k4) is négative or '"'Ô, is 0, die term has a maximum for d) 0. If {°\ is positive or ) + "'k^ (1 cos2(b) +=£
bonds
bonds j>i
i
*„,(V-W-ty
This term is présent in CVFF and CFF93. Since k is
asymmetric B.
bond
Bond-angle
-
stretching
bond
around
a
given
coupling (two bonds j
with bond i)
atom
(1.6.3.1.1)
tiiis term favours
positive,
site.
involved in the
angle i)
(2)
E V., := E
angles
l
reproduce
This term is used in CVFF, CFF93, MM2 and MM3 to
frequencies stretched
or
bond-angle C.
and
Bond-angle
me
bond
compressed.
effects in strained molécules where
length
Since k is
a.6.5.1.2)
tonds j
positive,
bond
lengtiiening
a
vibrational
bond-angle
is
is favoured when the
is reduced -
bond-angle coupling (angles j sharing one
common
bond witii
angle i)
( 10)
£9e
({e(,e;};{er,e;,feee,,y})
=
E *»-.* (tf-W-e,)
E angles
i
This term is présent in CVFF, CFF93 and MM3. It is used to
frequencies for coupled bending D.
Torsional-angle
-
bond
modes, k may be
positive
or
coupling (central / peripheral bonds j
hb MM ; i*/.( °WwWfflW--i>
(1.6.5.1.3)
angles j
reproduce
vibrational
négative.
involved in torsion i)
=
,,,.,„
(l)or(2) dihedrals
(l.O.D.l.t)
E ty'-bj) [%>,
E i
cos
bonds j
, +
(1.6.6.1.13)
:
bufferedfunction n-m
^
= '
m m
l+ôl""" f 1+Y
[
l
L0..+ÔJ 6i/+ô
A buffered 14-7 energy function has been
-
g"+y
(1.6.6.1.14)
n~m
proposed (Halgren, 1992)
Ebur-n-m(lr,J}^y-WJ),Rmm(iJ)})= atoms
atoms
where n=14, m=7, ô=0.07 and rare
gas
is at
expérimental
(0.996;-1.0006),
(1.6.6.1.2).
The
and
(reduced)
reduced intercept,
y=0.12,
tiiese parameters
being
obtained from
a
best fit to
data. Note tiiat witii tiiese values of ô and y, die minimum of
Ç(i,j),
mus
Equation (1.6.6.1.14) nearly
curvature at the
minimum,K(i,j),
is in this
case
is 0.89, both close to the Lennard-Jones value.
Chapter
1
1^(0^)
satisfies the conditions in
79.6, and die
52
Figure
In
1 6 the
with parameters
reduced energy functions mentioned above
vanous
corresponding
of the Lennard-Jones
curvature
to a curvature at
the
minimum
function), except me 9-6
van
54) and die 14-7 buffered function (curvature 79 6) As interactions is short and
tiiey
will
play
van
(curvature
an
essential rôle
of 72
are
displayed,
(reduced units, the
der Waals function
only
(curvature
be seen, die range of dièse
can
for direct
neighbour atoms
der Waals functions
at minimum
12
=
36, except for the 9-6 function)
14
16
20
Reduced distance pl{
Figure
1.6: Représentation ofthe
van
der Waals interactions,
in a
reduced form,
corresponding
to
Equation
(16616) with m=6 and n=12 or m=6 and n=9, Equation (1 6 1 1 10) with m=6 and f(ij)=13 77, Equation (16 11 12) with a(ij)=6, and Equation (16 114) with n=14, m=7, 8=0 07 and y=0 12 Due to the
întnnsically
small
magnitude
He to -2 3 [kJ/mol] for Xe-Xe), the =
1
are
likely to
may of
of die énergies involved
divergences
affect die overall energy
course not
function below Qit
=
m a minor
way
m
1 wdl influence die are
=
condensed
be true for gaseous Systems On the other
system When electrostaUc effects
(e
-0 1
[kJ/mol] for He-
between die différent functions above q,s
phase Systems
density and compressibihty
of
a
condensed-phase
présent, the balance between dus steep
repulsion and
the electrostatic mteraction will be the déterminant part of packmg forces Since
Chapter
1
This
hand, the steepness of the
van
der
53
determining parameters), just
Waals parameters (e,Rmm,curvature effective parameters, forms
give
can
effective
as
atomic
will
gas-phase
sum
seems
up for
a
an
pairs,
of
are
adjustment, condensed-phase
energetical
state. Since die small
large number
charges,
virtually any one of the above functional
reasonable results. Of course, after such
of the
représentation parameters
diat
so
der Waals parameters may not be suitable anymore to
van
neighbours
they can be adjusted
a
give
a
proper
contributions for nearest
proper choice for the
primordial, and combination rules (Section 1.6.6.2)
e
and
R^
should be considered
carefully. 1.6.6.2. Combination rules Because the définition of N atom types Waals interaction parameter sets for atom which
dépend
homonuclear
sets
on
&
al., 1984, Halgren, 1992), R^,,,
extent,
A.
although
Géométrie
RnJfJ) The
following
often
formally
meansfor
two rules are
Jones function and any
C6(ij) or
B.
a(ij)
Géométrie The
mean
for
following rules
KmJ'V)
=
e
md to
and
be calibrated
by studying
Since die
are
die
expérimental
interchangeable
M^WJ)
=
Equation (1.6.6.2.1)
Ca(ij)
to a
large
for the
(1.6.6.2.1) case
JCn(i,i) Cn(jj)
=
and
e(ij)
équivalent
mean
for die
JC^JV)+*„.
otherwise
atomic cutoff (AT) truncation scheme whereas
(2.1b)
cutoff (CG) truncation scheme.
(AT) scheme first. The force
consider die atom based truncation
by differentiating the négative
on atom
i is
of the interaction energy (2. la) with respect to r„
to
SQ'
j,J*l
where
J
(2"la)
complementary Heaviside function
describes
Equation (2.1a)
obtained
AT
Rc reads
R„ is die distance between die corresponding charge-group
( 1
a
distance
l
respectively dénote
H^
describes
at a
permittivity of the médium between the charges (usually one for free
respectively belong,
centres.
located at atomic sites and that ail
straight cutoff (SC) truncation
a
4ne,/,/cc)
dH(x;Rc)/dx
-
is the Dirac delta function. The second
infinitesimally
^
+
VSC
the
on
an
simulating
the
corresponding
Vsc 0FF
+
V0FF
AT
(2.5)
92
The second term,
Rg,
shifts the
V0FF,
such that it vanishes at
R,..
potential
gives rise
It
distances from
the system limit of at
Rc,
(i.e.
very
a
R,., expression (2.5)
die work
large
H(r,j;Rc)
will
required to bring charges together
cause
in (2.1a)
function S is called
a
and die
Rc,
dynamics,
which
artifacts in simulations. An alternative is to
replace the step
a
smoother
switching
replaced by force
a
et
in
S(r1];R„Rc)
unity for r.jSR, et
term in
et
R,
,,j,i
interaction
by differentiating die négative
water oxygens,
-1
(proportional
dipole-dipole
I and J. If die centres of the
present simulations the
(van Gunsteren & Berendsen, 1987,
reduce artifacts of
interaction
with respect to r,. Since the Heaviside function
charge groups
to
_3
charge groups
this force acts
lSr \ 4tie e,
dépends
on
')
(proportional
RB,
a
where atom
Nœ is
k
as
given
this
to
£ tùu({ru,kel,lej))
charge
charge
of
The
on
atomic sites, in die
atom
and
we
i
U
-£-
Ru
CG
*
-
die total number of
belongs
r"3).
by
hâve
"
assumed to be
to one
(2.8b)
R%
,
vector
a
(2.8a)
origin, r=0, solely the 1=0
die 1=1 term may generate
a
term
field
95
in
Equation (2.8b),
as
far
as
so
that
only thèse two terms of the multipole expansion
MD simulation is concerned
potential (frVu obtained
with r=Ô" and
î^ï
-•= E"
where the j-sum
équations,
we
ail
runs over
e,-e,
(2-10a)
t2(t2~ei)
q>
-
charges qt (including qj inside die cutoff sphère of i. In dièse potential (2.10a) due to die spherically by
in the continuum
functions of
derivative of
r so
that it cannot be
dielecttic continuum, die
following
neutral
charge density
charge groups (water molécules).
sphères,
thèse two
Clearly,
expected
inside die
polarization
sphère
induced in
quantities
has to be
as
the
on
zéro
everywhere
case
of
In diis case,
the most narural choice for
a
a
be valid in die
outside of die cutoff.
neutral System
consisting
fOT the whole System may be calculated
v~
-
£/ •
"
0 1
where the second groups are
preserving
the
neutrality
of the cutoff
by reversibly charging
die atoms
one
after
respective cutoff sphères
4*?r V1 f 1 *+ tq>H(R-^ H1ltotl 1
E
sum runs over
E2
KRF
>
Rc
as
if the atoms
were
scheme,
new
one
water
and the full
to
as one
still at
a
avoid any
of die atoms
distance Rc,
would still like to préserve the
R^.
When two
charge
me
validity
a
distance
Rc,
ail
charge pairs
at a
distance
Figure (2. lc,d)
is illustrated in
of
(2.8)),
as
is
r1J>Rc
for die
=
neutrality
one
pair of atoms
1), but also possibly
could be
spécial
case
of die
but avoid interaction
groups I and J hâve at least
Rc (expressed as H(min{rkl,keI,leJ};R(.)
beyond Rc,
some at
artificially replaced Rg^R,..
at
This leads to
following energy expression
V
=
vborn
+
*W
+
i E 2 ,
ë,-e,
i —
can
will be a
a water
die continuum also when die central atom
Rc+d(0-H)
k
charge group
a
tiiis
distance
as
Figure (2.1b),
to
however, the hydrogen of the central
that of die continuum and
a
As
We
Rgp in (2.16)
>
(AT) truncation scheme described below
atomic
at a distance below
die
ry
This may reduce die effect, but not eliminate it.
sphère (dius
distances
opposite dipole.
of distances
would hâve to be considered instead of solely the Born and
R,^
D. Treat any interaction at distances
cutoff
occurrence
completely inside die
an
charge-charge distances larger than R^.
belonging to it. done in
if it had
and the continuum.
as
Equation (2.8)
in
term.
occurrence
C.
molécule is as
O-H distance in
molécule would be off-centre witii respect
dipolar
water
gap between the furthest
hydrogen, Figure (2.1b).
expansion
a
interacts
water.
RRF2Rc+2d(0-H), a
happen
R„f beyond Rc. Ideally, according
A. increase
B.
even
région (lower left of the picture),
+
——
r„
E
j%
H(min{ru,kel ,lsJ} ;RC) "
—+C
\H(r;Rr)
e.-e.
ri
i +
-L
c
+
—ï—!
W,
4itE0E1
R. i £_+H
AT
(2.18)
[\-H(r ;« )]
seen from the Figure, the multipole moments of water molécule pairs close to Rc strongly affected by tiiis procédure. The leading effect can be roughly described as
be
progressive removal,
at
intermolecular distances close to
Chapter
2
Rc,
of the component of the two
100
dipoles along die intermolecular vector. Provided tiiat ail charge groups of
q1q]H(min{rk,keI,leJ};R )
ail
over
charge pairs
from différent
Noting additionaily diat H(min{rkl,keI,leJ};R(:)H(r1J;Rc)
=
are
charge
neutral, the groups is
Hir^R,.), Equation (2.18)
sum
zéro.
can
be
rewritten "AT "AT
,
a,ai
^EE
,y,i 4ite0E1
,
2
AT
E.-E
L r„
v
VSC
+
V '
DRF
(2.19)
*w
2ë,+E i R;
j£
2e2 ei +
V
+
'
and thus this scheme turns out to be based
OFF
on
atom-atom
an
distance criterion for
Equation (2.19) hâve been denominated VB0RN
truncation. The contributions occurring in
(Born term, Equation (2.11)), VSELF (interaction of a charge group witii self-induced dipolar reaction-field, Equation (2.14)), V^ (straight truncation, Equation (2.1a)), VDRF (dipolar
reaction-field) and VOFF (offset term). When e^e,, Equation (2.19) becomes Equation (2.5), and the SC/AT scheme is thus condition.
Apart from the term
a
particular case of the RF/AT
Vs^»
which accounts for the
groups, die interaction energy vanishes at ry r, and rt in
VBORN
and for
rigid charge
=
scheme
corresponding to this
self-energy of isolated charge
Rc. Again ignoring the dependence of R„ on
groups, the force
corresponding to
this interaction
is
E
ai°j
2(E2-ei)
4nE„e.
2e2+El
r
] »W AT
[f
2e,
Since the delta function term is
Additionaily,
R,
R
always
V6!
Rc
2e2+6l
/&
die total force vanishes at r,
R^R,.,
so
that cutoff artifacts
correction which is known
to
are
avoided to
a
hâve bénéficiai effects
a
large on
(2.20)
àW
zéro, energy conservation is satisfied
in the limit e2»e, and when
tiius, die interaction defined in (2.19) implicidy contains function,
]
physically
=
reasonable
exactiy. R,.,
and
shifting
extent, and the reaction-field
die dielectric
properties
of
liquids,
is included. Note diat for both the RF/CG and RF/AT from
R^
In botil cases,
RrfsRj.
seems
schemes, Rw may be chosen independently
reasonable both to account for the thickness of charge
groups at the cutoff distance and to limit
(RF/CG)
Chapter
2
occurrence
of
inter-charge
distances
101
whereas
beyond Rgp occurrence.
at
R^
if the choice
Rc by 0.1 [nm] simulated
Rrf^Rc is made.
were
Both
attempted hère of
properties
Comparison
2.3.3.
will increase
Riu7t_0,M,2A.
where r„(t) is the location ofthe oxygen atom of molécule between two stored frame
?0(t+At)
trajectory
frames
selected
die
was
as
average diffusion constant, D
linear
régression
corresponding
line
the
same
the
régressions
where
=
5„(t)
is
by monitoring,
T_0,Al,2At
2At
L_
(2.26)
tin_,
expression may vanish,
in which
case
c„(r)
The average distribution, c(r), and its standard déviation, o(c(r)), were
was
,A
^
For low values of r, the denominator of this
arbitrarily set to zéro.
the
of (fourty) water molécules a, die function
Pl '««f*
)r-0.
P*a
permittivity, ê(r), obtained
B.
a
+
is die number of molécules
dépendent dielectric
function
a
1
1
size
spherical volume.
a
=
([ X„(r)
using Equations
Appendix
in
that die unit cell is
with the volume of the
molécules
1
with
given
also
the Fourier transformation of
on
assumes
evaluated in
evaluate
=
are
truncated octahedral). This formula need not, however,
properties
to
e„(r)
where
is based
in three dimensions and
be valid when the fi and E tensors
used
permittivity s^ of me
reaction-field correction. Values of K calculated
(B. 19,21,24) with the present values of R,., Rrp and t^ convolution
quantities.
(see Appendix B). It may dépend on the cutoff radius, Rc,
die simulation
during
die standard déviation of thèse
calculating
constant
die simulations intoa number of
by dividing
obtained
e
.,.-
for which r„psr
at
time
and its standard déviation,
by averaging
the
curves
£„(r)
using the convention described above of including Chapter
2
a
over
t.
The average
a(E(r)),
over
core-
the set
the set of molécules
single periodic image into
110
averages,
one
criterion for
has
from
E„(rïRext)=E
comparison
The dielectric relaxation time water
molécules a, and for
moment
Equation (2.29b)
for any molécule
a.
I(r)
is
cores
was
estimated
by monitoring,
of increasing radii
r
for
a
séries of
,Ma(r,t+x).Ma(r,x) t V W^,
Afa(r,T)
=
p
E P. r„p
Core-size
dépendent
inverse of the
slope
function of t at fixed Estimâtes of
Finally, e^WTrfe],
states
(fourty)
around water molécule a, the
*.V.*)
with
van
useful
dipole
autocorrélation function
=
&
a
between différent electrostatic schemes.
TM(r)
r
régression
between t
were
=
0.5
[ps]
TM(r)
fitting
lines
averaged
and die time of first
properties,
le.
estimated for five of die simulations
calculated
were
the
obtained for successive intervais in
self-consistent dielectric
were
r
dielectric relaxation times, of linear
(2.31)
?„(T)
r
occurrence
In
négative
$(r,t)
as a
$(r,t)
0006
1 181
1 14
0 538
113
0 680
113
0 979
Q, [kJ/(mol ps) 0 480
[g/cm3]
ATM
298 1
AT/2
45
300 2
CG/4
300 8
T[K]
R„, see also Figure (2 7), D diffusion constant, Table 2 3, corrected the followmg values are corrected by perturbation to dielectnc self-consistency (convergence was reached wittun at most 20
scalar dielectnc permitûvity, defined
from Table 2
e
properties Properties calculated for the simulation box and summary of orner important properties, G of me T tensor, Equation (2 28), GK fmite System Kirkwood G factor, defined as Tr[£-TJ,
Table 2.4: Structural and dklectric
699±13 7(68 3±9 5)
12 5
66
114
E«y Ek £jx
C
58 9
59 9
661
E«x *-yy CM
649
2 442±0 19
2 602±0 25
GK
—
—
AV^IU/mol]
-017
-013
AV», [kJ/mol]
83 6
5 33
D[10'm2/s] 5 26
36
40
Corrected:
0 054 0 005
0 781
63 3±6 5 (63 1±6 2)
-04
605
XM[Psl
X, fPSl
CG/5 0 753
2 260±0 15
-0008
0 012
-0 010
0 743
27
(68 1±10 6)
12 1
847
0006
0 002
0 087
0 897
29
70 0±17 2
87
119
C
603
651
2 374±0 23
-0028
0 014
0 043
0 055
0061
E«* eyy Em
r
r
CG/2
0 734
0 815
•=„ E„ C„
GK
r
r
G„GffG„ O^ G„ Gp
123
average square
r
r, cos0
f
r,
with
(A.7a)
based truncation scheme (see
r,0 and 0. When (A.7a) is
ï and
hâve
*>
(A.4b) through
spherical by the
r,R-2l-l
introduced after die second summation in (A.4b,5b) to avoid the
range of
be calculated from
Defining
a
expressions also lead
the
(2.8a)
(/+l)e2+/e,
(A.3b) requires
of
use
(/^1)(E2-E1)
^^Ï'^O'
and
Note diat die
%
4716^,
-
-J-
)
(A.9b)
r
where matter
©^©(f^r). Note that when 0=0 or 0=ti, ëx cannot be defined uniquely. This does not since the ex component vanishes
(sin0=O
and
ôcb^ryô©^,
see
(A. 11b)). From
(A.4b) and using the relation
(l-x2)
J- P,[x]
=
-
lxP,[x]
+
IP, ,M
(A.10)
wefind
l*RF&=t t'icjr'-iptcotd] or
j
x
-
*"
=
(AUa)
M
v^ [Ê''c,/'''sin"1e(cose^,[cos0]-/', ,[cos0]) if 0*0,71 r\'n 10
otherwise
Chapter
2
(A-11
135
Note that die 1=0 term vanishes in both
Consequently, the prime
after the second summation
rj=0. Finally, ôdjgj^rVôO
when
and has been omitted in die summations.
cases
now means
0=O,Tt
vanishes for
3cos0/30 vanishes. Combining (A.9) and (A. 11) leads
1
-
c;,
4^
l
Substituting
Oa-^Rup. Finally, is not
a
(A.4b)
case
suitable centre for
cannot be used.
r
expansion
Equation (2.8b)
leads to
along
die
to =
any Une
o^r) in a séries
of
lim
can
thus be used also for r=0,
r°P0[cos 0] prime
prime
is one,
just
as
well
the
after the summation
reasoning
applied
may be
vanish in (A. 12) when
r
(8a)
sign
to
range of
origin
of Legendre
term
origin
angle
©(r^r) with the ?
E^r)
in the
can so
one can
and
write
vector. In
except die 1=0 term.
zéro
j-sum
has
r
this
Equation
=0. The
meaning
of the
The
meaning
of the
The
same
case.
in (2.8a) is extended to include this
goes towards 0,
sphère
polynomials,
(A. 13)
is extended to include mis
and it
validity
of the
die 1*0 terms and widi die convention that
excluding a
a
«„
s
r
**
6
forming
when
as
after the summation sign in
(A.12)
RR
s
r
(r) R(r'-r) P(f)
where R is the interaction tensor describing die
]
dipole-dipole interactions
Chapter
2
(B.6)
in the system. If
139
die electrostatic
used in the simulation is the
entity
is exact. If this
Equation (B.6)
entity is
charge (as
the
(B.6) will only include die (leading) dipolar groups,
for
(Ewald),
to
a
truncated at
by
a
depending
given
a
distance
can
be recast
R(v)
I(v) is the
correction tensor,
m
c
=
die
following
approximations
charge
one can see
dérivation
used in the
—{— vvi 47tE0Ej
=
Âv) l I(v)
between distances
In ail thèse cases,
+
R,
E(v),
C ]
and
R,.
where
v
(B.7)
interaction tensor and Q is
for which
me
reaction-field
are
v^
—!— 47tE()Ë1
=
V
zéro
tensor.
as
dipole-dipole
exact
expressions
y3
rT»
.
=
—!— 4lte0ei
-5
[3vYvô-ôy6v2]
v5
(B.8)
1
_J_J^ÙJ_1
=
47te„e, o
In
on
switched to
Rj,
reaction-field "interaction"
short notation for r'-r,
where
model), Equation
deal with the electrostatic interactions, the tensor may be either almost exact
or/and corrected is
Stockmayer fluid),
of the interaction between neutral
term
from symmetry arguments diat die inclusion of thèse terms in the would not affect die resuit. In both cases,
simulation
a
for die SPC water
higher-order multipoles. However,
the contribution from
neglecting
dipole (as
\
2e.+e, 2
Equation (B.7), f(v)
r
is
a
3
txRF
1
function
characterizing
the
(isotropic)
truncation
or
switching
ofthe interaction at selected distances. In the is
equal
to one, at least in
functions
are
a
finite
following, we will assume that this function neighbourhood of v=Ô (this may not hold when shifting
used). In die spécial
orthogonal vectors,
i. e.
an
case
where Q is infinité and
infinité collection of periodically
U, Equation (B.6), which has the form of
a
periodic along
three
replicated rectangular unit cells
convolution
intégral,
may be Fourier
transformed
P(k)
Since
=
Ë0is homogeneous in
c, (
£
space,
-
I )
we
[Eo(k)
hâve
Chapter
2
+
R(k) P(k)]
