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Research Collection

Doctoral Thesis

Classical molecular dynamics simulations at different levels ofresolutionforce field development and applications

Author(s): Winger, Moritz Christoph Ludwig

Publication Date: 2008

Permanent Link: https://doi.org/10.3929/ethz-a-005787304

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Diss. ETH No. 18076

Classical molecular dynamics simulations

at different levels of resolution: force field

development and applications

A dissertation submitted to the

ETH ZURICH

for the degree of

Doctor of Sciences

presented by

MORITZ CHRISTOPH LUDWIG WINGER

Dipl. Chem. ETH

born September 29, 1981

citizen of Austria

accepted on the recommendation of

Prof. Dr. Wilfred F. van Gunsteren, examiner

Prof. Dr. Ulrich W. Suter and Dr. Lorna J. Smith, co–examiners

2008

for Ilga

Acknowledgements

I would like to thank Wilfred van Gunsteren for giving me the opportunity to work in his re-

search group. It was a pleasing experience in both, professional and private ways. Sharing and

discussing ideas with Wilfred was always inspiring, and he knew how to make ETH an enjoyable

place to work. Next, I would like to express my gratitude to Jolande, Wilfred’s wife, for taking

such good care of us and Wilfred, and teaching us how to present. I am very thankful to Daniela,

the IGC secretary, for sharing many cups with me, and inviting me into her office when I needed

someone to talk. She always cheered me up and had an open ear for my problems. Talking to her

kept me happy during my PhD years. To Philippe Hunenberger I would like to give my thanks for

scientific discussions, and advice on my projects that made my work much clearer. Warm thanks

go to my co-examiner Dr. Lorna J. Smith for traveling from England to attend my defense. A

big thank you to all the people from the IGC group that made life at work so much easier and

more fun. I think the communication in the group is very good and if one has a problem, there is

always someone to help out. I think this is very important for successful science and this works

well in our group. Thanks also to all md++ and GROMOS++ developers. This thesis is dedicated

to my mother, Ilga. She is close to me in any situation I am in. She is a big part of this as well,

without her I would not be where I am today. Finally thank you to my brother and all my friends

who lived my life with me outside ETH.

i

ii Acknowledgements

Contents

Acknowledgements i

Zusammenfassung vii

Summary ix

Summa xi

Publications xiii

1 Introduction 1

1.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Classical MD simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Coarse-grained simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Applications of MD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Consensus Designed Ankyrin Repeat Proteins 13

2.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Material and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

iii

iv Contents

3 Molecular dynamics simulation of human interleukin-4: comparison with NMR

data and effect of pH, counterions and force field on tertiary structure stability 29

3.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Material and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3.1 Molecular dynamics simulations . . . . . . . . . . . . . . . . . . . . . . 32

3.3.2 Structural refinement with nuclear Overhauser effect restraints . . . . . . 35

3.3.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4.1 IL-4 at pH 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4.2 IL-4 at pH 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4.3 Effects of counterions at pH 6 . . . . . . . . . . . . . . . . . . . . . . . 44

3.4.4 Effects of counterions at pH 2 . . . . . . . . . . . . . . . . . . . . . . . 46

3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4 Force-field dependence of the conformational properties of α,ω-dimethoxypolyethylene

glycol 61

4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62


4.3.1 Molecular Dynamics Simulations . . . . . . . . . . . . . . . . . . . . . 64

4.3.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5 On the conformational properties of amylose and cellulose oligomers in solution 81

Contents v

5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82


5.3.1 Molecular Dynamics Simulations . . . . . . . . . . . . . . . . . . . . . 83

5.3.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.4.1 Double-stranded amylose in pure solvent . . . . . . . . . . . . . . . . . 85

5.4.2 Double-stranded amylose in ionic solution . . . . . . . . . . . . . . . . . 86

5.4.3 Single-stranded amylose and cellulose in ionic solution . . . . . . . . . . 87

5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6 On using a too large integration time step in molecular dynamics simulations of

coarse-grained molecular models 97

6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98


6.3.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7 Outlook 117

vi Contents

Zusammenfassung

In dieser Arbeit wird Molekuldynamik-Simulation angewendet um verschiedene Probleme in

der Strukturbiologie und der physikalischen Chemie zu untersuchen. Im ersten Kapitel wer-

den die Grundlagen der Molekuldynamik (MD) und die wichtigsten Gleichungen, die hier ihre

Anwendung finden, vorgestellt. Ausserdem wird der Einsatz von MD an mehreren Beispielen

gezeigt. Im zweiten und dritten Kapitel werden Biomolekule untersucht und deren Stabilitat

in Abhangigkeit von der Ladung der Seitenketten, dem pH-Wert und dem Kraftfeld beobachtet

und registriert. Im zweiten Kapitel zeigt sich, dass die Ladungsverteilung in einem Biomolekul

grossen Einfluss auf die Sekundarstruktur haben kann. Im dritten Kapitel wird die Auswirkung

des pH-Wertes, d.h. des Protonierungszustandes der Seitenkette, auf die Stabilitat eines simu-

lierten Molekuls dargestellt. Es zeigt sich, dass die Ionenstarke von Losungen ebenso Einfluss

auf die Proteinstabilitat hat. Im vierten Kapitel wird ein einfaches Polymer, das Polyethylen-

glykol, vorgestellt. Man weist auf die Wichtigkeit der Auswahl von Kraftfeldparametern hin,

weil das Verhalten der Losung in verschiedenen Losungsmitteln des untersuchten Molekuls

durch die Wahl der Parametersatze weitgehend beeinflusst wird. Das funfte Kapitel behandelt

den Unterschied zweier Zuckerpolymere, Amylose und Zellulose, in Bezug auf ihre Losungsstruk-

tur. Daruber hinaus wird in diesem Kapitel der Einfluss von Wasserstoffbrucken auf die Sta-

bilitat der Helix der Amylose untersucht. Das sechste Kapitel behandelt die grundlegenden

Merkmale der Parametrisierung in einem oft angewendeten MD-Simulationsverfahren, namlich

den grobkornigen Simulationen. Man untersucht die physikalisch-chemischen Eigenschaften als

Funktion der cut-off-Distanz und der unterschiedlichen Zeitschritte. Dabei werden bei der Wahl

eines grossen Zeitschrittes, der die Newtonschen Bewegungsgleichungen ins Spiel bringen soll,

vii

viii Zusammenfassung

hohe Energieflusse und Temperaturverlust registriert. Es werden die moglichen Ursachen fur

den Temperaturverlust erlautert und dargelegt, wo der grosse Zeitschritt zum Problem wird. Im

letzten Kapitel wird ein kurzer Ausblick auf die Fortsetzung der in dieser Arbeit vorgestellten

Forschung gegeben. Man diskutiert eine mogliche Parametrisierung einer Funktion der poten-

tiellen Energie, die Interaktionen in grobkornigen MD-Simulationen genauer beschreibt.

Summary

In this thesis molecular dynamics (MD) simulation is used to study different problems of struc-

tural biology and physical chemistry. In the first chapter the principles of MD and the most

important equations that MD uses are given, and some applications of MD are shown. In the

second and third chapter, biomolecules are studied, and their stability dependent on charge of

side chains, pH, and force field is monitored. The second chapter shows that charge distributions

throughout a biomolecule can have a great influence on secondary structure stability. The third

chapter shows the effect of pH, i.e. side-chain protonation state, on the stability of a simulated

molecule. Ionic strength of solutions is shown to have an effect on protein stability as well. In

Chapter 4 a study of a simple polymer, polyethyleneglycol, is presented. Here the importance of

the choice of force-field parameters is pointed out, since solution behaviour in different solvents

of the investigated molecule is largely affected by the choice of parameter set. Chapter 5 treats

the difference of two sugar polymers, amylose and cellulose, in terms of solution structure. Fur-

thermore, in this chapter, the influence of hydrogen bonds on helix stability of amylose is studied.

The sixth chapter is about a very basic parametrisation feature of a widely used MD simulation

technique: coarse-grained simulations. Physical chemical properties as function of cutoff radius

and timestep are investigated, and high energy fluctuations and the loss of energy are found where

a large timestep is used to integrate the Newtonian equations of motion. This chapter shows the

possible reasons for energy loss, and identifies where the large timestep becomes a problem. In

the final chapter a short outlook on the continuation of the research described in this thesis is

given, the possible parametrisation of a potential energy function that describes more accurately

interactions in coarse-grained MD simulations is discussed.

ix

x Summary

Summa

In hoc opere utitur simulatione dynamicae progressionis molecularis ad exquirendas varias quaes-

tiones biologiae structuralis chemiae physicaeque. In capite primo principia dynamicae pro-

gressionis molecularis (MD) et aequationes gravissimae hic adhibitae ostenduntur, in pluribus

exemplis demonstratus est praeterea usus dynamicae progressionis molecularis. In capitibus se-

cundo tertioque biomoleculis examinatis observata et enotata est stabilitas earum in obsequio

erga onera electrica serierum obliquarum, erga numerum pH, erga propagationem virium. In

capite secundo videtur distributio oneris electrici in biomolecula erga structuram secundam mul-

tum valere posse. In capite tertio descriptus est effectus numeri pH, id est status protonationis,

seriei obsequiae erga stabilitatem moleculae simulatae. Videtur liquorum vis particularum elec-

trice oneratarum erga stabilitatem proteinii multum valere. In capite quarto polymerum simplex,

Polyethylenglykol, descriptum est. Indicatur selectio condicionum propagationis virium inter-

esse, quia selectio gregis condicionum ad statum solvendi in variis mediis solventibus molec-

ulae examinatae late momenti habet. Caput quintum tractat discrimen duorum polymerorum

saccharorum, Amylose et Cellulose, referens ad structuram liquorum, exploratum est praeterea

momentum pontium hydrogenii ad stabilitatem helicis Amylosis. Caput sextum tractat propri-

etates principales parametrisationis in actione late exercita simulationis dynamicae progressio-

nis molecularis, id est simulationes crassi grani. Examinatae sunt qualitates physicae chemicae

quasi functiones intervalli interclusi et variorum graduum temporis. Ad inserendas aequationes

motuum Newtonii gradu magno delecto cernuntur fluvii energiae et damna caloris. Enucleatae

sunt causae damnorum caloris et repertum, ubi gradus temporis magnus quaestionem affert. In

capite ultimo datur prospectus continuationis quaestionum in hoc opere ostentatarum. Disputatur

xi

xii Summa

de parametrisatione possibili functionis energiae potentialis, quae interactiones in simulationibus

dynamicae progressionis molecularis crassi grani accuratius describat.

Publications

This thesis has led to the following publications:

Chapter 2:

Moritz Winger and Wilfred F. van Gunsteren,

“Use of molecular dynamics simulation for optimising protein stability: Consensus designed

ankyrin repeat proteins”

Helv. Chim. Acta. (2008) in press

Chapter 3:

Moritz Winger, Haibo Yu, Christina Redfield, and Wilfred F. van Gunsteren,

“Molecular dynamics simulation of human interleukin-4: comparison with NMR data and effect

of pH, counterions and force field on tertiary structure stability ”

Mol. Sim. 33 (2007), 1143-1154

xiii

xiv Publications

Chapter 4:

Moritz Winger, Alex H. de Vries, and Wilfred F. van Gunsteren,

“Force-field dependence of the conformational properties of α,ω-dimethoxypolyethylene gly-

col”

Mol. Phys. (2008) accepted

Chapter 5:

Moritz Winger, Markus Christen, and Wilfred F. van Gunsteren,

“On the conformational properties of amylose and cellulose oligomers in solution”

Int. J. Carb. Chem. (2008) accepted

Chapter 6:

Moritz Winger, Daniel Trzesniak, Riccardo Baron, Wilfred F. van Gunsteren,

“On using a too large integration time step in MD simulations of coarse-grained molecular mo-

dels”

Phys. Chem. Chem. Phys. 11 (2009), 1934-1941

Chapter 1

Introduction

1.1 General considerations

Different approaches can be used to investigate a given phenomenon. The most common one is

the experiment, a technique that has been used by humans for a long time. Through experiments

the observer can get a true representation of the process of interest, since an experiment is per def-

inition carried through under controlled and reproducible circumstances1. Based on experiments

hypotheses can be concluded, and laws and theories can be formulated. The second approach

to investigate a phenomenon is a simulation. A simulation is the imitation of a real system, that

functions1. In other words reality is modelled in a representative system. Simulations are applied

when conducting experiments is not possible, unethical or tedious and time consuming. In most

cases it is not possible to simulate instead of carrying out experiments, but it has become a more

and more popular way to succesfully avoid experiments. A simple example is the gymnast that

practices his moves first on soft underground or into the water before he takes them to the hard

floors of the gymnasium. He ”simulates” the real conditions without having to fear that he will

break a bone and risk his career as a sportsman. Under the circumstances of a competition, that

means real wooden floor, and the crowd that watches the performance, the gymnast will still have

to prove that he is able to succesfully complete his routine. If the simulation conditions are accu-

rate enough, the experiment, that is the competition, will be a success. We can see, experiments

1

2 Chapter 1.

are still needed to validate and confirm results gathered from simulations.

Computer simulations have become a very frequently used tool, not only in science, but in

many different fields, such as risk management, stock-market prediction, the development of

cars and planes, drug discovery and many more. This work focuses on computer simulation

of molecules using fundamental equations that describe interactions between atoms. A brief

introduction into molecular dynamics (MD) simulations will be given in the following section of

this chapter.

The projects described in this thesis use different MD simulations of different model resolu-

tions: In Chapters 2-5, a resolution has been chosen, in which each atom - apart from aliphatic

hydrogen atoms - represents a point in cartesian space, in Chapter 6 a less detailed resolution,

where one point represents a collection of atoms, has been chosen. Introductions to the different

model resolutions will be given later in this chapter.

1.2 Classical MD simulations

In this thesis the technique of classical molecular dynamics simulation2, or simply MD simu-

lation has been applied to study biomolecules and other, simpler molecules, at different model

resolutions. This section will give a basic introduction to the equations that the method of MD

relies on. Some other principles that are important for the understanding of this thesis will be

explained in greater detail.

The first thing that might be important to mention is that we do classical MD. That means

quantum effects are not considered, or only in a mean-field manner. Everything that underlies

the laws of quantum physics, e.g. the motion of electrons, is neglected. The inaccuracies arising

from that have to be corrected for. This is done by the parametrisation of a so-called force field,

which governs the main interactions between particles in MD. A force field consists of data

specifying charges of atoms, van der Waals parameters, bond lengths, bond angles, and more.

Different parameter sets are available3–11, throughout this thesis specific sets of parameters have

been used: those of the GROMOS force field10, 11.

In a molecular dynamics simulation we are interested in the interaction of atoms, which might

1.2. Classical MD simulations 3

be connected through chemical bonds. Each atom is represented by a sphere with position vector

r. The potential energy function that is calculated in an MD simulation,

U(r) = Ubonded(r)+Unon−bonded(r)+Uspecial(r), (1.1)

is only dependent on these positions. As we can see the potential energy function of our system

consists of three terms. The first term is the bonded term and it considers interactions within a

molecule. It comprises bond stretching, angle bending, torsional-angle rotation, and improper

dihedral angle distortion. The latter is responsible for keeping the proper hybridisation type for

particular groups of atoms. When deviating from an ideal reference value of a length or an angle

in these terms, the potential energy rises,

Ubonded(r) = ∑bonds

12Kb(b−b0)2 + ∑

bondangles

12Kθ(θ−θ0)2 + ∑

improperdihedralangles

12Kξ(ξ−ξ0)2+

∑dihedralangles

Kϕ[1+ cos(mϕ−δ)](1.2)

The letters b, θ, ξ, and ϕ are the bond lengths, bond angles, improper dihedral angles, and

torsional dihedral angles. The variables with a 0 as subscript are the reference or ideal values

and are parameters of the force field. Generally all interactions have a harmonic functional

form, except for the torsional dihedral-angle term which has a trigonometric form. The labels K

indicate force constants.

Non-bonded energies are calculated using two terms, the first one being the Lennard-Jones

term12 describing the van der Waals interaction13, and the second one being the Coulomb term14

dealing with the electrostatic interactions of partial charges of the atoms. The former term de-

scribes atomic repulsion due to atom-atom overlap and the attraction due to London dispersion

interactions15,

Unon−bonded(r) = ∑atompairs

(C12

r12 −C6

r6 )+ ∑atompairs

qiq j

4πε0ε1r. (1.3)

In Equation 6.3, r is the distance between the two atoms the non-bonded energy is calculated

for, 4πε0 is a constant, and ε1 denotes the relative dielectric permittivity. C12, C6, qi, and q j are

force-field parameters.

4 Chapter 1.

The third term in Equation 6.1 is optional and might represent an external potential, such as

a restraining potential, as is applied in Chapter 3 of this thesis. There, in the first 100ps of the

simulations atomic distances are restrained according to experimentally observed NOE distance

bounds between hydrogen atoms in a protein,

Vdr(ri j) =12Kdr(ri j− r0

i j)2 if ri j > r0

i j

0 if ri j ≤ r0i j

. (1.4)

Molecular dynamics simulation not only describes the energies of different states of a system, it

also describes the evolution of atomic coordinates in time. A trajectory is generated consisting of

positions and velocities. In order to propagate the atomic coordinates an integration of Newton’s

second law16 is necessary,

md2rdt2 =−∇U(r), (1.5)

where m denotes the particle’s mass and f = −∇U(r) the conservative force on the particle.

In MD simulations the integration is done numerically, for which there are several methods

available17–19.

The method would produce an ensemble with a fixed number of particles, a constant volume,

and constant energy. Since experiments are rather carried out under constant pressure or temper-

ature, it is also possible to generate these ensembles by applying methods that keep pressure and

temperature constant20–27.

1.3 Observables

When a MD simulation has been completed, it might be of interest to compare the collected data

with data from experiment. This section sketches how to calculate properties from simulation

trajectories.

In an experiment the measured value of an observable or property Q is the average < Q >meas

over time and space of Q that is produced by different molecular conformations. In order to get

the right value, Q has to be weighted by the probability P of a conformation to occur integrated

1.4. Coarse-grained simulations 5

over momenta and positions,

< Q >=Z Z

Q(p,r)P(p,r)dpdr. (1.6)

In MD it can be very difficult to sample all possible states within a single, finite simulation.

However, if configurations that are relevant for the average are properly sampled, < Q > can be

calculated from this finite set of configurations. The ergodic hypothesis assumes that the relevant

configurations are sampled during an MD simulation that is long enough. Averages are then

calculated with

< Q >MD= limτ→∞

1τ

Zτ

0Q(t)dt, (1.7)

with τ being simulation time and t being time. Calculating properties is what makes comparison

to experimental data possible, and thus allows validation of the results of an MD simulation.

1.4 Coarse-grained simulations

Coarse-grained MD simulations generally use less detailed resolution in their models, that means

a group of atoms is represented by one particle. This allows to simulate bigger systems on longer

timescales, since less interactions have to be calculated.

Popular coarse-grain models28–42 use a four-to-one mapping, i.e. four heavy atoms are rep-

resented by a single interaction center. For example28 four aliphatic carbon atoms or four water

molecules become one interaction center, four main types of interaction sites, polar, nonpolar,

apolar, and charged are distinguished, and the form of the interaction function is the same as

for all-atom MD simulation, a Lennard-Jones term as described in Equation 6.3 for non-bonded

interactions, and an additional Coulomb term like in Equation 6.3 for interactions involving

charged sites.

Bonded interactions are treated similarly as in all-atom MD simulations, bond- and angle

potentials are described by weak, harmonic functions,

Ubond =12

Kb(b−b0)2 (1.8)

6 Chapter 1.

and

Uangle =12

Kangle(cosθ− cosθ0)2. (1.9)

Simulation parameters are also different from all-atom simulations, most important an integration

timestep of 50 fs is used28 compared to a timestep of 2 fs that is used in standard all-atom MD

simulation. This issue will be discussed in greater detail in Chapter 6 of this thesis.

As stated earlier coarse-grained simulations are particularly useful for large systems on long

timescales. This kind of simulation gives a semi-quantitative picture of how a large system

behaves over time. Applications are the simulations of lipid bilayers and their self assembly,

membrane proteins and many more, the focus still remaining on large systems.

1.5 Applications of MD

MD simulations are often used to simulate biomolecules, such as proteins. In many cases NMR

or other structural data is available, that makes comparison to experiment possible. Not only

biomolecules, but polymers in general can be simulated, such as sugar-oligomers, or simply

polyethers. In all cases using an appropriately parametrized force field is an important issue.

Polymers can be broken down in simple molecules that are then parametrized to reproduce ex-

perimental data, such as density, heat of vaporisation, and free energy of solvation. Once a proper

force field has been found, the polymers can be put back together, and simulation will show, if

the set of parameter is also valid for the larger molecule.

MD allows to study interactions on a microscopic level, such as the influence of ions on

the stability of a biomolecule, and their coordination behaviour towards a solute. Techniques

like selective deprotonation or protonation of side chains can be applied to investigate protein

stability as a function of charge distribution along the molecule.

The number of applications of MD simulation is large, and with increasing computer power

this number will become even larger.

1.6. References 7

1.6 References

[1] Miriam-Webster Online Dictionary. Miriam-Webster Inc., 2006.

[2] D. C. Rapaport. The Art of Molecular Dynamics Simulation. Cambridge University Press,

New York, 1997.

[3] P. K. Weiner and P. A. Kollman. AMBER: Assisted model building with energy refine-

ment. A general program for modeling molecules and their interactions. J. Comput. Chem.,

2:287–303, 1981.

[4] B. R. Brooks, R. E. Bruccoleri, B. D. Olafson, D. J. States, S. Swaminathan, and

M. Karplus. CHARMM - A program for macromolecular energy, minimization, and dy-

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1.6. References 9

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12 Chapter 1.

Chapter 2

Consensus Designed Ankyrin Repeat

Proteins

2.1 Summary

In earlier work two highly homologous (87% sequence identity) ankyrin repeat (AR) proteins,

E3 5 and E3 19, were studied using molecular dynamics (MD) simulation. Their stabilities were

compared and it was found that the C-terminal capping unit is unstable in the protein E3 19,

in agreement with CD-experiments. The different stabilities of these two very similar proteins

could be explained by the different charge distributions among the AR units of the two proteins.

Here another AR protein, N3C, with yet another charge distribution has been simulated using

MD and its stability was analysed. In agreement with the experimental data, the N3C structure

was found to be less stable than that of E3 5, but in contrast to E3 19, secondary structure was

only slightly lost, while structurally N3C is closer to E3 19 than to E3 5. The results suggest that

a homogeneous charge distribution over the repeat units does enhance the stability of design AR

proteins in aqueous solution, which however may be modulated by the bulkiness of amino-acid

side chains involved in the mutations.

13

14 Chapter 2.

2.2 Introduction

Ankyrin repeat (AR) proteins are involved in protein-protein interactions in most species1–4.

The number of identified AR’s runs into the ten thousands. They are a part of thousands of

proteins. Usually they consist of 33 amino acids, each AR forming a structural module (β2,α2)

consisting of a β-turn, followed by two antiparallel α-helices and a loop connecting to the turn of

the next AR. The AR architecture allows for modification of the size and character of the binding

surface to a target protein with the aim of high-affinity binding. X-ray and NMR structures

of AR proteins have offered structural insights into the molecular basis for their wide variety

of biological functions. Molecular dynamics simulation studies may provide a more detailed,

structural, dynamic, and energetic insight into their properties.

In a previous simulation study by Yu et al.5 the highly homologous consensus designed

ankyrin repeat (AR) proteins E3 56 and E3 197 (Figure 2.1) were studied using molecular dy-

namics (MD) simulations with explicit water. CD-experiments6, 8 had indicated that the protein

E3 19 is significantly less stable than E3 5.

According to the simulation trajectories the difference in stability is mainly due to the differ-

ence in stability of the C-terminal capping AR, while the proteins have similar properties for the

internal ARs5 . Analysis of the charge redistribution when mutating E3 5 into E3 19 reveals that

the third internal AR, which is spatially closest to the C-terminal capping AR, becomes more

negatively charged. This explains the unfolding of the C-terminal capping in the MD simula-

tion of E3 19. Yu’s study illustrates the complementarity between experiment and simulation

when designing proteins with specific properties. Simulation studies offer detailed insight into

detailed energetic and structural properties of proteins in solution that are not accessible through

experiment. This makes design suggestions possible.

In the present study a third AR protein, N3C, is investigated using MD simulation in ex-

plicit water. N3C is again highly homologous to the previously studied two proteins, E3 5 and

E3 19 (87, 88% sequence identity, Table 2.1). The crystal structures of these two proteins have

a backbone (N, Cα, C) atom-positional root-mean square difference of 0.27 nm (Figure 2.1).

Structurally, N3C is closer to E3 19 than to E3 5, but the N3C structure seems experimentally

2.2. Introduction 15

a

f e

d c

b

CCap

NCap

AR3

AR1

AR2

0.22

0.42

0.44

0.27

0.27 0.42

0.46

0.42 0.30

0.32 0.42

0.42 0.58

Figure 2.1: Crystal X-ray structures of the proteins E3 56 (PDB: 1MJ0) (a), E3 197 (PDB:

2BKG) (c), N3C9 (e). Structures after 12 ns MD simulation: E3 5 (b), E3 19 (d), N3C (f).

Backbone (Cα, N, C) atom-positional RMS differences in nm between the different structures are

shown at the black arrows. The least squares superposition of structures involved all backbone

(Cα, N, C) atoms.

to be the most stable of the three, while E3 19 being least stable. The latter finding might be

explained by the observation that N3C has the lowest overall charge (-12 e) compared to that of

the other two proteins (-16 e) and its charge is moreover more evenly distributed over the five AR

units. Here we investigate the relative stability of the three proteins by MD simulation in order

to obtain insight into the features determining protein stability.

16 Chapter 2.

Table 2.1: Mutations that distinguish the proteins E3 5, E3 19 and N3C from each other.residue E3 5 E3 19 N3C

33 Thr Glu Lys

35 Asn Thr Lys

36 Asp Tyr Asp

38 Tyr Asp Tyr

46 Ser Arg Arg

47 Asn Val Glu

59 Asn Asn Ala

66 Ser Leu Lys

68 Leu Phe Lys

69 Thr Ser Asp

71 Ile Ser Tyr

79 Ala Lys Arg

80 Thr Arg Glu

92 His Tyr Ala

99 Tyr Asp Lys

101 Asn Thr Lys

102 Asp Ile Asp

104 His Ser Tyr

112 Lys Asp Arg

113 Tyr Thr Glu

125 His Tyr Ala

156 Gln - Gln

A MD simulation of N3C in aqueous solution was carried out and the obtained trajectory

was compared to those of the proteins E3 5 and E3 19, which were described in Yu et al.5. In

experimental unfolding experiments, the unfolding of the entire protein was measured9, here the

stability of the AR-units is investigated only.

2.3 Material and methods

Molecular Dynamics Simulations

MD simulations were performed with the GROMOS software10, 11 using the force-field para-

meter set 45A312, 13. The simulation of N3C and the ones of E3 5 and E3 19 by Yu et al.5 are

summarized in Table 2.2.

2.3. Material and methods 17

Table 2.2: Overview of the three 12 ns MD simulations of the different systems. E3 5: protein

E3 5 in aqueous solution, starting from the X-ray structure (PDB ID: 1MJ0); E3 19: protein

E3 19 in aqueous solution, starting from the X-ray structure (PDB ID: 2BKG). N3C: protein

N3C in aqueous solution, starting from the X-ray structure9.

Simulation label E3 5 E3 19 N3C

protein E3 5 E3 19 N3C

starting structure PDB ID: 1MJ0 PDB ID: 2BKG -

number of water molecules 9522 8790 15324

total charge [e] -16 -16 -12

Initial coordinates for N3C were taken from the X-ray structure of N3C (PDB: 2QYJ)9. The

mutation sites, that distinguish the three structures E3 5, E3 19 and N3C from each other, are

listed in Table 2.1. Ionization states of residues were assigned according to a pH of 8.0. The his-

tidine side chains were protonated at the Nε atom. The simple-point-charge (SPC) water model14

was used to describe the solvent molecules. In the simulations, water molecules were added

around the protein within a truncated octahedron with a minimum distance of 1.4 nm between

the protein atoms and the square walls of the periodic box. Since in the E3 5 and E3 19 simula-

tions no counterions were included5, we did not include them in the N3C simulation for reasons

of comparison. Moreover, we did perform 3 ns simulation of N3C including 12 Na+ ions, which

gave essentially the same results as the one without counterions. All bonds were constrained

with a geometric tolerance of 10−4 using the SHAKE algorithm15. A steepest-descent energy

minimization of the system was performed to relax the solute-solvent contacts, while position-

ally restraining the solute atoms using a harmonic interaction with a force constant of 2.5 · 104

kJ mol−1 nm−2. Next, steepest-descent energy minimization of the system without any restraints

was performed to eliminate any residual strain. The energy minimizations were terminated when

the energy change per step became smaller than 0.1 kJ mol−1. For the non-bonded interac-

tions, a triple-range method with cutoff radii of 0.8/1.4 nm was used. Short-range van der Waals

and electrostatic interactions were evaluated every time step based on a charge-group pairlist.

18 Chapter 2.

Medium-range van der Waals and electrostatic interactions, between (charge group) pairs at a

distance longer than 0.8 nm and shorter than 1.4 nm, were evaluated every fifth time step, at

which point the pair list was updated. Outside the longer cutoff radius a reaction-field approxi-

mation16 was used with a relative dielectric permittivity of 78.5. The center of mass motion of

the whole system was removed every 1000 time steps. Solvent and solute were independently,

weakly coupled to a temperature bath of 295 K with a relaxation time of 0.1 ps17. The systems

were also weakly coupled to a pressure bath of 1 atm with a relaxation time of 0.5 ps and an

isothermal compressibility of 0.7513 · 10−3 (kJ mol−1 nm−3)−1. 100 ps of MD simulation with

harmonic position restraining of the solute atoms with a force constant of 2.5 · 104 kJ mol−1

nm−2 were performed to further equilibrate the systems. The simulations E3 5, E3 19, and N3C

were each carried out for 12 ns. The trajectory coordinates and energies were saved every 0.5 ps

for analysis.

Analysis

Analyses were done with the analysis software GROMOS++18 and esra19. Atom-positional root-

mean-square differences (RMSDs) between structures were calculated by performing a rotational

and translational atom-positional least-squares fit of one structure on the second (reference) struc-

ture using a given set of atoms. Atom-positional root-mean-square fluctuations (RMSFs) over a

period of simulation were calculated by performing a rotational and translational atom-positional

least-squares fit of the trajectory structures on the reference structure (usually the first structure

of the period) using a given set of atoms. The secondary structure assignment was done using

the program DSSP, based on the Kabsch-Sander rules20. The percentages of intramolecular (n,

n-4)-hydrogen bonds that are involved in the formation of α-helices have been calculated using a

maximum distance criterion of 0.25 nm between the hydrogen atom and the acceptor atom, and

a minimum donor-hydrogen-acceptor angle criterion of 135◦.


Results

The atom-positional root-mean-square deviations (RMSDs) from the starting structures for the

atoms (N, Cα, C) in the simulations E3 5, E3 19, and N3C are shown in Figure 2.2 and have

been calculated for all residues (solid lines) and for the three internal AR residues only (dotted

lines). The protein E3 5 remains close to its crystal structure and converges to an RMSD of

about 0.25 nm. The simulations E3 19 and N3C do not converge to a constant value within the

12 ns of simulation, the protein N3C showing larger structural rearrangements than the protein

E3 19. The RMSDs of the internal ARs reach a value of 0.1 nm after about 2 ns and stay constant

throughout the whole simulation period. The stability of the internal repeat units proves that the

structural rearrangements occur either in the N- or C-terminal capping units.

Figure 2.2: Atom-positional root-mean-square deviations (RMSDs) of the backbone atoms (N,

Cα, C) of E3 5 (black), E3 19 (red), and N3C (green) with respect to the starting (X-ray) coor-

dinates in the simulations. The dotted lines represent the backbone RMSDs for the three internal

AR units of the corresponding protein. The least squares superposition of structures involved all

backbone (Cα, N, C) atoms (solid lines) and those of the three internal AR units (dotted lines).

20 Chapter 2.

The atom-positional root-mean-square fluctuations (RMSFs) for the Cα-atoms were calcu-

lated for the entire 12 ns of the trajectories (Figure 2.3). All simulations show reasonably small

fluctuations for the internal AR units as well as for the N-terminal capping unit. Larger fluc-

tuations are observed only in the C-terminal capping unit. This indicates that the latter unit is

relatively unstable.

Figure 2.3: Atom-positional root-mean-square fluctuations (RMSFs) of the Cα atoms of E3 5

(black), E3 19 (red), and N3C (green) over 12 ns of simulation. AR units are indicated by

horizontal black bars. The least squares superposition of structures involved all Cα atoms.

The secondary structure assignment in Figure 2.4 shows that the helical structural elements

are very stable for the simulation E3 5. In the simulation E3 19 secondary structure features of

the C-terminal capping unit vanish completely after 5 ns. In the simulation N3C the secondary

structure of the C-terminal AR unit is still in place for the majority of the residues. According to

these secondary structure time histories the elevated RMSDs and RMSFs originate from move-

ment of or between the two helices of the C-terminal capping repeat and less from destruction of

its helical secondary structure.


Figure 2.4: Secondary structure elements of E3 5 (left), E3 19 (middle), and N3C (right). α-

helix (black). Assignment according to20.

Backbone (n, n-4) hydrogen-bond occurrences for the C-terminal capping unit are tabulated

in Table 2.3.

22 Chapter 2.

Table 2.3: Occurrence (in %) of all (n, n-4) backbone hydrogen bonds (n is residue sequence

number) from the 12 ns MD simulations.

residue number E3 5 E3 19 N3C

Donor Acceptor occurrence (%) of

N-H O hydrogen bonds

142 138 88 9 74

143 139 87 13 74

144 140 78 13 74

145 141 81 31 38

146 142 77 11 -

147 143 64 - -

148 144 76 - -

149 145 - - -

150 146 - - 89

151 147 - 8 79

152 148 76 18 61

153 149 82 - 64

154 150 52 10 -

155 151 53 - -


Only the hydrogen bonds involved in the formation of an α-helix are shown. The (n, n-4)

hydrogen bonds support the statement drawn from the secondary structure analysis: secondary

structure of E3 5 is stable, the C-terminal capping in E3 19 unfolds, N3C retains most of its

helical features, though at somewhat lower percentages.

Final structures from the simulations are depicted in Figure 2.1. The unfolding of the C-

terminal AR unit in the simulation E3 19 is clearly seen, in the two other simulations parts of

this secondary structure are still visible. The backbone atom-positional RMSD values for the

various pairs of X-ray and simulated structures show that structurally N3C is closer to E3 19

than to E3 5.

In short we can conclude that the structures of E3 5 and N3C both are reasonably stable

throughout the simulation periods. The structure of the protein E3 19 proves to be unstable in

agreement with experiment.

Discussion

This work presents the results of a molecular dynamics simulation of the ankyrin repeat protein

N3C using the GROMOS force-field parameter set 45A3. Root-mean-square deviations and

fluctuations, as well as hydrogen bond analysis and secondary structure assignment has been

done to evaluate the protein’s stability. This data was compared with the data of two previously

simulated proteins, E3 19 and E3 5, in order to be able to investigate the stabilities of AR proteins

as a function of charge distribution over the repeat units. The protein E3 5 proves to be the most

stable of the simulated proteins, followed by the protein N3C, which loses a small amount of

secondary structure during the 12 ns of simulation, while E3 19 loses all secondary structure

in its C-terminal repeat unit. Yet, structurally N3C is closer to E3 19 than to E3 5. These two

observations from MD simulation confirm analoguous experimental ones6–9.

Considering the hypothesis that a more or less uniform distribution of the charges among the

AR units or a reduction of the total charge of the last internal AR unit would increase stability

one would expect that the protein N3C would be very stable in solution (Table 2.4), even more

stable than the protein E3 5.

24 Chapter 2.

Table 2.4: Charge distribution over the various ARs in the proteins E3 5, E3 19 and N3C (in e).

The atomic partial charges were taken from the GROMOS force-field parameter set 45A312, 13.

NCap: residues 1-32; AR1: residues 33-65; AR2: residues 66-98; AR3: residues 99-131; CCap:

residues 132-156.

NCap AR1 AR2 AR3 CCap Total

E3 5 -1 -4 -3 -3 -5 -16

E3 19 -1 -4 -1 -5 -5 -16

N3C -1 -2 -2 -2 -5 -12

This is not the case, although the protein retains most of its helical features in the C-terminal

capping AR.

Other factors than charge-charge interactions between AR units must be responsible for the

higher stability of E3 5 compared to N3C. Comparing the amino-acid sequences of the three

proteins we find 21 differences between E3 5 and E3 19, 18 differences between E3 5 and N3C

and again 21 differences between N3C and E3 19. This suggests more similarity between N3C

and E3 5 than with E3 19. Comparing the bulkiness of the side chains, N3C is more like E3 19

for AR1 and AR2 and lies in between E3 5 and E3 19 for AR3. This suggests that charge

considerations are not sufficient to explain protein stability, but that other factors such as polarity

and volume of side chains should also be accounted for.

2.4. References 25

2.4 References

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26 Chapter 2.


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[16] I. G. Tironi, R. Sperb, P. E. Smith, and W. F. van Gunsteren. A generalized reaction field

method for molecular dynamics simulations. J. Chem. Phys., 102:5451–5459, 1995.

[17] H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, A. Di Nola, and J. R. Haak.


1984.

[18] M. Christen, P.H. Hunenberger, D. Bakowies, R. Baron, R. Burgi, D.P. Geerke, T.N. Heinz,

M.A. Kastenholz, V. Krautler, C. Oostenbrink, C. Peter, D. Trzesniak, and W.F. van Gun-

steren. The GROMOS software for biomolecular simulation: GROMOS05 . J. Comput.

Chem., 26:1719–1751, 2005.

2.4. References 27

[19] V. Krautler, M. Kastenholz, and P. H. Hunenberger. The esra molecular mechanics analysis

package. available at http://esra.sf.net/, 2005.

[20] W. Kabsch and C. Sander. Dictionary of protein secondary structure - pattern-recognition

of hydrogen-bonded and geometrical features. Biopolymers, 22:2577–2637, 1983.

28 Chapter 2.

Chapter 3

Molecular dynamics simulation of human

interleukin-4: comparison with NMR data

and effect of pH, counterions and force

field on tertiary structure stability

3.1 Summary

The human protein interleukin-4 (IL-4) has been simulated at two different pH values, 2 and

6, with different amounts of counterions present in the aqueous solution, and with two differ-

ent force-field parameter sets using molecular dynamics simulation with the aim of validation

of force field and simulation set-up by comparison to experimental NMR data, such as proton-

proton NOE distance bounds, 3J(HN,HCα) coupling constants and backbone N-H order param-

eters. Thirteen simulations varying in the length from 3 to 7 ns are compared.

At pH 6 both force-field parameter sets used do largely reproduce the NOE’s and order pa-

rameters, the GROMOS 45A3 set slightly better than the GROMOS 53A6 set. 3J values predicted

from the simulation agree less well with experimental values. At pH 2 the protein unfolds, unless

counterions are explicitly present in the system, but even then the agreement with experiment is

29

30 Chapter 3.

worse than at pH 6. When simulating a highly charged protein, such as IL-4 at pH 2, the inclusion

of counterions in the simulation seems mandatory.

3.2 Introduction

During protein folding, a series of intermediate conformations are sampled prior to the appear-

ance of the native fold. This makes an accurate description of the unfolded and the intermediate

conformational states crucial to our understanding of protein folding1–3. Partial or complete de-

naturation of proteins can be achieved in vitro by various solution conditions, such as addition

of denaturant, extremes of pH, salt concentrations or temperatures4. There is much interest in

characterizing the non-native states formed under such conditions and comparing their proper-

ties with those of globular proteins. These studies provide insights into issues such as protein

three-dimensional structure, stability and folding5, 6. They also have significance with regard to

understanding diseases associated with protein misfolding and the aggregation of non-native pro-

tein species. Such diseases include cystic fibrosis, Alzheimer’s, and the spongiform encephalo-

phies7. Among the series of intermediate states, molten globule ensembles are of considerable

interest8, 9. A number of proteins (e.g. α-lactalbumin, carbonic anhydrase B, and β-lactoglobulin)

have been found to form molten globular states under mild denaturing conditions. Molten globule

ensembles are characterized by having a pronounced amount of secondary structure in a compact

state that lacks most of the specific tertiary interactions coming from tightly packed side-chain

groups.

Interleukin-4 (IL-4) is a pleiotropic type I cytokine, which plays a central role in the control

and regulation of the immune and inflammatory systems10. The most notable functions of IL-

4 include the development of T-helper cells to a type 2 cytokine-producing phenotype. IL-4

evokes a cellular response by promoting the formation of a heterodimeric receptor complex in

the plasma membrane11–13. The three-dimensional structure of IL-4 under native conditions

has been established in solution14, 15 and in crystals16–18. IL-4 is one of the four helix bundle

cytokines19 that are characterized by antiparallel juxtaposed helices A, B, C, D and two long end-

to-end loops, loops AB and CD, which are connected by a short β-sheet packed against helices


B and D. Human IL-4 has six cysteine residues which form three disulfide bonds (C3-C127,

C24-C65, C46-C99). The overall structure is highly compact and globular with a predominantly

hydrophobic core. Smith et al.20 have thoroughly compared the four independently determined

structures of human recombinant IL-4 and they found the core of the four helix bundle to be very

similar in all the structures, except for differences in loop regions that are known to be mobile

in solution21. At low pH (e.g. at pH ≈ 2.4), it was found experimentally that IL-4 undergoes

a partial unfolding transition22. Comparison of the NMR spectra of IL-4 at pH 5.6 and pH

2.4 shows that the pattern of chemical shifts and NOE’s is little changed for the majority of the

residues. Therefore, at low pH IL-4 appears to retain a highly ordered hydrophobic core in which

most, but not all, the secondary structure is preserved. However, extensive disorder exists in the

loop regions of the polypeptide chain that link these secondary structural elements; at pH 2.4

these mobile loop regions represent more than one third of the protein residues. This low pH

form has been described as a highly-ordered molten globule22.

Despite extensive experimental studies, detailed insight into the low pH molten globule state

of IL-4 is still lacking. For example, the diversity of conformations in the disordered loops

and the rearrangement of secondary structure between pH 5.6 and pH 2.4 have not been well

characterised.

Previously, MD simulation has been used to study the properties of the molten globule state

of various proteins, including hen egg white lysozyme23, bovine pancreatic trypsin inhibitor24,

α-lactalbumin25, 26. MD simulation was used in the present study to investigate in atomic detail

the structural and dynamic properties of IL-4 at low pH. Various properties obtained from simu-

lation trajectories at low pH will be compared with those at pH 6 and with the corresponding

experimental NMR data.

In practice, counterions are usually not included in protein simulations, mainly because they

diffuse slowly in the simulation and their initial positioning might have significant effects on

the simulation trajectory27. The underlying thought is that features, e.g. the counterion posi-

tional distribution, that will not converge within accessible simulation time are better omitted or,

in other words, included in a mean-field manner in the simulation. Secondly, in explicit water

simulations the water molecules of the first few coordination layers very effectively shield the

32 Chapter 3.

ions, as can be observed from radial distribution functions. Thus for proteins with a relatively

low overall charge with respect to their size or number of residues, omission of counterions

is a reasonable first-order approximation of co-solvent effects. In the literature, contradictory

results are reported regarding the effects of counterions on the stability of a protein fold in sim-

ulations28–31. Ibragimova and Wade28 found that explicit modelling of 0.2 M ionic strength is

necessary to maintain the stable structure for the YAP-WW domain. Martı-Renom et al.29 found

that inclusion of counterions contributes to preserve the native structure in the simulation of the

activation domain of procarboxypeptidase B. However, Drabik et al.31 found that simulations of

solvated proteins are only moderately sensitive to the presence of counterions. This sensitivity

was reported to be highly dependent on the starting structures and the different equilibration

procedures used.

For IL-4, the different pH values, mimicked by setting proper protonated states of the ioni-

zable residues, lead to highly charged proteins, + 11 e at pH 6 and + 27 e at pH 2, while the

129-residue protein is relatively small. Therefore, apart from pH effects, we also investigate the

effect of the counterions on the stability of this protein in the simulations.

The quality of a biomolecular simulation will depend on the force field used. Current widely

used biomolecular force fields are AMBER32–34, CHARMM35–37, OPLS-AA38, 39 and GRO-

MOS40–42. Not long ago, it was shown that all of these force fields severely underestimate the free

energy of hydration for series of small molecules that represent the amino acid side chains43–45

. For this reason a parametrisation of the GROMOS force field was carried out which led to the

53A6 parameter set46. Here we use both sets, the 45A342 and the 53A6 one, in order to see how

the new set derived from free energies of hydration and solvation will perform for IL-4.

3.3 Material and Methods

3.3.1 Molecular dynamics simulations

MD simulations were performed with the GROMOS software package41, 47 using the force-field

parameter sets 45A342 and 53A646. The 13 MD simulations are summarized in Table 3.1.

3.3. Material and Methods 33

Initial coordinates were taken from the X-ray structure of IL-4 (PDB entry: 1RCB)16. The

different pH values were mimicked by different protonation states of protonisable residues25:

Asp, Glu and C-terminus are not protonated at pH 6 while protonated at pH 2; His is protonated

at both pH’s except for His76 which at pH 6 is only protonated at Nε. The simple-point-charge

(SPC) water model48 was used to describe the solvent molecules. In the simulations, water

molecules were added around the protein within a truncated octahedron with a minimum dis-

tance of 1.4 nm between the protein atoms and the square walls of the periodic box. In some

of the simulations, ions (Cl− and Na+) were included. Cl− ions were introduced by repla-

cing the water molecules with the highest electrostatic potential and Na+ by replacing the water

molecules with the lowest electrostatic potential. The electrostatic potential at the oxygen atoms

of the water molecules was calculated by taking into account all the atoms within the spherical

shell of water with a cutoff 1.4 nm. The minimal distance between the protein atoms and the ions

was set to be 0.35 nm. All the bonds were constrained with a geometric tolerance of 10−4 using

Table 3.1: Overview of the simulations performed. Two GROMOS41, 47 force-field parameter

sets, 45A342 and 53A646 were used in the simulations. The pH-values indicate different

charge states for the ionizable (side-chain) moieties. In simulations pH6 45A3 REF and

pH2 45A3 REF initially 100 ps of MD simulation was performed with proton-proton NOE

upper-bound distance restraining, as in NMR structure refinement.

simulation force-field number of protein

label parameter set pH counterions total charge simulation length

[e] [ns]

pH6 45A3 45A3 6 - +11 4

pH6 53A6 53A6 6 - +11 4

pH6 45A3 REF 45A3 6 - +11 4

pH6 45A3 6Cl 45A3 6 6 Cl− +5 7

pH6 45A3 11Cl 45A3 6 11 Cl− 0 4

pH6 45A3 20Cl 9Na 45A3 6 20 Cl− 9 Na+ 0 7

pH6 45A3 30Cl 19Na 45A3 6 30 Cl− 19 Na+ 0 4

pH2 45A3 45A3 2 - +27 4

pH2 53A6 53A6 2 - +27 4

pH2 45A3 REF 45A3 2 - +27 4

pH2 45A3 16Cl 45A3 2 16 Cl− +11 3

pH2 45A3 21Cl 45A3 2 21 Cl− +6 3

pH2 45A3 27Cl 45A3 2 27 Cl− 0 4

34 Chapter 3.

the SHAKE algorithm49. A steepest-descent energy minimization of the systems was performed

to relax the solute-solvent contacts, while positionally restraining the solute atoms using a har-

monic interaction with a force constant of 2.5·104 kJ mol−1 nm−2. Next, steepest-descent energy

minimization of the system without any restraints was performed to eliminate any residual strain.

The energy minimizations were terminated when the energy change per step became smaller than

0.1 kJ mol−1. For the non-bonded interactions, a triple-range method with cutoff radii of 0.8/1.4

nm was used. Short-range van der Waals and electrostatic interactions were evaluated every time

step based on a charge-group pairlist. Medium-range van der Waals and electrostatic interac-

tions, between pairs at a distance longer than 0.8 nm and shorter than 1.4 nm, were evaluated

every fifth time step, at which time point the pair list was updated. Outside the longer cutoff ra-

dius a reaction-field approximation50 was used with a relative dielectric permittivity of 6651. The

initial velocities of the atoms were assigned from a Maxwell distribution at 50 K. 5 ps periods of

MD simulation with harmonic position restraining of the solute atoms with force constants of 2.5

104 kJ mol−1 nm−2, 2.0·104 kJ mol−1 nm−2, 1.5·104 kJ mol−1 nm−2, 1.0·104 kJ mol−1 nm−2,

0.5·104 kJ mol−1 nm−2 were performed to equilibrate further the systems at 50 K, 150 K, 250 K,

308 K and 308 K, respectively. During the equilibration, solvent and solute were independently,

weakly coupled to a temperature bath of the given temperature with a relaxation time of 0.1 ps52.

In the further simulations, the center of mass motion of the whole system was removed every

1000 time steps. The systems were also weakly coupled to a pressure bath of 1 atm with a re-

laxation time of 0.5 ps and an isothermal compressibility of 0.4575 × 10−3 (kJ mol−1 nm−3)−1.

The trajectory coordinates and energies were saved every 0.5 ps for analysis.

For a highly charged protein, the use of an appropriate method to treat long-range electro-

static interactions, such as lattice-sum or reaction-field methods, is mandatory. Both, lattice-sum

methods (Ewald, P3M, etc.) and reaction-field methods (dipolar, time-dependent, etc.) to treat

long-range electrostatic interactions cause artefacts in molecular simulations, the former because

of their artificial enhancement of periodic order and the latter because of their mean-field char-

acter. Since we judge the former artefacts to be more severe than the latter53–59 we applied the

reaction-field methodology in the present study.


3.3.2 Structural refinement with nuclear Overhauser effect restraints

Structural refinement with explicit solvent MD simulation was performed using instantaneous

nuclear Overhauser effect (NOE) upper-bound distance restraining60. In addition to the physical

potential energy, a restraining potential-energy term

Vdr(ri j) =12Kdr(ri j− r0

i j)2 if ri j > r0

i j

0 if ri j ≤ r0i j

(3.1)

is used, where Kdr is the force constant for the distance restraining, ri j is the instaneous distance

between atoms i and j, and r0i j is the reference distance, i.e. the NOE upper-bound distance. The

force constant Kdr was chosen to be 2000 kJ mol−1 nm−2. In the GROMOS force-field para-

meter sets 45A342 and 53A646, aliphatic hydrogen atoms are not explicitly treated but are part

of united atoms. We thus calculate interproton distances involving the aliphatic hydrogen atoms

by calculating virtual (for CH1 and prochiral CH2)61 and pseudo (for CH3)62 atomic positions

for these hydrogen atoms41. When comparing the experimentally derived distances with the

calculated proton-proton distances, pseudo-atom corrections involving equivalent or nonstereo-

specifically assigned protons should be included in the NOE-derived upper-bound distances. The

same corrections as used in the structure determination of IL-414, 22, 62 are used in our study.

3.3.3 Analysis

Analyses of the trajectory configurations were done with the analysis software GROMOS++63

and esra64.

Atom-positional root-mean-square differences (RMSDs) between structures were calculated

by performing a rotational and translational atom-positional least-squares fit of one structure on

the second (reference) structure using a given set of atoms. Atom-positional root-mean-square

fluctuations (RMSFs) over a period of simulation were calculated by performing a rotational

and translational atom-positional least-squares fit of the trajectories on the reference structure

(usually the first structure of the period) using a given set of atoms. The secondary structure

assignment was done using the program DSSP, based on the Kabsch-Sander rules65.

36 Chapter 3.

Comparisons to nuclear magnetic resonance (NMR) experimental data were made through

an analysis of proton-proton distances as compared to nuclear Overhauser effect (NOE) upper

bounds, of 3J(HN,HCα) coupling constants, and of 1H-15N order parameters. For pH 5.614,

1656 NOE upper bounds were available and for pH 2.4, 1450 NOE upper bounds22. Proton-

proton distances were averaged using 1/r3 averaging, using r =(⟨

r−3⟩)−1/3, corresponding to

a slowly tumbling molecule66. 3J-coupling constants were calculated from the simulations using

the Karplus relation67,3J(H,H) = acos2

θ+bcosθ+ c (3.2)

The parameters a, b, c used were: a = 6.51 Hz, b =−1.76 Hz, and c = 1.60 Hz68 for calculating3J(HN, HCα) values.

According to their definition, generalized order parameters S2 may be directly calculated

from a simulation using the long-time tail of the second-order Legendre function of the reorien-

tation correlation function of the N-H vector (~v):

S2 = limt→∞

C2(t) (3.3)

where

C2(t) = 〈P2(~v(τ) ·~v(τ+ t))〉τ

(3.4)

Here P2 is the second-order Legendre polynomial, P2(x) = 1/2(3x2− 1). The angular brackets

(<>) represent the average over the ensemble (trajectory). The unit vectors ~v(τ) and ~v(τ + t)

describe the orientation of the N-H vector at times τ and τ + t in relation to a fixed reference

frame. To construct this frame, the translational and overall rotational motions were removed

by a root-mean-square fitting of all backbone atoms onto the starting conformation. In practice,

expressions (3.3) and (3.4) are not very suitable to obtain accurate results, since the long-time

tail of a correlation function is generally plagued by poor statistics. Therefore, the alternative

formula69, 70

S2 =12

[3

3

∑α=1

3

∑β=1

< vα(t)vβ(t) >2t −1

](3.5)

is used, which involves trajectory averages of the elements vαvβ of the Cartesian tensor built

as a direct product of the Cartesian components of the unit vector ~v(t), and yields, therefore,

3.4. Results and discussion 37

more precise results. The average of S2 is either taken over all configurations at 0.5 ps intervals

of the whole simulation or by using an averaging window of 25 ps moving through the whole

simulation period71–73.

Diffusion constants have been calculated via the Einstein-formula

D = limt→∞

〈(r(t + τ)− r(τ))2〉τ6t

(3.6)

3.4 Results and discussion

3.4.1 IL-4 at pH 6

The atom-positional root-mean-square deviations (RMSDs) from the starting coordinates for the

atoms (N, Cα, C) in the simulations pH6 45A3, pH6 53A6, and pH6 45A3 REF are shown in

Figure 3.1.

0 1 2 3 4 5 6 7time [ns]

0

0.1

0.2

0.3

0.4

0.5

atom

-pos

ition

al R

MSD

[nm

]

pH6_45A3

pH6_53A6

pH6_45A3_REF

pH6_45A3_6Cl

pH6_45A3_11Cl

pH6_45A3_11Cl helix residues

pH6_45A3_20Cl_9Na

pH6_45A3_30Cl_9Na


Cα, C) of IL-4 with respect to the starting (X-ray) structure in the simulations pH6 45A3 (black),

pH6 53A6 (red), and pH6 45A3 REF (green), pH6 45A3 6Cl (blue), pH6 45A3 11Cl (brown),

pH6 45A3 11Cl helix residues (brown dotted), pH6 45A3 20Cl 9Na (purple), and pH6 45A3 -

30Cl 19Na (orange).

38 Chapter 3.

Figure 3.2: Secondary structure elements (left panels) and atom-positional root-mean-square

fluctuations (RMSFs) of the Cα atoms (right panels) of IL-4 over the simulations pH6 45A3

(upper panels), pH6 53A6 (middle panels), and pH6 45A3 REF (lower panels). α-helix (red),

310-helix (black), and β-strand (green).

The RMSD values for all three simulations reach a plateau value of 0.2 nm after 0.5 to 1 ns,

and fluctuate around that value afterwards. The atom-positional root-mean-square fluctuations

(RMSFs) for the Cα atoms were calculated for the whole 4 ns of the trajectories (Figure 3.2).

These three simulations show larger fluctuations in loop regions (i.e. loops AB and CD) while

smaller fluctuations in the helical parts.

Generally, the simulations using force-field parameter sets 45A342 and 53A646 show compa-

rable behaviour although the 53A6 parameter set seems to favour 310-helical structure slightly

more and α-helical structure slightly less than the 45A3 set. The secondary structure assignment

shows that the helical structural elements are very stable in these three simulations. Some re-

arrangement in the loops was observed. In Figure 3.3 the final conformations from these three

simulations are shown. The overall secondary structure is well preserved in the three simulations

with the different force-field parameter sets.


Figure 3.3: The X-ray structure (a) and NMR solution structure (b) of IL-4 are shown with the

four helices A-D indicated. Structures of IL-4 at pH 6 at the end of simulations pH6 45A3 (c),

pH6 53A6 (d), pH6 45A3 REF (e), pH6 45A3 6Cl (f), pH6 45A3 11Cl (g), and pH6 45A3 -

20Cl 9Na (h).

A summary of the NOE analysis can be found in Table 3.2, while NOE distance distributions

are displayed in Figure 3.9 of the appendix.

40 Chapter 3.

Table 3.2: Number of NOE upper distance bound violations and average violations in the

simulations for given time periods. At pH 6 a total number of 1656 NOE’s and at pH 2 a total of

1450 NOE’s were considered.

number of

Simulation label averaging NOE upper bound violations average violations

time period

[ns] ≥ 0.1 [nm] ≥ 0.2 [nm] ≥ 0.3 [nm] [nm]

pH6 45A3 1-4 19 1 1 0.004

pH6 53A6 1-4 30 3 2 0.006

pH6 45A3 REF 1-4 16 3 0 0.003

pH6 45A3 6Cl 1-7 23 3 1 0.004

pH6 45A3 11Cl 1-4 23 7 2 0.004

pH6 45A3 20Cl 9Na 1-7 36 12 9 0.007

pH6 45A3 30Cl 19Na 1-4 20 3 1 0.004

pH2 45A3 1-4 183 141 115 0.114

pH2 53A6 1-4 464 357 297 0.227

pH2 45A3 REF 1-4 131 45 7 0.019

pH2 45A3 16Cl 1-3 82 43 22 0.047

pH2 45A3 21Cl 1-3 84 58 55 0.038

pH2 45A3 27Cl 1-4 45 28 17 0.011

A total of 1656 NOE upper bounds were used in the analysis for pH 614. The three simu-

lations without counterions satisfy more than 98% of the experimental upper bounds within 0.1

nm. Applying NOE restraints for the first 0.1 ns (pH6 45A3 REF) only slightly reduces the num-

ber of violations and the average violation. The simulation pH6 45A3 shows somewhat lower

violations compared to the simulation pH6 53A6. One of the two observed larger NOE viola-

tions (larger than 0.3 nm averaged over 1.0-4.0 ns) using the 53A6 parameter set occurs in loop

AB (35AlaHα-38AsnHN) and the other is related to 63Thr in the loop between helix B and C.

Figure 3.3 shows the experimentally derived crystal structure (PDB entry: 1RCB16) and solution

structure (PDB entry: 1ITM14). There exists a large difference for the BC and CD loops between

the crystal structure and the solution structure. The simulations, which started from the crystal

structure, may not have allowed these loops to explore all their conformational possibilities such

that the observed NOE bounds are satisfied.


Three-bond 3J(HN,HCα) coupling constants have been calculated from the simulation en-

sembles and were compared against experimental values (see Table 3.3 and Figure 3.10 of the

appendix).

Table 3.3: Number of 3JHNα-coupling constants for which the absolute difference between

the experimentally determined 3JHNα-coupling constants and the calculated 3JHNα-coupling

constants averaged over the MD trajectories is larger than a given value. Only 68 3J-coupling

constants with exact experimental values have been considered.

Simulation label averaging number of J-values with

period [ns] |Jexp - Jsim|

≥ 1 Hz ≥ 2 Hz ≥ 3 Hz ≥ 4 Hz

pH6 45A3 1-4 27 12 3 0

pH6 53A6 1-4 26 11 6 1

pH6 45A3 REF 1-4 22 7 3 1

pH6 45A3 6Cl 1-4 25 12 6 1

pH6 45A3 11Cl 1-4 21 7 1 1

pH6 45A3 20Cl 9Na 1-7 24 10 2 1

pH6 45A3 30Cl 19Na 1-4 22 6 2 1

3J coupling constants were reported for 114 residues of which 46 were upper bounds14. The

two parameter sets 45A3 and 53A6 give comparable results. Applying NOE upper-bound dis-

tance restraining for the first 0.1 ns does not improve the agreement. Generally 3J(HN,HCα)

coupling constants are less well reproduced than other NMR properties74, which might have dif-

ferent reasons: (i) the time scale needed to achieve converged sampling for 3J-values from MD

simulations may be much beyond a few nanoseconds, (ii) the intrinsic problem of the accuracy of

the empirically calibrated Karplus relation, and (iii) the difficulty of obtaining reliable φ-torsional

angle potential-energy terms for amino-acid residues.

Backbone N-H order parameters calculated from the MD simulations averaged over 0.0-4.0

ns are compared with experimental data21 in Figure 3.4.

All three simulations without counterions show similar amplitudes and profiles with respect

to N-H mobility and reproduce the experimental variations. On average, the experimental order

parameters are slightly larger than those obtained from the simulations.

42 Chapter 3.

Figure 3.4: Backbone N-H order parameter S2 as a function of the residue sequence num-

ber. The averages were calculated over the entire simulation time. Black line: experimen-

tal values21; Red line: simulation pH6 45A3; Green line: simulation pH6 53A6; Blue line:

simulation pH6 45A3 REF. Brown line: simulation pH6 45A3 6Cl. Orange line: simulation

pH6 45A3 11Cl. Turquoise line: simulation pH6 45A3 20Cl 9Na. Purple line: simulation

pH6 45A3 30Cl 19Na. The four helices are indicated by horizontal black bars.

In short we can conclude that MD simulations at pH 6 with two force-field parameter sets both

reproduced various experimentally derived properties well, except for the 3J(HN,HCα) values for

which about one third showed violations of more than 1 Hz.

3.4.2 IL-4 at pH 2

The atom-positional RMSDs from the initial X-ray structure for the backbone atoms (N,Cα,C)

in the simulations pH2 45A3, pH2 53A6 and pH2 45A3 REF are shown in Figure 3.5.

The RMSD values increase to around 1.5 nm, a value much larger than those in the sim-

ulations at pH 6. This means that the starting structure is far from the equilibrium ensemble


0 1 2 3 4time [ns]

0

0.5

1

1.5

2

atom

-pos

ition

al R

MSD

[nm

]

pH2_45A3

pH2_53A6

pH2_45A3_REF

pH2_45A3_16CL

pH2_45A3_21Cl

pH2_45A3_27Cl


Cα, C) of IL-4 with respect to initial (X-ray) structure in the simulations pH2 45A3 (black),

pH2 53A6 (red), and pH2 45A3 REF (green), pH2 45A3 16Cl (blue), pH2 45A3 21Cl (brown),

and pH2 45A3 27Cl (orange).

sampled by MD at low pH. Since the starting structure was the X-ray crystal structure which was

determined at pH 6.0, we applied distance restraining with NOE bounds derived from the NOE

intensities determined at pH 222 for the first 0.1 ns (simulation pH2 45A3 REF). This did not

keep the structure stable, similar unfolding behavior was observed. The atom-positional RMSF

for the Cα atoms were calculated for the entire simulation periods (see Figure 3.6).

In all three simulations without counterions much larger fluctuations were observed com-

pared to those of the simulations at pH 6. A large amount of the secondary structure was lost in

all three simulations, which is in contrast to what has been found experimentally.

Not surprisingly, large NOE violations were observed in all three simulations without coun-

terions (Table 3.2) due to the unfolding of the compact structure. Both the 3J-coupling constants

and the N-H order parameters were not well reproduced in the simulations.

44 Chapter 3.


fluctuations (RMSFs) of the Cα atoms (right panels) of IL-4 in the simulations pH2 45A3 (upper

panels), pH2 53A6 (middle panels), and pH2 45A3 REF (lower panels). α-helix (red), 310-helix

(black), and β-strand (green).

3.4.3 Effects of counterions at pH 6

In the present study, the protonation states of the protonisable residues are assigned according to

the pKa of the side chain without taking into account the protein environment and the possible

variation of the local pH value. With this simple method, the total charges of IL-4 at pH 6 and

2 are + 11 e and + 27 e respectively. The high net charges, especially at pH 2 may have a

large effect on the stability of the protein. Addition of neutralizing counterions or additionally

increasing the ionic strength of the solution may reduce a dominating influence of the protein

charge on its behaviour.

In Figure 3.1 backbone (N, Cα, C) RMSDs with respect to the starting structure are shown for

the simulations pH6 45A3 6Cl, pH6 45A3 11Cl, pH6 45A3 20Cl 9Na, and pH6 45A3 30Cl 19Na.

Significant deviation from the starting structure is only observed for the simulation pH6 45A3 11Cl,


the others show values around 0.25 nm, as in the simulations without counterions. The deviation

for pH6 45A3 11Cl arises from structural changes between the helices (Figure 3.1).

As seen in Figure 3.7 the secondary structure features of the simulation pH6 45A3 11Cl

remain intact, the Cα-atom positional RMSF in the helix regions still being higher than in the

other three simulations at pH 6 with counterions (Figure 3.7).


fluctuations (RMSFs) of the Cα atoms (right panels) of IL-4 in the simulations pH6 45A3 -

6Cl, pH6 45A3 11Cl, pH6 45A3 20Cl 9Na, and pH6 45A3 30Cl 19Na. α-helix (red), 310-helix

(black), and β-strand (green).

Figure 3.3 shows the reference structure and the structure at the end of simulation pH6 45A3 11Cl.

The elevated RMSD and RMSF values seem to arise from a kink in helix C.

In Table 3.2 the NOE upper distance bound violations for the simulations with ions are shown.

The average violations are similar to those in the simulations without ions at pH 6, even though

there are more distance violations than in the simulation pH6 45A3.

In Table 3.3 the average 3J-coupling constant violations are shown. The simulations with

46 Chapter 3.

counterions show a similar number of violations as the simulations without ions, except at high

ionic strength, simulation pH6 45A3 30Cl 19Na. The backbone N-H order parameters (Figure

3.4) show similar profiles with respect to N-H mobility in all simulations.

Diffusion constants for chloride and sodium ions have been calculated (Table 3.4, appendix).

For chloride the values are of the same order of magnitude as the experimentally determined dif-

fusion constants of the ions at 298 K in water (DCl− = 2.03·10−5cm2/s75, DNa

+ = 1.33·10−5cm2/s76).

The sodium ions diffuse too fast in the simulations.

3.4.4 Effects of counterions at pH 2

The N, Cα, C atom-positional RMSDs for the simulations pH2 45A3 16Cl, pH2 45A3 21Cl,

and pH2 45A3 27Cl with respect to the starting structure are shown in Figure 3.5. The RMSD

values are significantly smaller than the values obtained from simulation without ions at the

same pH. The simulations pH2 45A3 16Cl and pH2 45A3 21Cl converge to an RMSD-value of

1 nm, the simulation pH2 45A3 27Cl to a value somewhat lower (0.7 nm). The protein’s se-

condary structure features (Figure 3.8) stay intact during the simulation period except for partial

unfolding of the A helix in simulations pH2 45A3 16Cl and pH2 45A3 21Cl and of the D helix

in simulations pH2 45A3 16Cl and pH2 45A3 27Cl. As expected the atom-positional RMS

fluctuations are smaller than in the simulations without counterions.

The NOE violations are summarised in Table 3.2. A total number of 1450 experimental upper

bounds has been used. Compared to the simulations without counterions there are significantly

fewer violations and the average violations are smaller for the ionic simulations. Yet, at pH 2 the

average violations and the number of violations are much larger than at pH 6. The N-H order

parameters also show sizeable deviations from experiment (results not shown).

3.5 Discussion

This work presents the results of molecular dynamics simulations of interleukin-4 using the

GROMOS force-field parameter sets 45A3 and 53A6. Comparison of simulated and measured

3.5. Discussion 47


fluctuations (RMSFs) of the Cα atoms (right panels) of IL-4 in the simulations pH2 45A3 6Cl

(upper panels), pH2 45A3 21Cl (middle panels), and pH2 45A3 27Cl (lower panels). α-helix

(red), 310-helix (black), and β-strand (green).

NMR data (NOE’s, 3J(HN,HCα)-values, N-H S2-values) has been done to evaluate the perfor-

mance of the more recent force-field parameter set 53A6.

Generally, at pH 6 the parameter sets 45A3 and 53A6 yield similar results, although the

former seems to reproduce experimental values slightly better. Use of NOE restraints during the

first 0.1 ns of simulation does not significantly improve agreement with experiment.

Counterions were added to the simulated system in order to monitor the protein stability as

a function of ionic strength at two different pH values. At pH 6 the protein stability seems to be

neither increased nor decreased significantly by the introduction of counterions. Atom-positional

deviations from the starting X-ray structure converge to similar values, except in the case where

helix C becomes kinked after the introduction of 11 chloride ions (pH6 45A3 11Cl). Atom-

postional fluctuations are also similar with and without counterions, secondary structure elements

48 Chapter 3.

are stable in both cases. The behaviour of simulation pH6 45A3 11Cl gives an indication that

the initial placement of counterions followed by an extensive MD equilibration phase with the

solute restrained is crucial. Otherwise the high charge of an ion being initially positioned close

to a charged protein atom may seriously alter the protein structure already at the beginning of the

simulation.

At low pH, however, the stability of IL-4 is dramatically increased by the introduction of

counterions as is reflected in the atom-positional deviations from the starting structure and the

secondary structure evolution. The presence of counteracting charges in the solution around the

protein’s net charge seems to infer stability by reducing the effective repulsive interactions be-

tween charges in the protein. In the present study no clear correlation between protein stability

and ionic strength could be observed. At pH 6 the agreement between simulation and experi-

ment degrades somewhat upon increasing the ionic strength of the solution, whereas at pH 2 the

opposite effect is observed.

Finally we conclude that ions have a positive influence on a highly charged protein’s sta-

bility in simulation if the initial placement of the ions is chosen well and their distribution is

equilibrated with MD with the solute restrained for a sufficiently long time.

3.6. Appendix 49

3.6 Appendix

Table 3.4: Calculated diffusion constants for the chloride and sodium ions.Simulation label DCl

− DNa+

10−5cm2/s 10−5cm2/s

pH6 45A3 6Cl 5.7 -

pH6 45A3 11Cl 2.1 -

pH6 45A3 20Cl 9Na 5.1 4.1

pH6 45A3 30Cl 19Na 2.6 2.2

pH2 45A3 16Cl 2.6 -

pH2 45A3 21Cl 1.3 -

pH2 45A3 27Cl 2.3 -

-0.5 0 0.5NOE violation [nm]

0

50

100

150

200

250

300

dist

ribu

tion

pH6_45A3


0

50

100

150

200

250

300

dist

ribu

tion

pH6_53A6

-0.5 0 0.5NOE violation[nm]

0

50

100

150

200

250

300

dist

ribu

tion

pH6_45A3_6Cl


0

50

100

150

200

250

300

dist

ribu

tion

pH6_45A3_11Cl


0

50

100

150

200

250

300

dist

ribu

tion

pH6_45A3_20Cl_9Na


0

50

100

150

200

250

300

dist

ribu

tion

pH6_45A3_30Cl_19Na

A B C

D E F

Figure 3.9: Occurrence of r−3-averaged 1H-1H NOE distances in six simulations of IL-4 with

respect to a set of NMR derived NOE distance bounds. Positive values represent violations of

the experimental NOE upper distance bounds. Simulations: (A) pH6 45A3, (B) pH6 53A6, (C)

pH6 45A3 6Cl, (D) pH6 345A3 11Cl, (E) pH6 45A3 20Cl 9Na, (F) pH6 45A3 30Cl 19Na.

50 Chapter 3.

Figure 3.10: Comparison of 3J(HN,HCα) coupling constants between the experimental value

and those averaged over 4 ns of simulation. The sets are divided into (i) the coupling constants

for helical residues with precise experimental values (black circle), (ii) the coupling constants

for helical residues with experimental upper bounds (red circle), (iii) the coupling constants

for non-helical residues with precise experimental values (green circle), and (iv) the coupling

constants for non-helical residues with experimental upper bounds (orange circle). Panel A:

simulation pH6 53A6; Panel B: simulation pH6 45A3 11Cl; Panel C: simulation pH6 45A3;

Panel D: simulation pH6 45A3 REF.

3.7. References 51

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60 Chapter 3.

Chapter 4

Force-field dependence of the

conformational properties of

α,ω-dimethoxypolyethylene glycol

4.1 Summary

A molecular dynamics (MD) study of α,ω-dimethoxypolyethylene glycol has been carried out

under various conditions with respect to solvent composition, ionic strength, chain length, force

field and temperature. A previous MD study on a 15-mer of polyethyleneglycol (PEG) sug-

gested a helical equilibrium structure that was stabilized by hydrogen bonding and bridging wa-

ter molecules. Experiments show that PEG is highly soluble in water, and indicate that clustering

is not favoured. In the present study using different force fields, the GROMOS force fields 45A3

and 53A6, a variation on the latter 53A6 OE, and a force field by Smith et al. produced different

results. For the GROMOS force fields 45A3 and 53A6 no helical structure was found, but forma-

tion of more or less compact random coils in aqueous solution due to hydrophobic interactions

was observed. For the other two force fields used, α,ω-dimethoxypolyethylene glycol stayed

flexible and more or less elongated in aqueous solution, more in agreement with experimental

observations and the previous MD study.

61

62 Chapter 4.

4.2 Introduction

Polyethylene glycol or polyethylene oxide (PEG, PEO) is regarded as one of the most important

polyethers1–6. It shows interesting behaviour in both solid and liquid state7–9. PEG is extremely

well soluble in water10, 11. It is also soluble in methanol, benzene, dichloromethane, but is in-

soluble in diethyl ether and hexane. It can be coupled to hydrophobic molecules to produce

non-ionic surfactants. According to experimental data, PEG dissolves not only in water but also

in numerous organic solvents12. Nuclear magnetic resonance and other spectroscopic measure-

ments13–17 show that PEG undergoes significant conformational change when going from water

to organic solvents. Light scattering experiments on high molecular weight PEG, for instance,

show that PEG has a more extended structure in water than in methanol10, 11. When moving two

surfaces with long PEG chains attached to them closer together, repulsive interactions between

the individual PEG chains could be observed in experimental studies18, 19. These experimental

observations call for an investigation of the conformational properties of PEG. A molecular-

dynamics (MD) computer simulation study of PEG20 suggests that the polymer forms a helical

structure in aqueous solution after 2 ns of simulation. The authors found the helix to be stabi-

lized by a network of hydrogen bonds formed between solute-oxygen atoms and water molecules

and between water molecules themselves. More recently, MD simulations of PEG-water solu-

tions of 1221 and 5422 ether repeat units for a range of compositions (1-32 polymer chains with

1152-100 water molecules) and temperatures (298-410 K) were reported. The trajectories were

compared to neutron scattering data22 and analysed in terms of hydration structure21 and dynam-

ics22. Another issue is the preference of PEG molecules to spatially cluster in particular solvents.

For low-molecular-weight PEG molecules spatial clustering observations are still controversially

documented in the literature. Some authors claim that clustering of PEG molecules is caused by

impurities23, whereas other authors think a chain-end effect is responsible for the clustering of

PEG24.

In the molecular dynamics study presented below, the compaction and aggregation properties

of α,ω-dimethoxypolyethylene glycol (PEG) have been investigated as function of solute com-

position, chain length, solvent composition, temperature, ionic strength, and, most importantly,


force field. A polymer of PEG, with the number of units n, is depicted in Figure 4.1.

Figure 4.1: Simulated polymers: α,ω -dimethoxypolyethylene glycol (PEG), α,ω -dimethoxy-

polyethylene glycol methoxylated every third residue (PEG ome), polyethylene glycol (PEG oh).

n indicates the number of repeating units.

Unfortunately, the values for n that can be investigated using MD simulation on sufficiently

long time scales to observe convergence is limited due to finite computing power available. Here

we use n values ranging from 2 to 46, i.e. low molecular weight PEG. In contrast, experimental

studies often involve high molecular weight PEG, i.e. n ≈ 103−104.

When comparing experimental data with results obtained from simulation, one should keep

in mind the differences between the simulated and experimentally investigated systems. First,

in experimental studies generally much longer PEG chains are present. Second, in the present

MD study the chain-terminal groups of the simulated molecules are methoxy-groups (Figure

4.1), whereas most experimental studies have been carried out with hydroxy-groups as chain-

terminal groups. Yet, we have used methoxy terminal groups rather than hydroxy ones, because

we wished to keep chain-end influence upon the conformational behaviour of the chain as small

as possible. This in view of the relatively short chain lengths that could be simulated with rea-

sonable computing effort and the fact that the hydrogen-bond donor strength of a hydroxyl group

is greater than that of a methoxy group.

64 Chapter 4.


4.3.1 Molecular Dynamics Simulations

MD simulations were performed with the GROMOS software package25–27 using the GROMOS

force-field parameter sets 45A328, 53A629, and 53A6 OE, which contains a modification of the

non-bonded interaction parameters (charges, van der Waals) of the ether oxygen (OE) in the

53A6 force field. The partial charge of the ether oxygen OE was changed from -0.32 e (53A6) to

-0.42 e (53A6 OE) and the partial charges of its adjacent CH2 united atoms were changed from

+0.16 e (53A6) to +0.21 e (53A6 OE). The attractive van der Waals parameter (C6(OE, OE))1/2

for the ether oxygen was changed from 0.04756 (kJ mol−1 nm6)1/2 for the 53A6 parameter set to

the value 0.06313 (kJ mol−1 nm6)1/2 for the 53A6 OE one. The repulsive van der Waals param-

eter (C12(OE, OE))1/2 for interactions with non-polar atoms was changed from 1.100×10−3 (kJ

mol−1 nm12)1/2 for 53A6 to the value 2.148×10−3 (kJ mol−1 nm12)1/2 for 53A6 OE, while the

value for interactions with polar atoms remained unmodified at 1.227×10−3 (kJ mol−1 nm12)1/2

for both parameter sets and the value for interactions with charged atoms was changed from

1.227×10−3 (kJ mol−1 nm12)1/2 for 53A6 to the value of 1.748×10−3 (kJ mol−1 nm12)1/2 for

53A6 OE. In addition, one other, non-GROMOS parameter set has been used, a parameter set

by Smith et al.30. The performed MD simulations are summarized in Table 4.1.


Table 4.1: Overview of the performed MD simulations on α,ω -dimethoxypolyethylene glycol

(polyethylene glycol, PEG). The label ”ome” indicates PEG with every third unit methoxylated.

The label ”oh” indicates PEG with hydroxy-groups at the chain termini. The labels ”cut” or

”cuts” indicate that the PEG chain is cut into a given number of separate pieces.

simulation chain number of end force-field simulation

label length counterions groups parameter set solvent length / ns

peg no ions 46 53A6 46 0 OCH3 53A6 H2O 5

peg no ions 46 53A6 OE 46 0 OCH3 53A6 OE H2O 5

peg no ions 46 Smith 46 0 OCH3 Smith H2O 4.5

peg no ions 23 45A3 23 0 OCH3 45A3 H2O 4.8


peg no ions 23 53A6 OE 23 0 OCH3 53A6 OE H2O 5

peg no ions 23 Smith 23 0 OCH3 Smith H2O 5

peg no ions 23 53A6 meoh 23 0 OCH3 53A6 CH3OH 4

peg no ions 23 53A6 OE meoh 23 0 OCH3 53A6 OE CH3OH 5

peg no ions 23 Smith meoh 23 0 OCH3 Smith CH3OH 5

peg no ions 23 53A6 dmso 23 0 OCH3 53A6 DMSO 4.4

peg no ions 23 53A6 ccl4 23 0 OCH3 53A6 CCl4 2.6


peg no ions 15 53A6 OE 15 0 OCH3 53A6 OE H2O 3.7


peg no ions 11 53A6 11 0 OCH3 53A6 H2O 5


peg no ions 11 Smith 11 0 OCH3 Smith H2O 3




peg ome no ions 23 53A6 23 0 OCH3 53A6 H2O 2.4

peg caso 23 53A6 23 45 Ca2+ , 45 SO2−4 OCH3 53A6 H2O 4.2

peg caso 23 53A6 OE 23 45 Ca2+ , 45 SO2−4 OCH3 53A6 OE H2O 3

peg caso 23 Smith 23 45 Ca2+ , 45 SO2−4 OCH3 Smith H2O 3.4

peg no ions 46 1cut 53A6 2×23 0 OCH3 53A6 H2O 2

peg no ions 46 1cut 53A6 OE 2×23 0 OCH3 53A6 OE H2O 3

peg no ions 46 1cut Smith 2×23 0 OCH3 Smith H2O 3

peg no ions 46 3cuts 53A6 2×12 and 2×11 0 OCH3 53A6 H2O 2


peg no ions 46 8cuts Smith 5×5 and 1×6 0 OCH3 Smith H2O 3


peg oh no ions 46 8cuts 53A6 5×5 and 1×6 0 OH 53A6 H2O 2

peg oh no ions 46 15cuts 53A6 14×3 and 2×2 0 OH 53A6 H2O 2

peg no ions 6 free energy h2o 53A6 6 0 OCH3 53A6 H2O 21× (0.03+0.16)

peg no ions 6 free energy meoh 53A6 6 0 OCH3 53A6 CH3OH 21× (0.03+0.16)

peg no ions 6 free energy h2o 53A6 OE 6 0 OCH3 53A6 OE H2O 21× (0.03+0.16)

peg no ions 6 free energy meoh 53A6 OE 6 0 OCH3 53A6 OE CH3OH 21× (0.03+0.16)

66 Chapter 4.

The initial structure of α,ω-dimethoxypolyethylene glycol was chosen to be an elongated

conformation modelled using the INSIGHTII software package (Accelrys Inc., San Diego, CA).

The simple-point-charge (SPC) water model31 was used to describe the solvent molecules. In

the simulations the solvent molecules were added around the solute within a cubic box with a

minimum distance of 1.4 nm between the solute atoms and the walls of the periodic box. In some

of the simulations, ions (SO42− and Ca2+) were included. The initial placement of the ions was

random. All the bonds and the geometry of the water molecules were kept fixed with a geomet-

ric tolerance of 10−4 using the SHAKE algorithm32. A steepest-descent energy minimisation

without any restraints of all systems was performed to relax the solute-solvent contacts. The

energy minimisations were terminated when the energy change per step became smaller than 0.1

kJ mol−1. For the non-bonded interactions, a triple-range method with cutoff radii of 0.8/1.4 nm

was used. Short-range van der Waals and electrostatic interactions were evaluated every time

step based on a charge-group pairlist. Medium-range van der Waals and electrostatic interac-

tions, between pairs at a distance longer than 0.8 nm and shorter than 1.4 nm, were evaluated

every fifth time step, at which time point the pair list was updated. Outside the longer cutoff

radius a reaction-field approximation33 was used with a relative dielectric permittivity of 6634.

Newton’s equations of motion were integrated using the leap-frog scheme and a time step of 2 fs.

The initial velocities of the atoms were assigned from a Maxwell distribution at 50 K. A 10 ps

period of MD simulation at constant volume was performed, followed by 100 ps of MD. Solvent

and solute were independently, weakly coupled to a temperature bath of the given temperature

with a relaxation time of 0.1 ps35. In the further simulations, the center of mass motion of the

whole system was removed every 1000 time steps. The systems were also weakly coupled to

a pressure bath of 1 atm with a relaxation time of 0.5 ps and an isothermal compressibility of

0.4575 × 10−3 (kJ mol−1 nm−3)−1. The trajectory coordinates and energies were saved every

0.5 ps for analysis.

To investigate hydrophobic interactions between and clustering behaviour of the chains, sim-

ulations of α,ω-dimethoxypolyethylene glycol (n = 23) with every third unit methoxylated (see

label ”ome” in Table 5.1) and simulations of α,ω-dimethoxypolyethylene glycol (n = 46) cut into

several pieces (see label ”cut” or ”cuts” in Table 5.1) were also performed. For the latter the final


configurations of the uncut chain simulations were used as starting configurations, followed by

one more round of energy minimisation and MD equilibration following the above mentioned

procedure. For polyethylene glycol the same starting structures were used for simulations with

the terminal methoxy-groups substituted by hydroxyl-groups.

The free enthalpy of solvation was calculated using thermodynamic integration (TI)36. To

remove α,ω-dimethoxypolyethylene glycol from the system all non-bonded interactions involv-

ing solute atoms were scaled down to zero in a stepwise manner as a function of the coupling

parameter λ. The free-enthalpy change corresponding to the removal of all solute non-bonded

interactions was then calculated by integrating the average value of the derivative of the Hamilto-

nian of the system with respect to λ. This integral was evaluated using 21 evenly spaced λ-values

with 30 ps of equilibration and 160 ps of data collection at each λ-value. Soft-core interaction

was used in order to avoid singularities in the non-bonded interaction function at particular λ-

values37. The free enthalpy of solvation was then calculated as the difference between the free

enthalpy change of letting the solute disappear in vacuo and the free enthalpy change of letting

the solute disappear in solution.

4.3.2 Analysis

Analyses were done with the analysis software packages GROMOS++27 and esra38. Radii of

gyration were calculated with respect to a given set of atoms to observe the level of compactness

of the simulated solute molecules. Structural information on solutions was obtained from atom-

atom radial distribution functions, g(r). Percentages of intermolecular hydrogen bonds have been

calculated using a maximum distance criterion of 0.25 nm between the hydrogen atom and the

acceptor atom, and a minimum angle criterion of 135◦ for the donor-hydrogen-acceptor angle. In

order to further investigate the compaction properties of the α,ω-dimethoxypolyethylene glycol

chains solute oxygen-oxygen distance distributions d(Oi-O j) and percentages of water molecules

bridging between solute oxygen atoms were calculated. The bridging waters have been deter-

mined using a maximum distance criterion of 0.75 nm between two bridged solute-oxygen atoms

and a maximum distance criterion of 0.36 nm between the solute-oxygen atoms and the bridging-

68 Chapter 4.

water oxygen atom.

4.4 Results

In Figure 4.2 the radii of gyration for α,ω-dimethoxypolyethylene glycol (n = 9, 11, 15, 23, 46)

are presented for the simulations peg no ions n (n = 9, 11, 15, 23, 46) and different force fields.

0

0.5

1

1.5

2

radi

us o

f gy

ratio

n [n

m]

53A6_278K53A6_313K53A6_OE_278KSmith_278K

1 2 3 4 5

0 1 2 3 4t [ns]

0

0.5

1

1.5

0 1 2 3 4 5

n=46 n=23 n=15

n=11 n=9

Figure 4.2: Radii of gyration of chains of different lengths (n) of α,ω -dimethoxypolyethylene

glycol simulated in water with the force field 53A6 at 278 K (black) and 313 K (red) and with the

force field 53A6 OE (blue) and that of Smith et al.30 (green), each at 278 K, as function of simu-

lation time (Simulations peg no ions n 53A6, peg no ions n 53A6 OE, peg no ions n Smith,

with n = 46, 23, 15, 11, and 9, see Table 4.1).

For the GROMOS 53A6 force field the radii of gyration of the chains of different lengths

converge to values between 0.50 and 0.75 nm. This indicates a compaction of the chains, keep-

ing in mind that the initial structures were chosen to be elongated. At 313 K the compact state is

reached faster than at low temperature due to faster sampling at higher temperatures. The simula-

tions of α,ω-dimethoxypolyethylene glycol with the force field 53A6 OE and that of Smith et al.

show larger, more fluctuating radii of gyration, suggesting that here the chains remain elongated

and do not form compact conformations.

4.4. Results 69

The radii of gyration for a chain of length n = 23 in Figure 4.3 show again the compacting

behaviour of α,ω-dimethoxypolyethylene glycol in water this time for the GROMOS 45A3 force

field.

0 1 2 3 4 5t [ns]

0

0.5

1

1.5

2

2.5

3ra

dius

of

gyra

tion

[nm

]45A3ome_53A6DMSO_53A6CCl

4_53A6

MeOH_53A6MeOH_53A6_OEMeOH_SmithCaSO

4_53A6

CaSO4_Smith

Figure 4.3: Radii of gyration of α,ω -dimethoxypolyethylene glycol (n = 23) for force field

45A3 and for various solute and solvent compositions at 278 K: black: peg no ions 23 45A3,

red: peg ome no ions 23 53A6, green: peg no ions 23 53A6 dmso, blue: peg no ions 23-

53A6 ccl4, yellow: peg no ions 23 53A6 meoh, brown: peg no ions 23 Smith meoh, grey:

peg no ions 23 53A6 OE meoh, purple: peg caso 23 53A6, turquoise: peg caso 23 Smith.

The GROMOS force fields 53A6 (Figure 4.2) and 45A3 (Figure 4.3) show similar behaviour,

they reach the same radius of gyration, and converge to a stable value equally quickly.

When adding Ca2+ and SO42− ions to the solution, α,ω-dimethoxypolyethylene glycol also

forms compact structures, the radii of gyration being similar to those of the simulations in pure

water (Figure 4.3, CaSO4). The methoxylated solute shows behaviour similar to that of the

unmethoxylated one (Figure 4.3, ome). In the solvents methanol, carbon tetrachloride, and

dimethylsulfoxide the polymer does not show the compacting behaviour observed in water (Fig-

ure 4.3, CCl4, DMSO). The radii of gyration do fluctuate much more than in water and around a

value that is twice as large as the one in water. The force fields 53A6 OE and that by Smith et

al.30 perform differently from the GROMOS 45A3 and 53A6 force fields in water, but behave

70 Chapter 4.

similarly in methanol. For the simulations in methanol the polymer seems to be elongated in all

cases.

In Figure 4.4 the radial distribution functions, g(r), between different atom pairs are shown

for the GROMOS force field 53A6 and the force field by Smith et al.30. The solute oxygen atoms

(OE) show no coordination with calcium (Ca) or sulfate ions (SO4) in any of the simulations, so

their g(r)’s are not shown.

5

10

15

peg_caso_53A6 278Kpeg_caso_53A6 313Kpeg_caso_53A6_OE 278K

0 1 2 3 4distance [nm]

0.5

1

1.5

2

2.5

3

g(r)

1 2 3 4

B

C

A

D

Ca-SO4

Ca-OW

OW-SO4

OW-OE

Figure 4.4: Radial distribution functions g(r) for the simulation of α,ω -dimethoxypoly-

ethylene glycol (n = 23) in calcium sulfate solutions: black: peg caso 23 53A6 at 278 K, red:

peg caso 23 53A6 at 313 K, green: peg caso 23 53A6 OE at 278 K. The four plots show the

g(r) between calcium and sulfate oxygen atoms (A), calcium and water oxygen (OW) atoms (B),

sulfate and water oxygen atoms (C), and water oxygen atoms and solute oxygen (OE) atoms (D).

The g(r) of calcium and sulfate in Figure 4.4A shows interesting behaviour: at higher tem-

perature there is higher coordination of sulfate to calcium in the first two solvation shells. This

indicates entropically favoured clustering of these ions. All other radial distribution functions

have similar shapes at the two different temperatures. Calcium and sulfate are solvated by water

(OW), the first solvation shell of calcium being more structured (Figures 4.4B and 4.4C). The

coordination of solute oxygen atoms (OE) with water oxygen atoms is not significant. This is

the case for the GROMOS force fields as well as for the force field by Smith et al.30. The Smith

4.4. Results 71

force field shows slightly more structured solute oxygen solvation in the first solvation shell at

278 K.

The results of the hydrogen-bond and bridging-water analyses do not bring forward any sig-

nificant percentage of water molecules that stay coordinated between different solute-oxygen

atoms (Results not shown).

In Figure 4.5, solute oxygen-oxygen distance distributions d(Oi-O j) are shown for the sim-

ulations of α,ω-dimethoxypolyethylene glycol. Distance distributions have been calculated for

oxygen atoms belonging to units i and j that have a given distance n = |i− j| ≥ 2 along the chain

from each other.

0

0.5

1

1.5

2

peg_no_ions_23_53A6 278 K

peg_no_ions_23_53A6 313 K

peg_no_ions_23_53A6_meoh 278 K

peg_no_ions_23_53A6_meoh 313 K

peg_no_ions_23_53A6_OE 278 K

peg_no_ions_23_Smith 278 K

0 1 2 3 4distance [nm]

0

0.5

1

1.5

dist

ribu

tion

0 1 2 3 4 0 1 2 3 4 5

A B

D

C

E F

n = 2

n = 20

n = 8n = 4

n = 12 n = 18

Figure 4.5: Distance distributions d(Oi-Oi+n) for different separations (n = 2 (A), n = 4 (B), n=

8 (C), n = 12 (D), n = 18 (E), n =20 (F)) of solute oxygen atoms along the solute chain for sol-

vation in water and in methanol at 278 K and 313 K. Simulations: black: peg no ions 23 53A6,

red: peg no ions 23 53A6 meoh, blue: peg no ions 23 OE, green: peg no ions 23 Smith. Solid

lines: 278 K, dashed lines 313 K.

Especially for the oxygen atoms that are further away along the chain the difference between

the simulations in water and those in methanol becomes striking. Larger distances have a larger

72 Chapter 4.

probability to occur in methanol, illustrating the non-compact conformations of the polymer in

methanol. The illustrations confirm the suggestion that with the force field 53A6 OE and that of

Smith et al.30 the structure is more elongated, showing wider distributed end-to-end lengths.

The tendency for compaction of the polyethylene glycol chains when simulated using the

53A6 force field was investigated by cutting the larger chains into smaller segments and ob-

serving the relative motion of the latter as expressed by the radius of gyration of the cluster of

segments. The radii of gyration calculated from the 53A6 simulations of an α,ω-dimethoxypoly-

ethylene glycol (n = 46) molecule cut into pieces of different sizes are shown in Figure 4.6. The

radius of gyration calculated over all atoms reaches a value of about 0.75 nm in the case of 1,

3 and 8 cuts for α,ω-dimethoxypolyethylene glycol, a value very similar to the value obtained

from the simulations of the intact chain.

0 1 2 3time [ns]

0.6

0.8

1

1.2

1.4

1.6

1.8

2

radi

us o

f gy

ratio

n [n

m]

1 cut 278 K1 cut 313 K3 cuts 278 K3 cuts 313 K8 cuts 278 K8 cuts 313 K15 cuts 278 K15 cuts 313 K8 cuts oh 278 K8 cuts oh 313 K15 cuts oh 278 K15 cuts oh 313 K

15_cuts_oh 313 K

15_cuts_oh 278 K

15_cuts 313 K

Figure 4.6: Radii of gyration for the simulations of α,ω -dimethoxypolyethylene gly-

col (n = 46) (peg no ions 46 53A6) and polyethylene glycol cut into 2, 4, 9, and 16

pieces (peg no ions 46 1cut 53A6, peg no ions 46 3cuts 53A6, peg no ions 46 8cuts 53A6,

peg no ions 46 15cuts 53A6, peg oh no ions 46 8cuts 53A6, peg oh no ions 46 15cuts 53A6)

at 278 K and 313 K.

α,ω-dimethoxypolyethylene glycol cut into 16 pieces shows diffusive behaviour at 313 K

after 2.5 ns. The simulations of polyethylene glycol behave similarly in the case of 8 cuts,

4.5. Discussion 73

only when going to smaller chain lengths (15 cuts) the solutes show diffusive behaviour, even

at 278 K. After 1 ns of simulation the cluster members start to move away from each other. Of

course it is possible to extend the simulations beyond 3 ns to test whether larger segments will

dissociate on longer time scales, but because the 53A6 force field shows too much compaction,

a further investigation of those properties does not seem warranted. Because the 53A6 OE and

Smith force fields show no compaction for single chains, an investigation of clustering of smaller

segments makes no sense.

In Table 4.4 free enthalpies of solvation for α,ω-dimethoxypolyethylene glycol (n=6) are

listed. Free enthalpies have been calculated using the force fields 53A6 and 53A6 OE in the

solvents water and methanol.

Table 4.2: Free enthalpies of solvation of α,ω -dimethoxypolyethylene glycol (n = 6) calculated

using the GROMOS force fields 53A6 and 53A6 OE in water and methanol at 278 K and 1 atm.

system force field ∆Gsolv,water [kJ/mol] ∆Gsolv,methanol [kJ/mol]

α,ω -dimethoxypolyethylene glycol (n = 6) 53A6 -5.6 -43.7

α,ω -dimethoxypolyethylene glycol (n = 6) 53A6 OE -86.12 -86.03

Using force field 53A6 α,ω-dimethoxypolyethylene glycol is better solvated in methanol than

in water, whereas using 53A6 OE force field it is better solvated in water than in methanol. The

too small free enthalpy of hydration found for the 53A6 force field may explain the compaction

of the PEG chains observed in the 53A6 simulations in water.

4.5 Discussion

In the present work α,ω-dimethoxypolyethylene glycol of different chain lengths and compo-

sition has been simulated under various conditions: different force fields, temperatures, and

solvents.

The GROMOS force fields 45A3 and 53A6 induce similar behaviour, giving α,ω-dimethoxy-

polyethylene glycol a rather hydrophobic character, forming compact structures when simulated

74 Chapter 4.

in water. The compact structures are not formed due to water molecules bridging between the so-

lute oxygen atoms of different ethylene oxide units as is concluded from bridging water analyses.

The helical structure found in the molecular dynamics study by Tasaki20 was not observed here.

This difference may be due to the different type of force and simulation set-up used in20. An

indication of the hydrophobic behaviour of α,ω-dimethoxypolyethylene glycol using the 45A3

and 53A6 force fields is that in less polar solvents, such as methanol, DMSO, and carbon tetra-

chloride, the molecule does not form a compact aggregate. This observation is confirmed by

the solute-oxygen-oxygen distance distribution analysis: α,ω-dimethoxypolyethylene glycol’s

solute oxygen atoms tend to separate further from each other in solvents that are less polar than

water. Free enthalpies of solvation show that different oligomers of α,ω-dimethoxypolyethylene

glycol are dissolved better in methanol than in water using the 53A6 force field. When simu-

lating clusters of α,ω-dimethoxypolyethylene glycol molecules of different number and size of

the molecules, the molecules tend to stay aggregated except for very short chains and very small

clusters. When simulating clusters of α,ω-dimethoxypolyethylene glycol trimers and dimers the

clusters fall apart. The interaction of α,ω-dimethoxypolyethylene glycol with calcium and sul-

fate ions is not significant at all, the ions rather interact with each other than with PEG’s oxygen

atoms.

The behaviour of the simulated PEG chains as observed for the GROMOS 45A3 and 53A6

force fields seems different from the behaviour documented from experiment in the literature.

We find compact structures for molecules that are well soluble in water10, 11. Light scattering of

high molecular weight PEG suggests that PEG is less extended in methanol than in water10, 11,

which is the opposite of what was found in the present study of low molecular weight PEG.

Another discrepancy between simulation and experiment is that the simulated short PEG chains

show tendencies to form clusters, whereas an experimental study suggested that PEG chains

forcibly moved towards each other show repulsive interactions18, 19. These discrepancies be-

tween simulation and experiment are resolved when using the GROMOS 53A6 OE force field

or the one by Smith et al.30. They produce markedly different results from the GROMOS 45A3

and 53A6 force fields. Using the former, the PEG chains prefer elongated conformations, in

water as well as in methanol, thus reproducing experimentally observed data more accurately.

4.5. Discussion 75

The different simulation results, compaction, aggregation, and free enthalpies of solvation, offer

a consistent picture of short-chain PEG conformational properties in solution. The GROMOS

53A6 OE parameter set is a clear improvement over the 53A6 one. We note that the parameters

of the GROMOS 53A6 force field used were optimised to reproduce thermodynamic properties

of a series of small non-polar and polar compounds representing segments of biomolecules28, 29.

Their solvation free energies in a variety of solvents, such as the ones used here, do match very

well their experimental counterparts39. However, the calibration set of small compounds did not

comprise ether moieties, which may explain the poor behaviour of the 53A6 parameter set for

PEG. The results of our comparison of four force fields for PEG illustrate the importance of using

an appropriate force field in molecular simulation to understand structural behaviour of flexible

molecules.

76 Chapter 4.

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[2] P.-A. Albertsson. Partiton of Cell Particles and Macromolecules. Wiley, New York, 1986.

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[6] J. M. Harris, editor. Poly(Ethylene Glycol) Chemistry. Plenum, New York, 1992.

[7] P. Molyneux, editor. Water-soluble Synthetic Polymers: Properties and Uses. CRC Press,

Boca Raton, FL, 1983.

[8] E. E. Bailey Jr. and J. V. Koleske, editors. Alkylene Oxides and Their Polymers: Surfactant

Science Series. Marcel Dekker, New York, 1991.

[9] E. E. Bailey Jr. and J. V. Koleke. Poly (ethylene oxide). Academic Press, 1976.

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4.6. References 77

[11] K. Devanand. Polyethylene oxide does not necessarily aggregate in water. Nature, 343:739–

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[12] S. Kawaguchi, G. Imai, J. Suzuki, A. Miyahara, T. Kitano, and K. Ito. Aqueous solution

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78 Chapter 4.

[21] G. D. Smith and D. Bedrov. A molecular dynamics simulation study of the influ-

ence of hydrogen-bonding and polar interactions on hydration and conformations of a

poly(ethylene oxide) oligomer in dilute aqueous solution. Macromolecules, 35:5712–5719,

2002.

[22] O. Borodin, F. Trouw, D. Bedrov, and G. D. Smith. Temperature dependence of water

dynamics in poly(ethylene oxide)/water solutions from molecular dynamics simulations

and quasielastic neutron scattering experiments. J. Phys.Chem. B, 106:5184–5193, 2002.

[23] M. Polverari and T. G. M. van de Ven. Association-induced polymer bridging by

poly(ethylene oxide)cofactor flocculation systems. J. Phys. Chem., 100:13687–13695,

1996.

[24] R. Kjellander and E. Florin. Water structure and changes in thermal stability of the system

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4.6. References 79




[30] Dimitri Bedrov, Matthew Pekny, and D. Grant Smith. Quantum-chemistry-based force field

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80 Chapter 4.



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ChemPhysChem, 7:671–678, 2006.

Chapter 5

On the conformational properties of

amylose and cellulose oligomers in solution

5.1 Summary

Molecular dynamics (MD) simulations were used to monitor the stability and conformation of

double-stranded and single-stranded amylose and single-stranded cellulose oligomers containing

9 sugar moieties in solution as a function of solvent composition, ionic strength, temperature, and

methylation state. This study along with other previous studies suggests that hydrogen bonds

are crucial for guaranteeing the stability of the amylose double helix. Single-stranded amylose

forms a helical structure as well, cellulose stays highly elongated throughout the simulation

time, a behaviour that was also observed experimentally. In terms of coordination of solute

hydroxyl groups with ions, amylose shows entropy-driven coordination of calcium and sulfate

ions, whereas cellulose-ion coordination seems to be enthalpy-dominated. This indicates that

entropy considerations cannot be neglected when explaining the structural differences between

amyloses and celluloses.

81

82 Chapter 5.

5.2 Introduction

Amylose and cellulose are linear polymers of glucose linked with 1,4-bonds. The main difference

is the anomeric configuration: amylose’s glucose units are linked with α(1 → 4) glycosidic

bonds, whereas cellulose’s monomeric units are linked by β(1→ 4) glycosidic bonds. This

different kind of bonding causes amylose to form helical structures and cellulose to form straight

polymer chains.

Amylose occurs in different forms, the A, B, V, and other forms1. The A and B forms

both feature left-handed helices with six glucose units per turn and seem to differ only in the

packing of the starch helices. The V form of amylose is obtained through co-crystallisation with

compounds such as iodine, DMSO, alcohols or fatty acids2, 3. The helical conformations of B-

and V-amylose show differences4–6.

Cellulose is a linear polymer and is the most abundant natural polymer on earth. Cellulose’s

structure and properties have been investigated extensively, but still there are uncertainties about

cellulose’s crystal structure7–9. X-ray scattering and electron diffraction experiments show that

cellulose forms aggregates to sheet-like structures with the cellulose molecules in elongated con-

formation.

These structural differences are the reason why amyloses and celluloses have very different

physical and biological properties10. Amylose is poorly soluble in water and forms suspensions,

in which its helicity is preserved. Cellulose fibers are insoluble in water.

Computer simulations of these systems can be done to get atomic-level insight into the be-

haviour of these molecules and to perform computational studies of their properties. A Monte-

Carlo computer simulation of double-helical amylose in water by Eisenhaber and Schuler11 sug-

gests that a left-handed antiparallel double helix fits best to the structure of liquid water. Regular

water bridges forming a network around the duplex were observed.

Previous molecular dynamics (MD) studies on amylose and cellulose have been done by Yu

et al.12. Fragments of these molecules were methylated at different positions and their stabilities

were monitored. The authors concluded that single helices are more destabilized by methylation

of amylose’s O-2 and O-3 moieties than by methylation at O-6, but that the methylation of O-6


destabilizes double helices more.

In the present work simulations of double-stranded and single-stranded amylose and single-

stranded cellulose oligomers consisting of 9 sugar moieties are analyzed and the stability of

particular structures of these molecules is studied as a function of type of solvent, ionic strength,

temperature, and methylation state.


5.3.1 Molecular Dynamics Simulations

MD simulations were performed with the GROMOS software package13–15 using the force-field

parameter set 53A616. Sugar parameters have been optimised for the GROMOS force field by

Lins et al.17. The MD simulations performed for molecules consisting of 9 sugar units are

summarized in Table 5.1.

Table 5.1: Overview of the performed MD simulations of the different double-stranded (dou) or

single-stranded (sin) carbohydrate systems: amylose (amy), and cellulose (cel).

simulation simulated solvent T / K number of simulation

label system counterions length / ns

amy dou H2O amy H2O 313 0 5.5

amy noHB dou H2O amy hydrogen bonds excluded H2O 313 0 2.1

amy met dou H2O amy methylated H2O 313 0 5.5

amy dou DMSO amy DMSO 313 0 5.2

amy met dou DMSO amy methylated DMSO 313 0 7

amy dou H2O caso amy H2O 313 10 Ca2+, 10 SO2−4 7

amy sin H2O caso single-stranded amy H2O 278 / 313 10 Ca2+, 10 SO2−4 8

cel sin H2O caso single-stranded cel H2O 278 / 313 8 Ca2+, 8 SO2−4 10

Amylose’s and cellulose’s initial coordinates were generated with the INSIGHTII software

package (Accelerys Inc., San Diego, CA). Amylose was modeled into a double helix. Initial

solute conformations of the amylose double-helix were the regular helices built from the data by

Imberty et al.18. This model is characterized by torsional angles of φ = ∠(O5,C1,O4′,C4′) =

84 Chapter 5.

84◦ and ψ = ∠(C1,O4′,C4′,C5′) = −145◦. To further investigate the role of hydrogen bonds

in helix stability, simulations with all non-bonded interactions between solute atoms involved in

hydrogen bonds turned off were carried out (label: noHB), as well as simulations with hydroxy-

groups involved in hydrogen bonds methylated (label: met). In the topologies for the simulations

with excluded hydrogen bonds, Lennard-Jones interactions between hydrogen-bonded atoms

have been set to zero. For the simulations of single-stranded amylose one strand was removed

from the double-helical starting structure. The simulations involving cellulose oligomers started

from an extended conformation with the angles φ = −120◦ and ψ = −120◦. The simple-point-

charge (SPC) water model19 was used to describe the solvent molecules. In some simulations

DMSO20 was used as a solvent (label: DMSO). In the simulations the solvent molecules were

added around the solute within a rectangular box for (double-stranded) amylose and cellulose

with a minimum distance of 1.4 nm between the solute atoms and the walls of the periodic box.

In the simulations roto-translational constraints have been used21. In some of the simulations in

aqueous solution, ions (SO42− and Ca2+) were included (Table 5.1). The initial placement of the

ions was random. All the bonds and the bond angles of the solvent molecules were constrained

with a geometric tolerance of 10−4 using the SHAKE algorithm22. A steepest-descent energy

minimization without any restraints of all systems was performed to relax the solute-solvent

contacts. The energy minimizations were terminated when the energy change per step became

smaller than 0.1 kJ mol−1. For the non-bonded interactions, a triple-range method with cut-

off radii of 0.8/1.4 nm was used. Short-range van der Waals and electrostatic interactions were

evaluated every (time) step based on a charge-group pairlist. Medium-range van der Waals and

electrostatic interactions, between pairs at a distance longer than 0.8 nm and shorter than 1.4 nm,

were evaluated every fifth (time) step, at which (time) point the pair list was updated. Outside

the longer cutoff radius a reaction-field approximation23 was used with a relative dielectric per-

mittivity of 6624. The initial velocities of the atoms were assigned from a Maxwell distribution

at 50 K. For amylose four 50 ps periods of MD simulation with harmonic position restraining of

the solute atoms with force constants of 1.0·104 kJ mol−1 nm−2, 1.0·103 kJ mol−1 nm−2, 1.0·102

kJ mol−1 nm−2, and 5.0 kJ mol−1 nm−2 were performed to equilibrate further the systems at 50

K, 100 K, 200 K, and 278/313 K, respectively. For cellulose five 50 ps periods of MD simulation

5.4. Results 85

with harmonic position restraining of the solute atoms with force constants of 2.5·104 kJ mol−1

nm−2, 1.0·104 kJ mol−1 nm−2, 1.0·103 kJ mol−1 nm−2, 1.0·102 kJ mol−1 nm−2, and 5.0 kJ

mol−1 nm−2 were performed to equilibrate further the systems at 50 K, 100 K, 200 K, 278/313

K, and 278/313 K, respectively. During the equilibration, solvent and solute degrees of freedom

were independently, weakly coupled to a temperature bath at the given temperature with a relax-

ation time of 0.1 ps25. In the further simulations, the center of mass motion of the whole system

was set to zero every 1000 time steps. The systems were also weakly coupled to a pressure bath

of 1 atm with a relaxation time of 0.5 ps and an isothermal compressibility of 0.4575 × 10−3 (kJ

mol−1 nm−3)−1. The trajectory coordinates and energies were saved every 0.5 ps for analysis.

5.3.2 Analysis

Analyses were done with the analysis software packages GROMOS++15 and esra26. Radii of gy-

ration were calculated to observe the level of compactness of the simulated molecules. Structural

information on ionic solutions was obtained from the radial distribution functions g(r). For amy-

lose, percentages of inter- and intramolecular hydrogen bonds were calculated using a maximum

distance criterion of 0.25 nm between the hydrogen atom and the acceptor atom, and a minimum

angle criterion of 135◦ for the donor-hydrogen-acceptor angle.

5.4 Results

5.4.1 Double-stranded amylose in pure solvent

The radii of gyration of amylose in different solvents and methylation states are shown in Figure

5.1.

Exclusion of interactions between atoms that form hydrogen bonds do not influence the be-

haviour of the radii of gyration significantly. Hydrogen-bond percentages are shown in Table

5.2.

Excluding the hydrogen bond interaction in the simulation makes hydrogen bonds as ana-

lyzed vanish. Simulating in DMSO increases the solute-solute hydrogen bonding, higher per-

86 Chapter 5.

0 1 2 3 4 5 6 7time [ns]

0.8

1

1.2

1.4

1.6

1.8

2

radi

us o

f gy

ratio

n [n

m]

amy_dou_H2O

amy_noHB_dou_H2O

amy_met_dou_H2O

amy_dou_DMSOamy_met_dou_DMSOamy_dou_H

2O_caso 278 K

amy_dou_H2O_caso 313 K

Figure 5.1: Radius of gyration as function of time for the simulations of double-stranded amylose

under various conditions, such as different temperatures, solvents, ionic strengths, and methyla-

tion states. The labels of the simulations are defined in Table 5.1.

centages can be observed. Methylation of the structure reduces solute-solute hydrogen-bonding

to zero. Still the structures of the methylated and unmethylated case yield similar radii of gyra-

tion (Figure 5.1). Their structures are different, however, comparing the final structures (Figure

5.2). Amylose with hydroxy groups maintains a helix-like structure (Figure 5.2A and 5.2C),

whereas the methoxylated structure falls apart (Figure 5.2B and 5.2D).

5.4.2 Double-stranded amylose in ionic solution

Amylose adopts lower radii of gyration at 313 K than at 278 K (Figure 5.1). Hydrogen bond

analysis (Table 5.3) shows lower hydrogen-bond percentages at higher temperature, but on the

other hand more favourable sulfate coordination to solute hydroxyl groups than at lower temper-

ature.

5.4. Results 87

Table 5.2: Hydrogen-bond percentages (>10%) for the simulations of double-stranded amylose

in pure water and in DMSO and for methylated molecules as well as for the (alchemical)

molecules with excluded hydrogen-bond interactions all at 313 K. (molecule number: sugar unit

number: atom name)

Solute-solute hydrogen bond percentage hydrogen bonding

in water in DMSO

Donor Acceptor excluded methylated methylated

atom atom H-bonds structure structure

1 : 1 : H3 1 : 2 : O2 71 0 0 90 0

1 : 2 : H3 1 : 3 : O2 69 0 0 93 0

1 : 3 : H3 1 : 4 : O2 35 0 0 92 0

1 : 4 : H3 1 : 5 : O2 18 17 0 94 0

1 : 5 : H3 1 : 6 : O2 24 0 0 93 0

1 : 6 : H3 1 : 7 : O2 37 0 0 92 0

1 : 7 : H3 1 : 8 : O2 46 0 0 69 0

1 : 8 : H3 1 : 9 : O2 41 0 0 85 0

2 : 1 : H3 2 : 2 : O2 40 0 0 84 0

2 : 2 : H3 2 : 3 : O2 38 0 0 87 0

2 : 3 : H3 2 : 4 : O2 47 0 0 92 0

2 : 4 : H3 2 : 5 : O2 58 0 0 93 0

2 : 5 : H3 2 : 6 : O2 41 0 0 93 0

2 : 6 : H3 2 : 7 : O2 26 0 0 91 0

2 : 7 : H3 2 : 8 : O2 32 0 0 90 0

2 : 8 : H3 2 : 9 : O2 37 0 0 85 0

In Figure 5.3 the radial distribution functions for the different atom pairs are shown.

The affinity of sulfate to calcium is higher at higher temperature, at least for the first solva-

tion shell (Figure 5.3A). This suggests that the coordination of calcium and sulfate is entropy-

controlled. Both calcium and sulfate coordinate to the solute’s hydroxyl groups that are not on

the inner side of the helix (O2 and O3), sulfate to a slightly larger extent than calcium. The

coordination of sulfate to solute hydroxyl groups seems also to have entropic contributions, in

all solvation shells higher coordination at higher temperature was observed. Water has a higher

affinity to calcium ions than to sulfate ions.

5.4.3 Single-stranded amylose and cellulose in ionic solution

Figure 5.4 shows the radii of gyration for the simulations of single-stranded amylose and cellu-

lose.

88 Chapter 5.

A B

C D

E F

Figure 5.2: Final structures of double-stranded amylose simulated in water (amy dou H2O (A),

amy met dou H2O (B)) and in DMSO (amy dou DMSO (C), amy met dou DMSO (D)), and

of single-stranded amylose and cellulose in calcium sulfate solution (amy sin H2O caso (E),

cel sin H2O caso (F)).

Cellulose’s radii of gyration show higher values than single-stranded amylose’s. According to

that observation, cellulose stays highly elongated during the simulation, whereas single-stranded

amylose forms compact structures, which however show sizeable fluctuations. This is reflected

in the final structures of the two molecules (Figures 5.2E and 5.2F). Both molecules behave

similarly at the different temperatures. Radial distribution functions of the two molecules are

depicted in Figures 5.5 and 5.6.

Comparing the coordination of calcium with sulfate, similar behaviour in both simulations,

5.5. Discussion 89

Table 5.3: Inter- and intramolecular hydrogen bond percentages (>5%) for the simulation of

double-stranded amylose in ionic solution at the temperatures 278 K and 313 K. (molecule

number: sugar unit number: atom name)

hydrogen bond occurrence hydrogen bond occurrence

Donor atom Acceptor atom 278 K 313 K Donor atom Acceptor atom 278 K 313 K

1 : 1 : H3 1 : 2 : O2 77 9 1: 1 : H6 4: SO4 38 -

1 : 2 : H3 1 : 3 : O2 76 17 1: 8 : H2 1: SO4 48 -

1 : 3 : H3 1 : 4 : O2 44 27 1: 8 : H3 1: SO4 47 -

1 : 4 : H3 1 : 5 : O2 16 31 1: 9 : H2 5: SO4 50 -

1 : 5 : H3 1 : 6 : O2 19 32 1: 9 : H3 5: SO4 35 -

1 : 7 : H3 1 : 8 : O2 52 29 1 : 1 : H2 1: SO4 - 41

1 : 8 : H3 1 : 9 : O2 29 29 1 : 1 : H2 4: SO4 - 27

1 : 9 : H3 1 : 9 : O2 45 28 1 : 1 : H3 1: SO4 - 25

2 : 1 : H3 2 : 2 : O2 77 44 1: 3 : H6 3: SO4 - 45

2 : 2 : H3 2 : 3 : O2 79 77 1: 4 : H2 3: SO4 - 18

2 : 3 : H3 2 : 4 : O2 58 22 1: 4 : H3 2: SO4 - 18

2 : 4 : H3 2 : 5 : O2 10 37 1: 6 : H2 5: SO4 - 19

2 : 5 : H3 2 : 6 : O2 18 27 1: 6 : H3 5: SO4 - 38

2 : 6 : H3 2 : 7 : O2 37 50 1: 7 : H2 5: SO4 - 13

2 : 7 : H3 2 : 8 : O2 35 57 1: 7 : H3 5: SO4 - 23

2 : 8 : H3 2 : 9 : O2 48 35 1 : 8 : H6 5: SO4 - 18

of amylose and cellulose, can be observed. Both plots show higher calcium-sulfate affinity at

higher temperature. Comparing the coordination of calcium and sulfate ions with solute hydroxyl

groups, the two sugars show a rather different temperature dependence: the shapes of the curves

in Figures 5.5 and 5.6 are similar, but in the case of amylose, ionic cordination is favoured at

higher temperature, while in the case of cellulose, at lower temperature.

5.5 Discussion

In this work, double-stranded and single-stranded amylose and single-stranded cellulose have

been simulated under various conditions regarding solvent composition, ionic strength, and tem-

perature. In order to observe the role of hydrogen bonds in helix formation and the formation of

compact structures of double-stranded amylose, different approaches have been used to disrupt

the hydrogen-bond network formed by the two strands of amylose. First, the interactions be-

tween donor and acceptor atoms have been switched off, in another approach all hydroxy-groups

involved in hydrogen bonds have been methylated.

90 Chapter 5.

5

10

15

20

278 K313 K

0 1 20

5

10

15

20

1 2 1 2 1 2

A B C D E

F G H I

Ca-SO4 Ca-O

WCa-O2 Ca-O3 Ca-O6

OW

-SO4

O2-SO4 O3-SO

4O6-SO

4

Figure 5.3: Radial distribution functions for different atom pairs of the simulation of double-

stranded amylose in calcium sulfate solution at the two temperatures 278 K (black line) and 313

K (red line). Calcium with sulfate (A), calcium with water oxygen (B), calcium with amylose

oxygen O2 (C), calcium with amylose oxygen O3 (D), calcium with amylose oxygen O6 (E),

sulfate with water oxygen (F), sulfate with amylose oxygen O2 (G), sulfate with amylose oxygen

O3 (H), sulfate with amylose oxygen O6 (I).

From this study it can be concluded that hydrogen bonds are important for the stability of

the amylose double-helix. Introducing methoxy-groups instead of hydroxy-groups hinders the

formation of a helix. Changing the solvent from water to DMSO, increases interstrand hydrogen-

bond stability which can be explained by a lack of competition between hydrogen bonds with

water molecules and the hydroxy-groups of amylose. At higher temperature double-stranded

amylose in calcium sulfate solution shows less interstrand hydrogen bonding and forms more

compact structures. Another observation made in this work is that the coordination of calcium

with sulfate is entropy-driven, higher coordination at higher temperatures was found. Also the

coordination of ions with the solute hydroxyl groups shows such behaviour.

Single-stranded amylose and cellulose are similar molecules that differ in the nature of their

linking glycosidic bonds. Cellulose was found to stay in an elongated conformation, whereas

single-stranded amylose forms more compact structures. Another striking difference between

5.5. Discussion 91

0 2 4 6 8 10time [ns]

0.8

1

1.2

1.4

1.6

1.8

2

radi

us o

f gy

ratio

n [n

m]

amy_sin_H2O_caso 278 K

amy_sin_H2O_caso 313 K

cel_sin_H2O_caso 278 K

cel_sin_H2O_caso 313 K

Figure 5.4: Radii of gyration for the ionic solutions of single-stranded amylose and cellulose at

the temperatures 278 K and 313 K. Black: amy sin H2O caso at 278 K. Red: amy sin H2O caso

at 313 K. Green: cel sin H2O caso at 278 K. Blue: cel sin H2O caso at 313 K.

the two molecules is the sensitivity of the coordination of ions to the solute hydroxyl groups

to the temperature. The coordination of calcium and sulfate ions to solute hydroxyl groups

seems to be entropy-driven in the case of amylose, whereas in the case of cellulose it is enthalpy

dominated. This suggests that entropy considerations cannot be neglected when explaining the

structural differences between amyloses and celluloses.

92 Chapter 5.

0

5

10

15

20

278 K313 K

0 1 2distance [nm]

0

5

10

15

20

g(r)

1 2 1 2 1 2

A B C D E

F G H I

Ca-SO4

Ca-OW Ca-O2 Ca-O3 Ca-O6

OW

-SO4

O2-SO4 O3-SO

4O6-SO

4

Figure 5.5: Radial distribution functions for different atom pairs of the simulation of single-

stranded amylose in calcium sulfate solution at the two temperatures 278 K (black line) and 313

K (red line). Calcium with sulfate (A), calcium with water oxygen (B), calcium with amylose

oxygen O2 (C), calcium with amylose oxygen O3 (D), calcium with amylose oxygen O6 (E),

sulfate with water oxygen (F), sulfate with amylose oxygen O2 (G), sulfate with amylose oxygen

O3 (H), sulfate with amylose oxygen O6 (I).

5.5. Discussion 93

0

5

10

15

20

278 K313 K

0 1 2distance [nm]

0

5

10

15

20

g(r)

1 2 1 2 1 2

A B C D E

F G H I

Ca-SO4

Ca-OW Ca-O2 Ca-O3 Ca-O6

OW

-SO4

O2-SO4

O3-SO4

O6-SO4

Figure 5.6: Radial distribution functions for different atom pairs of the simulation of cellulose

in calcium sulfate solution at the two temperatures 278 K (black line) and 313 K (red line).

Calcium with sulfate (A), calcium with water oxygen (B), calcium with cellulose oxygen O2 (C),

calcium with cellulose oxygen O3 (D), calcium with cellulose oxygen O6 (E), sulfate with water

oxygen (F), sulfate with cellulose oxygen O2 (G), sulfate with cellulose oxygen O3 (H), sulfate

with cellulose oxygen O6 (I).

94 Chapter 5.

5.6 References

[1] A. Buleon, P. Colonna, V. Planchot, and S. Ball. Starch granules: Structure and biosynthe-

sis. Int. J. Biol. Macromol., 23:85–112, 1998.

[2] G. Rappenecker and P. Zugenmaier. Conformation and packing analysis of polysaccharides

and derivatives VIII. Detailed refinement of the crystal-structure of Vh-amylose. Carbo-

hydr. Res., 89:11–19, 1981.

[3] M. C. Godet, H. Bizot, and A. Buleon. Crystallization of amylose-fatty acid complexes

prepared with different amylose chain lengths. Carbohydr. Polym., 27:47–52, 1995.

[4] H. Saito, J. Yamada, T. Yukumoto, H. Yajima, and R. Endo. Conformational stability of

V-amyloses and their hydration induced conversion to B-type form as studied by high-

resolution solid-state 13C NMR-spectroscopy. Bull. Chem. Soc. Jpn., 64:3528–3537, 1991.

[5] A. Imberty, A. Buleon, V. Tran, and S. Perez. Recent advances in knowledge of starch

structure. Starch/Starke, 43:375–384, 1991.

[6] M. B. Cardoso, J.-L. Putaux, Y. Nishiyama, W. Helbert, M. Hytch, N. P Silveira, and

H. Chanzy. Single crystals of V-amylose complexed with α-naphthol. Biomacromolecules,

8:1319 1326, 2007.

[7] A. C. O’Sullivan. Cellulose: the structure slowly unravels. Cellulose, 4:173–207, 1997.

[8] P. Zugenmaier. Conformation and packing of various crystalline cellulose fibers. Prog.

Polym. Sci., 26:1341–1417, 2001.

[9] Y. Nishiyama, H. Chanzy, M. Wada, J. Sugiyama, K. Mazeau, V.T. Forsyth, C. Riekel,

M. Mueller, B. Rasmussen, and P. Langan. Synchrotron x-ray and neutron fiber diffraction

studies of cellulose polymorphs. Advances in X-ray Analysis, 45:385–390, 2002.

[10] J. F. Robyt. Essentials of Carbohydrate Chemistry. Springer-Verlag, New York, 1998.

5.6. References 95

[11] F. Eisenhaber and W. Schulz. Monte Carlo simulation of the hydration shell of double-

helical amylose: A left-handed antiparallel double helix fits best into liquid water structure.

Biopolymers, 32:1643–1664, 1992.

[12] H. B. Yu, M. Amann, T. Hansson, J. Kohler, G. Wich, and W. F. van Gunsteren. Effect of

methylation on the stability and solvation free energy of amylose and cellulose fragments:

A molecular dynamics study. Carbohydr. Res., 339:1697–1709, 2004.










Chem., 26:1719–1751, 2005.




[17] R. D. Lins and P. H. Hunenberger. A new GROMOS force field for hexoyranose-based

carbohydrates. J. Comput. Chem., 26:1400–1412, 2005.

[18] A. Imberty, H. Chanzy, S. Perez, A. Buleon, and V. Tran. The double-helical nature of the

crystalline part of A-starch. J. Mol. Biol., 201:365–378, 1988.




96 Chapter 5.

[20] Hayan Liu, Florian Muller-Plathe, and W. F. van Gunsteren. A force field for liquid

dimethyl sulfoxide and physical properties of liquid dimethyl sulfoxide calculated using

molecular dynamics simulation. J. Am. Chem. Soc., 117:4363–4366, 1995.

[21] A. Amadei, G. Chillemi, M. A. Ceruso, A. Grottesi, and A. Di Nola. Molecular dynamics

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Comput. Phys., 23:327–341, 1977.







1984.



Chapter 6

On using a too large integration time step

in molecular dynamics simulations of

coarse-grained molecular models

6.1 Summary

The use of a coarse-grained (CG) model that is widely used in molecular dynamics simulations

of biomolecular systems is investigated with respect to the dependence of a variety of quantities

upon the size of the used integration time step and cutoff radius. The results suggest that when

using a non-bonded interaction cutoff radius of 1.4 nm a time step of maximally 10 fs should be

used, in order not to produce energy sinks or wells. Using a too large time step, e.g. 50 fs with

a cutoff of 1.2 nm, as is done in the coarse-grained model of Marrink et al. (J. Phys. Chem. B

108 (2004) 250 and B 111 (2007) 7812), induces errors due to the linear approximation of the

integrators that are commonly used to integrate the equations of motion. As a spin-off of the

investigation of the mentioned CG models, we found that the parameters of the CG water model

place it at physiological temperatures well into the solid phase of the phase diagram.

97

98 Chapter 6.

6.2 Introduction

Coarse-grained (CG) models for molecular simulation, in which groups of atoms are modelled

as one particle or coarse-grained atom, are very popular these days1, 2. Their greatest advantage

is the reduction of computational power required, since fewer interaction sites have to be taken

into account. This makes simulations of larger systems at longer timescales feasible. Different

CG models have been proposed, see for example3–18. A much used one for lipids and water is the

one developed by Marrink et al.10, 14 and its extension to proteins15. In some applications19, 20

and CG model parametrisations10, 14, 15 rather large time steps ∆t of 30 - 50 fs are used in the

numerical propagation of the solutions of the Newtonian equations of motion. These values are a

factor 20 larger than the values of 0.5 - 2.0 fs used for all-atom models21, 22. The use of large time

steps in CG models is justified by invoking the smoother energy function or surface compared

to all-atom or fine-grained (FG) model functions. However, a ∆t difference of a factor 20 seems

too large considering that only about 4 non-hydrogen real atoms are mapped onto one CG atom

in the most common CG model for lipids and water10, 14.

Since in MD Newton’s equations of motion are integrated forward in time using discrete

time steps ∆t and truncated Taylor series expansions of the positions r(t) and velocities v(t) of

the atoms at the succeeding time points tn, the maximum size of the time step ∆t is restricted

by the size of the (higher-order) time derivatives of r and v that are omitted from the Taylor

expansions in the particular integration algorithm used. The time derivatives of r(t) and v(t)

involve derivatives of the forces f (t) with respect to time, because the acceleration a(t) of an

atom equals the force on it divided by its mass. The change of the force with time, therefore,

depends on the ruggedness of the atomic or molecular interaction function V (r): the smoother it

is, the smaller the change in f with distance ∆r covered will be.

Thirty years ago, the time step ∆t that is compatible with a reasonably accurate integration of

the equations of motion for typical all-atom biomolecular interaction functions has been analyzed

and was found to be 0.5 - 2 fs21–23. Since the CG model of10, 14 involves the same functional

form for the interaction function, a r−12, r−6, and r−1 distance dependence of the non-bonded

interaction, it is unlikely that it can be used with a ten or more times longer time step ∆t. Here,


we investigate this issue and show that using a typical CG model time steps larger than 10 fs lead

to very poor integration of the equations of motion, resulting in average values of properties, e.g.

temperatures, that are different from the correct ones. Taking a most widely used CG model10, 14

as example, we analyze for systems consisting of alkanes and of water their properties as func-

tion of the size of ∆t, of the size of the cut-off radius Rc for non-bonded interactions, and of the

strength of the coupling of the system to a heat bath to stabilize the temperature. The proper-

ties are thermodynamic ones, density, heat of vaporisation, excess free energy, conformational

entropy (alkanes), and solvation free energy, a dynamic one, diffusion, and structural ones: ra-

dial distribution functions and for alkanes, bond-angle, torsional-angle, and end-to-end of chain

distance distributions.

During our investigation of the timestep issue, in which we used the CG water model of10, 14,

we found that this model constitutes a poor approximation of liquid water, because its parameters

are such that at physiological temperatures the model is well into the solid part of its phase

diagram.


The coarse-grained (CG) model of Marrink et al.10, 14, 15 has been formulated in terms of a

lennard-jones function which is modified by using a particular shift function as implemented

in the GROMACS24 molecular simulation program. Since this shift function was erroreously de-

scribed in the GROMACS manual and also erroreously implemented in the GROMACS program,

as has been extensively investigated and reported in25, we did not want to use exactly the same

(inconsistent) shift function, but the shift function as implemented in the GROMOS05 biomolec-

ular simulation software26. To this end we investigated for which values of the parameters Rs

and Rc, the distances between which the non-bonded interaction is shifted, the GROMOS05

shift function would mimick the effect of the GROMACS one best25. The original CG model

of Marrink10, 14, 15 has been parametrized using Rs = 0.9 nm, Rc = 1.2 nm and the (inconsistent)

GROMACS shift function. Using Rs = 0 nm and Rc = 1.4 nm and the GROMOS05 shifting

function the non-bonded interaction energy function is virtually identical to the one using GRO-

100 Chapter 6.

MACS and Marrink’s parameters, see Figure 6 of25. Thus we use GROMOS05 shifting with Rs

= 0 nm and Rc = 1.4 nm when evaluating the CG model of10, 14, 15.

The non-bonded interaction parameters for water (W) and carbon (C) coarse-grained atoms

are C12(W,W ) = 2.324 ×10−3 kJ mol−1 nm12, C6(W,W ) = 0.2156 kJ mol−1 nm6, C12(C,W )

= 0.8366 ×10−3 kJ mol−1 nm12, C6(C,W ) = 0.07761 kJ mol−1 nm6, and C12(C,C) = 1.580

×10−3 kJ mol−1 nm12, C6(C,C) = 0.1466 kJ mol−1nm6. The bonded energy terms were as

in10, 14. The bond- and bond-angle motions were modelled using harmonic functions: Kb = 1250

kJ mol−1nm−2, and Kθ = 25 kJ mol−1, and b0 = 0.47 nm and θ0 = 180◦10. No torsional potential

energy term was used10. The atomic masses were 72 amu for CG carbon and 72 amu for CG

water.

The equations of motion were integrated using the standard leap-frog algorithm, which gen-

erally only uses the first two terms on the right hand side of the equations27

v(tn +∆t/2) = v(tn−∆t/2)+m−1 f (tn)∆t +124

m−1 f (tn)(∆t)3 +O((∆t)5) (6.1)

r(tn +∆t) = r(tn)+ v(tn +∆t/2)∆t +1

24m−1 f (tn +∆t/2)(∆t)3 +O((∆t)5) (6.2)

where the time derivative of a quantity f is denoted as f . The non-bonded interaction pairlist

was updated every 5 steps. Periodic boundary conditions in a cubic box were applied. If the

temperature was to be held constant, this was done by weak coupling to a temperature bath (Tre f

= 298 K) with relaxation times τT = 0.1 ps, 1ps, or 10 ps28. For water the box volume was 343

nm3 and for hexadecane 216 nm3. In the simulations with constant pressure the systems were

in addition weakly coupled to a pressure bath28 of 1 atm with a relaxation time of τp = 5τT and

an isothermal compressibility of 0.4575 × 10−3 (kJ mol−1 nm−3)−1. Different systems were

investigated: pure CG water, pure CG hexadecane (C16), and 1 CG hexadecane molecule in CG

water, see Table 6.1. The CG simulations of liquid water and liquid hexadecane were carried

out with constant volume and energy (NVE), at constant volume and temperature (NVT), and at

constant temperature and pressure (NPT), the solution of hexadecane in water only at constant

temperature and pressure (NPT). The lengths of the simulations were 1260 ps (NVE) and 1890

ps (NPT).


Thermodynamic integration was used to compute the excess free energies of the liquids and

the free enthalpies of solvation using the procedure described in25. Twenty-one equidistant val-

ues of the coupling parameter λ were used with 63 ps equilibration and 630 ps sampling for each

λ value.

Table 6.1: Overview of the performed MD simulations on coarse-grained water and coarse-

grained alkanes.

system property held

constant

number of molecules

water hexadecane

3200 0 NVE, NVT, NPT

0 343 NVE, NVT, NPT

1192 1 NPT

6.3.1 Analysis

The simulations with constant energy (NVE) were performed to evaluate how well the total

energy (Etot) was conserved. To this end, the fluctuations, defined as ∆E =〈[E − 〈E〉]2〉1/2,

of the total energy (Etot) were compared to those of the potential (Epot), and kinetic energy

(Ekin). Simulating at constant total energy (NVE), ∆Etot should be significantly smaller than

∆Ekin21, 23. If the total energy fluctuations are not equal to zero (or very small), energy sinks

or sources are present in the simulation. These can have a number of different origins: finite

machine precision of the computer used, time integration errors due to truncation of higher-order

(in ∆t) terms in the (leap-frog) algorithm, use of spatial (bond-length) constraints formulated in

terms of Lagrange multipliers21, 29, use of a non-bonded interaction cut-off radius, temperature-

bath coupling, pressure-bath coupling, and use of non-conservative forces, such as in Langevin

dynamics or when changing force-field parameters as a function of time as in local-elevation

MD30 or when applying time-averaged restraining in structure refinement of biomolecules31.

One may estimate the heat flow ∆Q into the system induced by the mentioned sinks or sources

102 Chapter 6.

using the following expressions.

To estimate the integration error in the standard leap-frog scheme, one may evaluate the next-

order corrections to the positions and velocities27:

∆rn+1 =1

24m−1 fn+1/2(∆t)3 =

124m

( fn+1− fn)(∆t)2 (6.3)

and

∆vn+ 12=

124

m−1 fn(∆t)3 =1

24m( fn+1−2 fn + fn−1)(∆t) (6.4)

where we have used the short-hand notation fn for f (tn).

Use of constraints (SHAKE)29 will also change the energy of the system because of nu-

merical resetting of the coordinates from the unconstrained positions rncn+1 to the (constrained)

positions rn+1. The constraint force f cn is

f cn = m(rn+1− rnc

n+1)/(∆t)2 (6.5)

and the work done becomes

∆Q = f cn (rn+1− rn). (6.6)

The energy change due to the use of a cut-off radius is

∆Q = −Epot(atoms leaving the cut−o f f sphere)+

Epot(atoms entering the cut−o f f sphere). (6.7)

In constant temperature simulations, the thermostat will supply and take energy from the

system. Using weak coupling to a temperature bath of temperature Tre f , as described in28, we

find

∆Q = Ekin(a f ter velocity scaling) − Ekin(be f ore velocity scaling)

=N

∑i=1

12

miv2i [λ2

sc−1] (6.8)

with

λsc = [1+2cd f

v

kB

∆tτT

[Tre f ‘

T (t−∆t/2)−1]]1/2, (6.9)


where cd fv is the heat capacity per degree of freedom and kB Boltzmann’s constant. In practice

the factor 2cd fv /kB is set to 128.

When simulating at constant pressure Pre f energy flow is also induced by the barostat. In that

case we have using weak coupling28

∆Q = Pre fV [µ3sc−1] (6.10)

with

µsc = [1− βτ∆tτp

[Pre f −P(t)]]1/3, (6.11)

where βτ is the isothermal compressibility of the system and V its volume. If the scaling factors

λsc or µsc are not equal to one, energy sources or sinks are present.

For the systems consisting of pure liquids, coarse-grained hexadecane and water, energy fluc-

tuations, the higher-order terms of the leap-frog integrator algorithm, average temperature, aver-

age volume, average potential energy and the average of the squared temperature scaling factor

λsc and the average of the cubed pressure scaling factor µsc have been calculated. Additionally

properties such as excess free energy, for both water and hexadecane, and angle distributions,

dihedral angle distributions, and end-to-end of chain distances for hexadecane have been calcu-

lated32. The free enthalpy of solvation of hexane in water was computed too.

6.3.2 Results

Figures 6.1 and 6.2 show the results for the simulations of water and hexadecane at constant

volume and energy. If we use as criterion for a not too inaccurate integration of the equations of

motion that the fluctuations of the total energy should be less than one fifth of the fluctuations of

the kinetic or potential energies21–23, we see that it is not fulfilled using time steps larger than 5

fs for a cutoff radius Rc = 0.8 nm, and using time steps larger than 10 fs for a cutoff radius Rc

= 1.4 nm. As expected, using a larger cutoff radius the higher-order terms (6.3) and (6.4) in the

leap-frog scheme become relatively more significant when integrating the equations of motion.

We also observe that using the same time step and cut-off radius the total energy conservation

is better for hexadecane than for water, which is due to the stronger non-bonded interaction of

104 Chapter 6.

water. In order to be able to use the standard leap-frog scheme, a timestep of 10 fs or smaller has

to be used at a cutoff radius of 1.4 nm. We note that for the CG water and CG alkane models

of10, 14 a timestep of 10 fs is equivalent to timesteps of 0.0056 (water) and 0.0047 (alkanes) in

reduced units. This also shows that a 50 fs time step is unacceptably long.

1000

2000

∆E [

kJ m

ol-1

]

Etot

Epot

Ekin

Rc = 0.8 nm R

c = 1.2 nm R

c = 1.4 nm

0.10.120.140.160.180.2

<|v

|> [

nm p

s-1]

xyz

2e-054e-056e-058e-05

<|∆

r|>/∆

t [nm

ps-1

]

0 0.01 0.02 0.03 0.04 0.050

0.0010.0020.0030.0040.005

<|∆

v|>

[nm

ps-1

]

0.01 0.02 0.03 0.04 0.05∆t [ps]

0.01 0.02 0.03 0.04 0.05

Figure 6.1: Energy fluctuations (total Etot solid lines, potential Epot dashed lines, kinetic Ekin

dotted lines) and higher-order leap-frog terms (x-, y-, z-components) calculated over the final

630 ps of the 1260 ps simulations of liquid water at NVE as a function of the cutoff radius Rc

and time step ∆t. The three colums represent the different cutoff radii Rc 0.8 nm, 1.2 nm, and 1.4

nm. First row: energy fluctuations, second row: averaged absolute velocities, 〈|v|〉, third row:

averaged absolute third-order term (Eq. 6.3) for positions divided by ∆t, fourth row: averaged

absolute third-order term (Eq. 6.4) for velocities.

When coupling the system to a heat bath and a pressure bath (NPT), the thermostat and

barostat will supply or withdraw energy from the system in order to keep it at the reference

temperature Tre f and reference pressure Pre f . Figures 6.3 and 6.4 show this effect for liquid

water and liquid hexadecane, respectively. The first row panels of Figures 6.3 and 6.4 show

that beyond a timestep of 20 fs, the reference temperature (298 K) is not conserved anymore,

because the thermostat cannot keep up with the energy loss of the system due to integration


100

200

300

400

∆E [

kJ m

ol-1

] Etot

Epot

Ekin

Rc = 0.8 nm R

c = 1.2 nm R

c = 1.4 nm

0.08

0.1

0.12

0.14<

|v|>

[nm

ps-1

]

xyz

2e-05

4e-05

6e-05

8e-05

<|∆

r|>/∆

t [nm

ps-1

]

0 0.01 0.02 0.03 0.04 0.050

0.00050.001

0.00150.002

0.00250.003

<|∆

v|>

[nm

ps-1

]

0.01 0.02 0.03 0.04 0.05∆t [ps]

0.01 0.02 0.03 0.04 0.05

Figure 6.2: Energy fluctuations and higher order leap-frog terms calculated over the final 630

ps of the 1260 ps simulations of liquid hexadecane at NVE as a function of the cutoff radius Rc

and time step ∆t. The three colums represent the different cutoff radii Rc 0.8 nm, 1.2 nm, and 1.4

nm. First row: energy fluctuations, second row: averaged absolute velocities, 〈|v|〉, third row:

averaged absolute third-order term (Eq. 6.3) for positions divided by ∆t, fourth row: averaged

absolute third-order term (Eq. 6.4) for velocities.

errors leading to cooling of the system by up to 20 K. The scaling factors λ2sc (6.9) and µ3

sc (6.11)

show deviations from the ideal value 1 at larger time steps (Rows 2 and 3). The results suggest

that using a cut-off radius of 1.2 nm or 1.4 nm, the time step should be smaller than 20 fs.

Physical-chemical properties such as average temperature, density, potential energy, excess

free energy and diffusion constant are shown in Table 6.2 for selected settings of cutoff radius

and time step. Most properties are significantly affected by the use of time steps beyond 10 fs.

Results indicate that for Rc=1.2 nm or 1.4 nm the time step should not be longer than 10 fs.

In Figure 6.5 the radial distribution functions g(r) are shown for the 1890 ps long simulations

of the pure liquids (NPT) for five different combinations of the cutoff radius Rc and time step ∆t.

The structure of the liquids is little affected by the use of too large time steps. We note that the

g(r) for CG water indicates that the liquid is frozen. At 298 K, the system of N = 3200 CG water

106 Chapter 6.

260270280290300

<T

> [

K]

τT = 0.1 ps

τT = 1 ps

τT = 10 psR

c = 0.8 nm R

c = 1.2 nm R

c = 1.4 nm

02000400060008000

<V

> [

nm3 ]

-80000-60000-40000-20000

0

<E

pot>

[kJ

mol

-1]

11.00021.00041.00061.0008

<λ sc

2 >

0 0.01 0.02 0.03 0.04 0.050.99920.99940.99960.9998

11.0002

<µ sc

3 >

0.01 0.02 0.03 0.04 0.05∆t [ps]

0.01 0.02 0.03 0.04 0.05

Figure 6.3: Average temperature T , volume V , potential energy Epot , squared temperature scal-

ing factor λsc (Eq. 6.9), and cubed pressure scaling factor µsc (Eq. 6.11) calculated over the final

630 ps of the 1260 ps simulations of liquid water at NPT as a function of the cutoff radius Rc and

time step ∆t. The temperature was held constant using weak coupling with τT = 0.1 ps (solid

lines), 1 ps (dashed lined), and 10 ps (dotted lines). The three colums represent the different

cut-off radii Rc = 0.8 nm, 1.2 nm, and 1.4 nm.

particles with ε = 5.0 kJ mol−1, σ = 0.47 nm and a volume V = 343 nm3 has a reduced density

ρ∗ = 0.97 and a reduced temperature T ∗ = 0.50, for which it is well into the solid state region of

the known phase diagram of a Lennard-Jones system33. This freezing of the CG model of water

has been observed before14 and is thus unavoidable. It makes this CG model less suitable for

biomolecular systems.

Figure 6.6 shows the bond-angle θ and torsional-angle ϕ distributions for the simulations of

pure hexadecane (NPT) at constant temperature and pressure. Different combinations of cutoff

radius and time step do not influence the distributions significantly.

The corresponding end-to-end distance distributions are shown in Figure 6.7. Again the

different combinations of Rc and ∆t do not influence the behaviour of the system significantly.


270280290300

<T

> [

K]

τT = 0.1 ps

τT = 1 ps

τT = 10 psR

c = 0.8 nm R

c = 1.2 nm R

c = 1.4 nm

150200250300

<V

> [

nm3 ]

-20000-15000-10000

-50000

<E

pot>

[kJ

mol

-1]

0.99991.00021.00051.0008

<λ sc

2 >

0 0.01 0.02 0.03 0.04 0.050.99960.9998

11.00021.0004

<µ sc

3 >

0.01 0.02 0.03 0.04 0.05∆t [ps]

0.01 0.02 0.03 0.04 0.05

Figure 6.4: Average temperature T , volume V , potential energy Epot , squared temperature scal-

ing factor λsc (Eq. 6.9), and cubed pressure scaling factor µsc (Eq. 6.11) calculated over the

final 630 ps of the 1260 ps simulations of liquid hexadecane at NPT as a function of the cutoff

radius Rc and time step ∆t. The temperature was held constant using weak coupling with τT =

0.1 ps (solid lines), 1 ps (dashed lined), and 10 ps (dotted lines). The three colums represent the

different cut-off radii Rc 0.8 nm, 1.2 nm, and 1.4 nm.

In Table 6.3, the free enthalpy of solvation (∆Gsolv) of one CG hexadecane molecule in a

solution of 1192 CG water molecules is shown. The values do not vary significantly with Rc and

∆t.

Table 6.3 also illustrates the distortive effect that may arise when a system that consists of two

components that do not readily exchange kinetic energy and that are each subject to a different

sink or source of energy, is coupled to only one temperature bath instead of to two separate baths,

i.e. one for each component. The temperature of the system is related to the temperature of the

components, e.g. solute and solvent, by the relation

Tsystem = fsoluteTsolute + fsolventTsolvent (6.12)

where fsolute is the fraction of solute degrees of freedom and fsolvent the fraction of solvent

108 Chapter 6.

Table 6.2: Physical-chemical properties of pure CG water and pure CG hexadecane using NPT

(τT = 0.1ps, τp = 0.5 ps) and excess free energies using NVT (τT = 0.1ps) thermodynamic

boundary conditions. The NPT data are averaged over the final 630 ps of the 1890 ps long

simulations.

NPT NPT NPT NPT NPT NVT

Rc [nm] ∆t[ f s] 〈T 〉[K] 〈ρ〉[kg m−3 ] 〈Epot 〉[ kJ mol−1 ] D [cm2s−1] Scon f [ kJ mol−1K−1 ] ∆Fexc[kJmol−1 ]

water: 1.4 5 298 1102 -98228 2.6 ×10−8 - -

1.4 10 297 1102 -98245 1.9 ×10−8 - 22.1

1.4 50 281 1093 -97213 2.6 ×10−8 - -

1.2 10 298 1066 -84747 6.5 ×10−6 - -

1.2 20 295 1067 -84712 7.4 ×10−7 - 18.3

1.2 50 282 1069 -84927 6.7 ×10−7 - 15.2

hexadecane: 1.4 5 298 1156 -20971 1.3 ×10−5 171 -

1.4 10 298 1156 -20952 1.1 ×10−5 172 -

1.4 50 288 1157 -20983 3.5 ×10−5 174 -

1.2 10 298 1064 -16530 5.8 ×10−5 173 -

1.2 20 297 1063 -16543 1.8 ×10−5 171 -

1.2 50 289 1063 -16559 1.6 ×10−5 173 -

Table 6.3: Physical-chemical properties of hexadecane in water from NPT simulation. The

solute and solvent degrees of freedom are either coupled to two separate heat baths or jointly to

one bath. The temperatures are averages over the final 630 ps of the 1890 ps long simulations.

2 baths 1 bath

Rc [nm] ∆ t [fs] ∆Gsolv 〈T 〉[K] 〈Tsolute〉[K] 〈Tsolvent 〉[K] 〈T 〉[K] 〈Tsolute〉[K] 〈Tsolvent 〉[K]

C16 in H2O 1.4 10 -1.9 297 297 297 284 296 284

1.2 20 0.4 296 307 296 258 293 258

1.2 50 -1.4 283 292 283 200 287 200

0.8 50 0.1 298 297 298 72 284 71

degrees of freedom in the system. If the system is coupled to one temperature bath of temperature

Tre f and the solute picks up more energy than the solvent, e.g. due to integration errors or cut-off

noise, we will find that

Tsolvent < Tsystem ≈ Tre f < Tsolute (6.13)

withTsolute−Tsystem

Tsolvent−Tsystem=− fsolvent

fsolute. (6.14)

If the solvent picks up more energy than the solute, the roles of solute and solvent in Eqs. (6.13)

and (6.14) are exchanged. In all-atom simulations of proteins in water, the latter is the case,


1

2

3

g(r)

Rc = 1.4 nm, ∆t = 10 fs

Rc = 1.4 nm, ∆t = 50 fs

Rc = 1.2 nm, ∆t = 10 fs

Rc = 1.2 nm, ∆t = 20 fs

Rc = 1.2 nm, ∆t = 50 fs

0 1 2distance [nm]

0

0.5

1

1.5

2

2.5

g(r)

CG water

CG hexadecane

Figure 6.5: Radial distribution functions g(r) calculated from the final 630 ps of the 1890 ps

simulations of liquid water and liquid hexadecane at NPT for different combinations of the cutoff

radius Rc and time step ∆t.

because the H2O molecules are generally more prone to cut-off noise than the relatively less

mobile protein and the protein bond lengths are generally constrained. For the CG hexane in CG

water simulation reported in Table 6.3, the opposite is the case: the single-particle Lennard-Jones

water is less prone to finite integration time step noise than the CG hexadecane molecule with its

bond-length and bond-angle vibrations, leading to an artificial cooling of the solvent and heating

of the solute. This effect disappears when each component is separately coupled to a temperature

bath.

6.3.3 Discussion

Coarse-grained (CG) models have become more and more popular in recent years and thus it has

become more important that these are correctly parametrised. The results reported here suggest

a reparametrisation of one very basic feature of the CG model of Marrink et al.10, 14, 15: the

time step that is used to integrate the Newtonian equations of motion. A time step of ∆t=50 fs,

that has been suggested and used by Marrink et al.10, 14 is considered too large, since at NVE

conditions the total energy fluctuations are significantly larger than the fluctuations in the kinetic

110 Chapter 6.

0 50 100 150 200θ [degrees]

0

0.005

0.01

0.015

0.02

0.025

0.03

Prob

abili

ty

0 100 200 300ϕ [degrees]

Rc = 1.4 nm, ∆t = 10 fs

Rc = 1.4 nm, ∆t = 50 fs

Rc = 1.2 nm, ∆t = 10 fs

Rc = 1.2 nm, ∆t = 20 fs

Rc = 1.2 nm, ∆t = 50 fs

Figure 6.6: Bond-angle θ distributions and torsional-angle ϕ distributions calculated from the

final 630 ps of the 1890 ps simulations of hexadecane at NPT for different combinations of the

cut-off radius Rc and time step ∆t.

and potential energies. We note that the use of a shifting function masks the errors introduced

by the use of a cutoff distance for non-bonded interactions. The ratio of total energy fluctuations

over kinetic energy fluctuations becomes smaller, but the physical model does not become a

better approximation of the real molecular system. Physical-chemical properties are affected by

the use of a long time step, most strikingly the average temperature of the system, that is lowered

by 20 K in the case of using time steps of 50 fs.

Our results show that for a cutoff radius of Rc = 1.4 nm a maximum of time step ∆t = 10 fs

should be used in coarse-grained model simulations based on Lennard-Jones interaction func-

tions in order to avoid energy flow in or out of the system.

Analysis of the CG water model of10, 14, 15 shows that the Lennard-Jones parameters are such

that at physiological temperatures the model is well into the solid region of the phase diagram,

making it less suitable for use in biomolecular simulations.


0 0.5 1 1.5 2 2.5 3distance [nm]

0

0.5

1

1.5

2

2.5

3

Prob

abili

ty

Rc = 1.4 nm, ∆t = 10 fs

Rc = 1.4 nm, ∆t = 50 fs

Rc = 1.2 nm, ∆t = 10 fs

Rc = 1.2 nm, ∆t = 20 fs

Rc = 1.2 nm, ∆t = 50 fs

Figure 6.7: Distribution of end-to-end of chain distances calculated from the final 630 ps of the

1890 ps simulations of liquid hexadecane at NPT for different combinations of the cut-off radius

Rc and time step ∆t.

112 Chapter 6.

6.4 References

[1] M. Muller, K. Katsov, and M. Schick. Biological and synthetic membranes: What can be

learned from a coarse-grained description? Phys. Rep., 434:113–176, 2006.

[2] Gregory A. Voth. Coarse-Graining of Condensed Phase and Biomolecular Systems. CRC

Press, Florida, USA, 2009.

[3] B. Smit, P. A. J. Hilbers, K. Esselink, L. A. M. Rupert, N. M. van Os, and A. G. Schlijper.

Computer simulations of a water/oil interface in the presence of micelles. Nature, 348:624–

625, 1990.

[4] R. Goetz and R. Lipowsky. Computer simulations of bilayer membranes: Self-assembly

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116 Chapter 6.

Chapter 7

Outlook

The field of molecular dynamics simulation has grown vastly in recent years, bringing forward

many different techniques and applications. A critical point in simulation is the relation to experi-

ment. On the different resolution levels different challenges have to be taken on, to overcome the

discrepancies between simulation and experiment.

In the second chapter, a protein is studied, where the charge distribution throughout the

molecule influences the molecule’s stability. The fundamental problem here is that the electro-

static interaction of a charged atom with an infinite, charged environment is a divergent integral,

see Figure. This makes it very difficult, if not impossible, to obtain unambiguous interactions for

systems containing charge monopoles. In the third chapter, ionic strength of the solvent governs

a protein’s stability. A question in molecular dynamics that has not been fully answered yet is

the influence of ions or charges in general on protein stability.

In the presented chapters it can be seen that stability is affected by electrostatic interactions

with ions in solution or the protonation or deprotonation of amino acid side chains. More sys-

tematic studies will have to be done to further investigate the influence of charged molecules on

biomolecules.

Chapter 4 shows the influence of force-field parametrisation of simple polymers on their be-

haviour in solution. The work presented there underlines the importance of a correct parametri-

sation of simple molecules in order to reproduce experimental behaviour in terms of solvation

117

118 Chapter 7. Outlook

W.F.van Gunsteren/Stradbroke Island 090908/1

Treatment of long-range forces

Interaction energy summed over space:

Converges (absolutely) if with

Alternatives: - Summation over a (periodic) lattice

- Cut-off plus reaction field due to continuum

dipole-dipole

charge-dipole

charge-charge

ForceEnergyInteraction type

2

2

2 3

3 4

1 1

1 1

r 1

1

1 1

1

1 1

r

r

r r

r

U

r

f

r

r r

r2 2

* *U r f r

0

2( )*4Total

r dU rU r !"

= #

1( )~ 3U r

r!

! >

behaviour and structure. Different force fields produce very different results. In the future a large

range of molecules that can be simulated by MD will be asked for, and an appropriate parametri-

sation of small molecules is crucial for simulating larger systems such as lipid bilayer membranes

and proteins. A significant amount of effort will have to be done to produce parameters for all

the molecules that are going to be simulated in future times.

In Chapter 6 of this thesis the very basic parametrisation of a coarse-grained force field has

been investigated. Still, some experimental properties of the studied liquids, water and alkanes,

are not yet sufficiently well reproduced. The logical consequence is that the shape of the potential

energy function has to be given a closer look. The potential energy function is likely to be

smoother than that of a standard Lennard-Jones one. One strategy to find a functional form that

fits better, is to take an all-atom simulation and find clusters of four molecules that are similar to

the mapped interaction sites in the coarse-grained model, and calculate the interaction energies of

two of these clusters as a function of their distance. The function that can be fitted to the obtained

curve can then be used for further coarse-grained simulations. Furtheron parametrisation of

repulsive and attractive non-bonded interaction terms will have to be done to correctly reproduce

experimental thermodynamic and structural data.

In MD, work has to be continued in many different places, force-field parametrisation being

just one example. New applications of this simulation technique will be found and made possible

119

with increasing computer power that will ask for the development of new methods and strategies.

120 Chapter 7. Outlook

121

Curriculum Vitae

Personal Data

Name: Moritz Christoph Ludwig Winger

Date of Birth: September 29, 1981

Place of Birth: Vienna, Austria

Citizenship: Austrian

Education

1986-1990 Primary School, Vienna, Austria

1991-1999 Secondary School, Schottengymnasium Wien, Vienna, Austria

1999-2004 Diploma Studies Chemistry, ETH Zurich, Zurich, Switzerland

2005-2008 PhD studies, Laboratory of Physical Chemistry, ETH Zurich, Zurich, Switzerland

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