Molecular Dynamics Simulations of Biomimetic Carbohydrate ...416145/FULLTEXT01.pdfII Preface The work presented in this thesis has been carried out at the Department of Theoretical

ROYAL INSTITUTE

OF TECHNOLOGY

Molecular Dynamics Simulations of

Biomimetic Carbohydrate Materials

Qiong Zhang

(张琼)

Doctoral Thesis in Theoretical Chemistry and Biology

School of Biotechnology

Royal Institute of Technology

Stockholm, Sweden 2011

© Qiong Zhang, 2011

ISBN 978-91-7415-966-0

ISSN 1654-2312

TRITA-BIO-Report 2011:12

Printed by Universitetsservice US-AB,

Stockholm, Sweden, 2011

献给志军及我的父母

To Zhijun and my parents

I

Abstract

The present thesis honors contemporary molecular dynamics simulation methodologies which

provide powerful means to predict data, interpret observations and widen our understanding

of the dynamics, structures and interactions of carbohydrate systems. With this as starting

point my thesis work embarked on several cutting edge problems summarized as follows.

In my first work the thermal response in crystal cellulose Iβ was studied with special

emphasis on the temperature dependence of the crystal unit cell parameters and the

organization of the hydrogen bonding network. The favorable comparison with available

experimental data, like the phase transition temperature, the X-ray diffraction crystal

structures of cellulose Iβ at room and high temperatures, and temperature dependent IR

spectra supported our conclusions on the good performance of the GLYCAM06 force field for

the description of cellulose crystals, and that a cautious parameterization of the non-bonded

interaction terms in a force field is critical for the correct prediction of the thermal response in

cellulose crystals.

The adsorption properties of xyloglucans on the cellulose Iβ surface were investigated in

my second paper. In our simulations, the interaction energies between xyloglucan and

cellulose in water were found to be considerably lower than those in vacuo. The van der

Waals interactions played a prevailing role over the electrostatic interactions in the adsorption.

Though the variation in one side chain did not have much influence on the interaction energy

and the binding affinity, it did affect the structural properties of the adsorbed xyloglucans.

The interaction of the tetradecasaccharide XXXGXXXG in complex with the hybrid

aspen xyloglucan endo-transglycosylase PttXET16-34 was studied in the third paper. The

effect of the charge state of the “nucleophile helper” residue Asp87 on the PttXET16-34

active site structure was emphasized. The results indicate that the catalysis is optimal when

the catalytic nucleophile is deprotonated, while the “helper” residue and general acid/base

residue are both protonated.

In my forth paper, the working mechanism for a redox-responsive bistable [2]rotaxane

based on an α-cyclodextrin ring was investigated. The umbrella sampling technique was

employed to calculate the free energy profiles for the shuttling motion of the α-cyclodextrin

ring between two recognition sites on the dumbbell of the rotaxane. The calculated free

energy profiles verified the binding preferences observed experimentally. The driving force

for the shuttling movement of the α-cyclodextrin ring was revealed by the analysis of the free

energy components.

keywords: molecular dynamics simulation, carbohydrate, cellulose, xyloglucan, cyclodextrin

II

Preface

The work presented in this thesis has been carried out at the Department of Theoretical

Chemistry and Biology, School of Biotechnology, Royal Institute of Technology, Stockholm,

Sweden and Laboratory for Advanced Materials and Institute of Fine Chemicals, East China

University of Science and Technology, Shanghai, P. R. China.

List of papers included in the thesis

Paper I A Molecular Dynamics Study on the Thermal Response of Crystalline Cellulose

Iβ

Zhang, Q.; Bulone, V.; Ågren, H.; Tu, Y.

Cellulose 2011, 18, 207–221.

Paper II The Adsorption of Xyloglucan on Cellulose: Effects of Explicit Water and Side

Chain Variation

Zhang, Q.; Brumer, H.; Ågren, H.; Tu, Y.

(submitted for publication)

Paper III Molecular Dynamics Simulations of a Branched Tetradecasaccharide Substrate

in the Active Site of a Xyloglucan endo-transglycosylase

Mark, P.; Zhang, Q.; Czjzek, M.; Brumer, H.; Ågren, H.

Mol. Sim. 2011 (accepted)

Paper IV Working Mechanism for a Redox Switchable Molecular Machine Based on

Cyclodextrin: A Free Energy Profile Approach

Zhang, Q.; Tu, Y.; Tian, H.; Zhao, Y-L.; Stoddart, J. F.; Ågren, H.

J. Phys. Chem. B 2010, 114, 6561–6566.

Comments on my contribution to the papers included

I, as the first author, took the major responsibility for papers I, II, and IV.

I, as the coauthor, was responsible for the calculations, discussion and revision of paper III.

III

List of main publications not included in the thesis

Paper I Molecular Dynamics Simulations of Local Field Factors

Zhang, Q.; Tu, Y.; Tian, H.; Ågren, H.

J. Chem. Phys. 2007, 127, 014501.

Paper II Molecular Dynamics Simulations of Polycarbonate Doped with Lemke

Chromophores

Zhang, Q.; Tu, Y.; Tian, H.; Ågren, H.

J. Phys. Chem. B 2007, 111, 10645-10650.

Paper III Electric Field Poled Polymeric Nonlinear Optical Systems: Molecular

Dynamics Simulations of Poly(methyl methacrylate) Doped with Disperse Red

Chromophores

Tu, Y.; Zhang, Q.; Ågren, H.

.J. Phys. Chem. B 2007, 111, 3591-3598.

Paper IV Photochromic Spiropyran Dendrimers: ‘Click’ Syntheses, Characterization,

and Optical Properties

Zhang, Q.; Ning, Z.; Yan, Y.; Qian, S.; Tian, H.

Macromol. Rapid Commun. 2008, 29, 193-201.

Paper V ‘Click’ Synthesis of Starburst Triphenylamine as Potential Emitting Material

Zhang, Q.; Ning, Z.; Tian, H.

Dyes and Pigments 2009, 81, 80-84.

Paper VI Dye-sensitized Solar Cells Based on Bisindolylmaleimide Derivatives

Zhang, Q.; Ning, Z.; Pei, H.; Wu, W.

Front. Chem. China 2009, 269-277.

Paper VII Modulation of Iridium(III) Phosphorescence via Photochromic Ligands: a

Density Functional Theory Study

Li, X.; Zhang, Q.; Tu, Y.; Ågren, H.; Tian, H.

Phys. Chem. Chem. Phys, 2010, 12, 13730-13736.

IV

Acknowledgements

It is my great pleasure to extend my sincere acknowledgements to all those who have

contributed in one way or another to the accomplishment of my thesis work in the past few

years.

First of all, I wish to express my deep felt gratitude toward my supervisor, Prof. Hans Ågren,

for giving me this precious opportunity to work in our department and introducing me to the

field of molecular modeling. I appreciate all the freedom that you gave to me on choosing the

projects of interest. I am indebted for your helpful advice, strong support, and constant

encouragement, without which I would never have completed the work.

I would like to give my heartfelt thanks to my supervisor, Prof. He Tian, in East China

University of Science and Technology, for leading me into the research realm, for

recommending me to study in this group and for all your support on both my work and life.

I am truly thankful to my co-supervisor Dr. Yaoquan Tu for your patient guidance, instructive

discussions and considerable help in many ways. The nice collaborations and constructive

discussions with Prof. Vincent Bulone, Prof. Harry Brumer and Dr. Pekka Mark on the

cellulose and xyloglucan related projects, Prof. J. Fraser Stoddart and Dr. Yan-Li Zhao on the

molecular machine project, and Dr. Marten Ahlquist on the on-going enzymatic catalysis

project are highly acknowledged. Great discussions with my friends and colleagues,

especially Xin Li, Yu Kang, Dr. Ying Zhang, Dr. Shilu Chen and Dr. Rongzhen Liao are

greatly appreciated.

I am sincerely grateful to Prof. Yi Luo for your valuable guidance and positive advice on both

my work and life, for sharing your own experience and your view of life and for being so

considerate and generous to all of us. I appreciate a lot Prof. Boris Minaev, Prof. Kersti

Hermansson Dr. Zilvinas Rinkevicius and Dr. Paweł Sałek for your excellent lectures. Prof.

Faris Gel’mukhanov and Prof. Fahmi Himo are especially acknowledged for your particular

interest in Chinese culture and for all the fun. I would also like to thank Dr. Ying Fu, Dr. Olav

Vahtras and Staffan Hellström for your strong support and kind help on the laboratory life.

I am grateful to all colleagues and friends in our department for all the fruitful discussions and

advice, the pleasant working environment, and the good fun. There are so many of you but

please allow me to write down your names here: Dr. Feng Zhang, Dr. Shilu Chen, Dr.

Rongzhen Liao, Dr. Yuejie Ai, Dr. JunJiang, Dr. Tiantian Han, Dr. Kai Fu, Dr. Wenhua

Zhang, Dr. Na Lin, Dr. Bin Gao, Dr. Guangde Tu, Dr. Jicai Liu, Dr. Kailiu, Dr. Yanhua Wang,

V

Dr. Hui Cao, Dr. Katia J. de Almeida, Dr. Kathrin Hopmann, Xin Li, Yu Kang, Lu Sun, Dr.

Ying Zhang, Dr. Sai Duan, Dr. Hao Ren, Xing Chen, Dr. Yuping Sun, Xiao Cheng, Dr.

Xiaofei Li, Dr. Keyan Lian, Dr. Ke Zhao, Dr. Weijie Hua. Dr. Qiang Fu, Fuming Ying, Dr.

Xifeng Yang, Dr. N. Arul Murugan, Dr. Kestutis Aidas, Bogdan Frecus, Xiangjun Shang,

Zhihui Chen, Dr. Junkuo Gao, Dr. Xiaohao Zhou, Ying Hou, Dr. Hongmei Zhong, Jiayuan Qi,

Chunze Yuan, Qiu Fang, Guangjun Tian, Li Li, Bonaman Xin Li, Hongbao Li, Xiuneng Song,

Yong Ma, Lili Lin, Quan Miao, Ying Wang and Yongfei Ji ...... Thank all of you for all those

happy, colorful, and unforgettable memories! Special thanks to my close friends – Feng,

Yuejie, Yu, Xin Li and Ying Zhang, for sharing happiness and sadness in our life, for the

strong support and great encouragement and for our forever friendship.

I would like to thank the China Scholarship Council for financial support and the Swedish

National Infrastructure for Computing (SNIC) for providing powerful computational

capability.

Finally, my wordless gratitude goes to my husband Zhijun and my parents for your constant

support, forever love and everything!

VI

Abbreviations

AFM Atomic force microscopy

AMBER Assisted Model Building with Energy Refinement

Asp Asparate

CD Cyclodextrin

CHARMM Chemistry at HARvard Molecular Mechanics

ESP Electrostatic potential

Glc Glucose

Glu Glutamate

LINCS Linear Constraint Solver

MD Molecular Dynamics

NPT Isothermal-isobaric (ensemble)

NVE Microcanonical (ensemble)

NVT Canonical (ensemble)

OPLS Optimized Potentials for Liquid Simulations

PBC Periodic boundary condition

PMF Potential of mean force

RESP Restrained ElectroStatic Potential

SMD Steered molecular dynamics

TTF Tetrathialfulvalene

TZ Triazole

WHAM Weighted Histogram Analysis Method

XET Xyloglucan endo-transglycosylase

XG Xyloglucan

VII

Contents

1 Introduction ...................................................................................................................... 1

2 Molecular Dynamics Algorithms...................................................................................... 9

2.1 Basic Principle – the Newton’s Equation of Motion .................................................. 9

2.2 Numerical Integration Algorithms ............................................................................10

2.2.1 The Verlet Algorithm ......................................................................................10

2.2.2 The Leap-frog Algorithm................................................................................11

2.2.3 The Velocity-Verlet Algorithm........................................................................11

2.3 Periodic Boundary Condition ...................................................................................12

2.4 Treatment of Non-bonded Interactions .....................................................................13

2.5 Constraint Algorithms ..............................................................................................15

2.6 Temperature Coupling and Pressure Coupling Methods............................................16

2.6.1 Temperature Coupling Method .......................................................................16

2.6.2 Pressure Coupling Method..............................................................................18

3 Force Field Models ..........................................................................................................21

3.1 The Potential Energy Function Form........................................................................21

3.2 Force Field Parameterization....................................................................................25

3.3 Force Fields for Carbohydrates ................................................................................27

4 Potential of Mean Force ..................................................................................................29

4.1 Definition of Potential of Mean Force ......................................................................29

4.2 Methods for the Potential of Mean Force Calculation ...............................................31

4.2.1 Umbrella Sampling.........................................................................................31

4.2.2 Steered Molecular Dynamics Simulations.......................................................33

5 Summary of Papers .........................................................................................................37

5.1 Thermal Response of Crystalline Cellulose Iβ (Paper I) ...........................................37

VIII

5.2 Adsorption of Xyloglucan on Cellulose (Paper II) ....................................................42

5.3 Carbohydrate-Enzyme Interaction (Paper III) ...........................................................45

5.4 Working Mechanism of an α-cyclodextrin Based Rotaxane (Paper IV).....................48

6 Conclusions and Outlook.................................................................................................55

1

Chapter 1

Introduction

Face the past with the least lamentation

Face the present with the least waste

With the most dream to face future

Nature, through billions of years of trial and error, has evolved with effective and marvelous

solutions to innumerable complicated real-world problems. Humans have learnt from and

have been inspired by a variety of phenomena in nature. As the industrial age advances into

the biological era, we are able to penetrate deeper into nature’s secrets with the aid of modern

scientific techniques, to understand and learn more from her elegant designs. Biomimetics is a

field of science that investigates biological structures and processes for their use as models for

the development of artificial systems. [1] Biomimetic approaches are promising in the

development of new high-performance and environment-friendly materials. In this thesis,

properties related to biomimetic carbohydrate materials are investigated by molecular

dynamics (MD) simulation technique in atomic detail.

Carbohydrates, like other biomolecules such as proteins and nucleic acids, play important

roles in living things. Carbohydrate is the common name of a class of organic compounds

including polyhydroxy aldehydes, polyhydroxy ketones, and substances which yield such

compounds upon hydrolysis. Carbohydrates gained their name since the simplest sugars have

the empirical formula (CH2O)n, where n is three or more. Carbohydrates are classified by size

into mono-, di-, oligo- and polysaccharides. The open-chain and the closed-ring forms of a

monosaccharide often coexist, the latter being more common. The most occurring

carbohydrate rings are six-membered rings (pyranoses) and five-membered rings (furanoses).

There are several stereocenters in a monosaccharide and it is assigned as D- or L-

Chapter 1 Introduction

2

configuration according to the orientation of the asymmetric carbon most distant from the

carbonyl group. Take glucose for example, in a standard Fischer projection when the hydroxyl

groups on C2, C4, and C5 are on the right it is a D-glucose, otherwise it is an L-glucose (Figure

1.1 upper). A monosaccharide can also be classified as α- and β- anomers on the basis of the

position of the hydroxyl group on the anomeric carbon (C1) relative to the CH2OH group

bound to C5. They are either on the opposite sides in the α-anomer or the same side in the

β-anomer (Figure 1.1 lower).

Figure 1.1 (upper) The D- and L-glucose. (lower) The α- and β-glucose.

The plant cell wall is a dynamic biocomposite material, comprised of load-bearing

paracrystalline cellulose fibrils embedded in a matrix of cross-linking non-cellulosic

polysaccharides, (glyco)proteins, and polyphenolics, which undergoes continuous changes

during tissue differentiation and plant growth. [2] The cell wall accommodates a variety of

mechanical requirements during plant growth and demonstrates physical and mechanical

properties comparable to those of the best synthetic materials. In addition, declining oil

resources impose a need for seeking sustainable and renewable substitutes for

petrochemical-based materials. The cell walls represent highly sophisticated composite

materials that can provide inspiration on the design and fabrication of carbohydrate-based

materials that combine environmental friendliness and biocompatibility with high

performance and increased functionality for applications in advanced packaging, electronic

devices, vehicles and sports equipment, etc. [1]

Cellulose is the main constituent of plant cell walls and is the most abundant natural

polymer on Earth. Each year, photosynthesis converts more than 100 billion metric tons of

CO2 and H2O into cellulose and other plant products. [3] Cellulose is a linear polymer of

(1→4)-linked D-glucosyl units (Figure 1.2). Due to the low density and excellent load-bearing


3

properties, cellulose becomes an important raw material for many industries, such as paper,

wood, textiles and many other materials. [4, 5]

In plants, the extended glucan chains in cellulose interact with one another by hydrogen

bonds to form cellulose sheets. These sheets are further stacked on top of each other via van

der Waals interaction and weak hydrogen bonds. Thereby, a network of up to an average of 36

cellulose chains makes up the crystalline core of the cellulose fibril. The crystalline and

paracrystalline (amorphous) cellulose core is associated with hemicellulose (mainly

xyloglucan) and other polymers to form linear structures of high tensile strength known as

microfibrils. Each microfibril is about 10 to 20 nm in diameter. These crystalline microfibrils

act as the main load-bearing components of the cell wall.

Cellulose Chain

Figure 1.2 The hierarchical structure of cellulose microfibril. Schematic representation of a cellulose chain

comprising atom labeling and the definition of different torsion angles: Φ: O5-C1-O4’-C4’; Ψ:

C1-O4’-C4’-C5’; ω: O5-C5-C6-O6; τ2: C3-C2-O2-HO2; τ3: C4-C3-O3-HO3; τ6: C5-C6-O6-HO6.

Cellulose is polymorph and adopts various possible packing arrangements, with either

parallel chains (cellulose I, IIII, and IVI) or antiparallel chains (cellulose II, IIIII, and IVII) (for

a review of cellulose see ref. [6]) The so-called native cellulose corresponds to cellulose I and

occurs in the form of two different allomorphs represented by two distinct crystal phases

designated as I and Iβ, showing triclinic and monoclinic symmetry, respectively. The

main structural difference between cellulose I and Iβ is the relative position of the chains to

each other. Allomorph Iβ is thermodynamically more stable and can irreversibly be

transformed from I by annealing. The proportions of I and Iβ phases vary with the origin of

cellulose. Cellulose in higher plants (woody tissue, cotton etc.) consists mainly of the Iβ phase


4

whereas primitive organisms (bacteria, algae etc.) are enriched in the I phase. [7], [8] The

structures of different forms of crystalline cellulose have been elucidated with the aid of the

synchrotron X-ray and neutron fiber diffraction techniques. [9, 10, 11, 12, 13]

The properties of most cellulose-based materials used industrially are modified by

chemical treatment during the manufacturing process, for example to enhance the brightness

and to increase the smoothness and hydrophobicity of paper surfaces. However, the hydrogen

bond network could be destroyed and hence the structural integrity and mechanical strength

of the cellulose microfibril will be weakened by extensive chemical modification on the

hydroxyl groups. [14, 15, 16] Hence, in applications where the superb load bearing properties of

cellulose are required, coating, instead of direct chemical modification, is used to alter the

surface and the bulk mechanical properties of cellulose. In biocomposites, cellulose is usually

blended in a man-made polymeric matrix to achieve good mechanical properties. [17] However,

one major problem often encountered is the poor interfacial interactions between the natural

fibers and the polymeric matrix.

In 2004, a novel chemoenzymatic approach for the efficient incorporation of chemical

functionality onto cellulose surfaces was developed in the laboratory of Wood Biotechnology

at KTH. [18] This creative idea is inspired by the organization of the plant cell wall in which

the combination of strength and interfacial stability is achieved by coating cellulose with

xyloglucan (XG) that can interact with other cell wall polymers and proteins. The main

principle of this biomimetic method is to perform chemo-enzymatic synthesis on XG instead

of the hydroxyl groups on the cellulose surface. The modification is realized by using a

transglycosylating enzyme xyloglucan endo-transglycosylase (XET, E.C.2.4.1.207) to join

chemically modified XG oligosaccharides to XG, which has a naturally high affinity to

cellulose. [18]

Therefore, understanding the structural properties of cellulose, the interaction details of

the cellulose-XG assembly and the catalytic details of the enzyme XET on the substrate XG is

essential to improve our knowledge of the ultra-structural organization and function of the

plant cell wall and can further inform the use of XG in applications for biomimetic cellulose

modification. Due to a paucity of experimental methods for high-resolution (atomic-level)

structural analysis of poorly-crystalline solid materials, [19] computer simulations provide a

powerful alternative to explore atomic-level interpretation.

Thermal behavior is one of the most fundamental properties of solid-state materials.

Progress in the investigation of the thermal behavior of neat cellulose Iβ has been made more

recently by X-ray diffraction [20, 21] and temperature-dependent IR measurements [22, 23] using


5

highly crystalline cellulose samples. Temperature-dependent IR measurements have been

performed to explore the temperature-dependent changes in hydrogen bonds in the crystal,

and to elucidate the transition process at the functional group level. [22] The thermal response

of crystalline cellulose I is further explored at the atomic level by MD simulation in Paper I.

Figure 1.3 Common structural motifs of xyloglucan and xylogluco-oligosaccharides: XXXG (x = 0, y = 0,

z = 0), XLXG (x = 1, y= 0, z = 0), XXLG (x = 0, y = 1, z = 0), XLLG (x = 1, y = 1, z = 0) and XXFG (x =

0, y = 1, z = 1). The arrow indicates the site for endo-glucanase cleavage.

Contemporary models of the primary cell wall of dicots and diverse monocots suggest

that the non-covalent network formed by insoluble cellulose chains and the non-crystalline

polysaccharide XG is a central contributor to wall strength and extensibility. [2, 24, 25] XG is a

complex branched polysaccharide consisting of a cellulose-like β-(1→4)-glucan main chain

regularly substituted with branching -(1→6) xylosyl residues. Branching is both extensive

and highly regular, with xylose substitution at 50% to 75% of the glucosyl units, depending

on the source. [26, 27] The branching chains on XG dramatically change the physical properties

of the polymer compared to cellulose; XG is highly water soluble and cannot form ordered

crystalline microfibrils like cellulose. [28] In XGs typical of many dicots, the repeating motif is

based on the XXXG heptasaccharide. Further galactosylation and fucosylation give rise to

other common oligosaccharide units such as XXLG, XLXG, XLLG and XXFG. A single-letter

linear notation is widely used to represent the complex branching pattern of XG (See Figure

1.3). [29]

XG molecules can adsorb onto the cellulose surface to form a sheath around the cellulose

microfibrils preventing them from direct aggregation. Some XG molecules are trapped in or

between the microfibrils. Other XG molecules are believed to span the gap between different

cellulose microfibrils, providing a flexible and strong network in the primary cell wall. The


6

excellent review by Brumer et al. [30] presented a detailed overview on the knowledge of XG

structures, solution properties, and the mechanisms of interaction of XGs with cellulose. It has

long been suggested that the XG-cellulose interaction may involve alignment of the

polysaccharide chains. [31] Early molecular modeling adopted this hypothesis, although studies

differed on whether interchain association is mediated by glucose-glucose ring stacking

interactions or side-on hydrogen bonds. [32, 33, 34] However, a contemporary MD simulation by

Hanus and Mazeau has seriously questioned the necessity of a parallel chain alignment in the

XG-cellulose interaction. [35] XGs with different starting orientations can bind to all five

unique crystal faces of cellulose Iβ with similar binding energies. [35] One potential concern

with the work of Hanus and Mazeau is that the binding simulations were performed in vacuo,

i.e. in the absence of solvating water molecules. In paper II, we have re-examined the

XG-cellulose Iβ interaction in the presence of explicit water molecules.

The cleavage and non-hydrolytic re-arrangement of XG are catalyzed in vivo by XETs.

Thus XETs have the potential to modify the cellulose XG network and are most likely

involved in the reconstruction of cell walls during the plant growth and development. [36] XET

is related to the retaining glycosyl hydrolases. It catalyzes in the first reaction step the

cleavage of a glycosidic bond, which leads to the formation of a covalent enzyme-glycosyl

intermediate. In the second step, XET transfers the enzyme-glycosyl intermediate to a

carbohydrate acceptor. [ 37 , 38 , 39 ] The interaction between the tetradecasaccharide

XXXGXXXG in complex with the hybrid aspen xyloglucan endo-transglycosylase

PttXET16-34 was studied in Paper III.

A molecular machine is defined as a number of discrete molecular components which

perform mechanical-like movements (output) in response to specific external stimuli (input). [40] The most complex molecular machines are found within cells. These include motor

proteins, such as kinesin, which moves cargo inside cells away from the nucleus along

microtubules, myosin, which is responsible for muscle contraction, and dynein, which

produces the axonemal beating of motile cilia and flagella. [41] Construction of artificially

molecular machines is inspired by these biological machines and has been an active area of

supra-molecular research.

Cyclodextrins (CDs) are one kind of attractive ring components in the construction of

molecular machines. CDs are a family of cyclic oligosaccharides with sugar molecules bound

in a ring. The most common CDs are made up of six, seven and eight D-glucopyranosyl

residues (named as α-, β-, and γ-CD respectively) linked by α-1,4 glycosidic bonds. The

glucose residues have the 4C1 (chair) conformation. CDs can be topologically represented as


7

toroids with the larger and the smaller openings of the toroid corresponding to the secondary

and primary hydroxyl groups, respectively. The working mechanism for a redox-responsive

bistable [2]rotaxane incorporating an α-CD was studied by MD simulations in Paper IV.

Figure 1.4 The structure of α-CD


8

9

Chapter 2

Molecular Dynamics Algorithms

MD simulation is a technique used to evolve the motions of nuclei in a system composed of

nuclei and electrons. In MD simulation, the rapid motion of the electrons is averaged out

assuming that they adjust instantaneously to the comparatively slow motion of the heavy

nuclei (Born-Oppenheimer approximation). Thereby, the Schrödinger equation can be

separated into a time-dependent equation for the nuclei and a time-independent equation for

the electronic degrees of freedom. In classical MD simulations, the quantum aspects of the

nuclear motion are also neglected so that classical mechanics can be used to describe the

motion of atoms. Each nucleus propagates according to the Newton’s equation of motion

under the potential provided by all the nuclei and electrons in the system, thereby the structure

and other properties of the whole system are calculated.

2.1 Basic Principle – the Newton’s Equation of Motion

For a system of N atoms, the potential energy, U(r1, r2,..., rn), is a function of the coordinates

of these N atoms. To a good approximation, nuclei behave as classical particles and their

dynamics can thus be simulated by numerically solving the Newton’s equation of motion:

2

2= i

i i

i

dUm

dt

rF

r (2.1)

with 2

2

i ii

d d

dt dt

r va (2.2)

Here U is the potential energy of the system at time t. Fi is the force on atom i. mi and ri

Chapter 2 Molecular Dynamics Algorithms

10

are the mass and the coordinate of atom i. ai and vi are the acceleration and the velocity of

atom i, respectively.

2.2 Numerical Integration Algorithms

To describe the atomic motion, there are several finite-difference methods developed for the

numerical integration of the Newton’s equation of motion. In mathematics, finite-difference

methods are numerical methods for approximating the solutions to differential equations by

approximating derivatives with finite difference equations. The most popular integration

methods in MD simulations are the Verlet algorithm [42] and its velocity-explicit variants,

leap-frog [43] and velocity Verlet algorithms. [44, 45]

2.2.1 The Verlet Algorithm

Give the position at time t r(t), the position at a short time (Δt) later is represented by the

Taylor expansion:

2 32 3 4

2 3

( ) ( ) ( )1 1( ) ( ) ( )

2! 3!i i i

i i

d t d t d tt t t t t t O t

dt dt dt

r r rr r (2.3)

Substitute Δt with –Δt in eq. (2.3), the position at a short time (Δt) earlier is written as:

2 32 3 4

2 3

( ) ( ) ( )1 1( ) ( ) ( )

2! 3!i i i

i i

d t d t d tt t t t t t O t

dt dt dt

r r rr r (2.4)

Adding eqs. (2.3) and (2.4) gives the recipe for updating the configuration from the

current position and acceleration, and the previous position in the Verlet algorithm:

22

2

( )( ) 2 ( ) ( ) i

i i i

d tt t t t t t

dt

rr r r

22 ( ) ( ) ( )i i it t t t t r r a

(2.5)

Due to its simplicity and moderate requirement on memory, the Verlet algorithm was

widely used. The algorithm is exactly reversible in time. As seen from the Taylor expansions,

The error in the atomic positions is of the order of O(Δt4). Moreover, the new positions are obtained


11

by adding a small number ai(t)Δt2 to a difference between two large numbers (2ri(t) −

ri(t−Δt)). This may further lead to truncation errors due to finite precision. The Verlet

algorithm has another disadvantage that velocities do not appear explicitly, which is a

problem in connection with generating ensembles with constant temperature. The velocities

can be obtained via the following formula with an error of the order of O(Δt2).

( ) ( )( )

2i i

i

t t t tt

t

r rv (2.6)

2.2.2 The Leap-frog Algorithm

The leap-frog algorithm made improvement on the Verlet algorithm with respect to the

numerical aspect and the explicit velocity. The equations for propagating the atoms in the

leap-frog algorithm are given by:

( ) ( ) ( )2

i i i

tt t t t t

r r v (2.7)

( ) ( ) ( )2 2

i i i

t tt t t t

v v a (2.8)

A disadvantage is that the position and velocity updates are out of phase by half a time

step. Therefore, the exact kinetic energy contribution to the total energy cannot be calculated

at the same time as the positions are defined. The velocities at time t can be computed from:

)

2

1()

2

1(

2

1)( ttttt iii vvv

(2.9)

2.2.3 The Velocity-Verlet Algorithm

The disadvantage of the leap-frog algorithm is remedied in the velocity-Verlet algorithm,

where the position and velocity are updated by:

21( ) ( ) ( ) ( )

2i i i it t t t t t t r r v a (2.10)


12

1( ) ( ) ( ( ) ( ))

2i i i it t t t t t t v v a a (2.11)

The leap-frog algorithm is utilized in the GROMACS program. [46, 47] The velocity-Verlet

algorithm has become available in the GROMACS program since version 4.5. [48, 49]

2.3 Periodic Boundary Condition

MD simulations are performed on a limited number of particles at the cost of a reasonable

computational resource. In modern MD simulations, the number of particles in the simulated

system can reach a million, however, it is still far small compared with that in a macroscopic

piece of matter. In addition, the ratio between the number of surface particles and the total

number of particles would be much larger than in reality, which would lead to unwanted

surface artifacts in an MD model.

Figure 2.1 Two-dimensional representation of periodic boundary condition.

To solve this problem, the periodic boundary condition (PBC) is commonly applied in

MD simulations, where the central simulated box is surrounded by the periodic images of

itself. The arrangement and movement of particles in the periodic images are exactly the same

as in the central simulated box, which means that the particles move out from the box on one

side would re-enter the box instantaneously from the opposite side. This also ensures the

constant number of particles in the simulated system. However, caution needs to be exercised

to minimize the artifacts caused by the artificial periodicity if one wishes to simulate

non-periodic systems, such as liquids or solutions. [50, 51] The box shapes are limited to


13

space-filling boxes to ensure seamless connection between boxes, such as triclinic unit cells,

which cover cubic, rhombic dodecahedra, and truncated octahedra, etc. GROMACS is based

on triclinic unit cells.

2.4 Treatment of Non-bonded Interactions

Under PBC, the interactions between particles in different boxes are allowed. The number of

pair-wise interactions is increased to an enormous quantity because not only the interactions

between “real” particles but also those between “real” and “image” particles have to be

considered. The calculations of the two non-bonded terms (electrostatic and van der Waals

interactions) make up the computationally most expensive part in the integration step. In the

classical MD simulation, the inter-atomic interaction is often described by an empirical

potential defined in the force field. Most commonly, the van der Waals interaction is described

by the Lennard-Jones potential and the electrostatic interaction is modeled by interaction

between point charges according to Coulomb’s law (for details please refer to Chapter 3).

Usually, in order to enhance the computational efficiency, a cut-off radius (rcut) of about

1.0-1.5 nm is introduced for the explicit calculation of short-range non-bonded interactions.

The minimum image convention states that only one – the nearest – image of each particle is

considered for short-range non-bonded interaction terms, which also implies that the rcut used

to truncate non-bonded interactions may not exceed half the shortest box vector. [49]

With regard to the Lennard-Jones potential for the van der Waals interactions, the use of

cut-off requires long-range correction only for the dispersion term (−4εσ6/r6), since the

repulsion term (4εσ12/r12) decays so fast with respect to the inter-atomic distance that it can be

safely neglected beyond rcut. For a plain cut-off, the long-range correction for the

Lennard-Jones potential takes the form of

62

, 6

1 44 ( )

2cut

LJ LRC

r

V N r g r drr

(2.12)

where ρ=N/V is the particle density. If one assumes that the radial distribution function g(r)

equals 1 beyond rcut, the long-range correction simply reduces to:[45, 49]

6

, 3

8

3LJ LRC

cut

NV

r

(2.13)


14

On the contrary, for the long-range electrostatic interactions, the use of a cut-off is known

to cause severe artifacts due to its slow 1/r decay. [52, 53] Such long-range artifacts are

prevented by the use of lattice summation methods such as the Ewald summation methods [54]

capable of consideration of the full electrostatics in an efficient manner.

The idea of the Ewald methods is that the original electrostatic potential generated by

point charges is split into two separate parts: one short-range term that sums quickly in real

space and one long-range term that sums quickly in reciprocal space. An appropriate

Coubomb cut-off distance is chosen to separate the short- and long-range electrostatic

interactions. Two neutralizing charge distributions with exactly the same quantity but opposite

charge centered at the same position as each point charge are introduced within the Coubomb

cut-off distance, thereby yielding exactly the original point charge distribution (Figure 2.2).

The real space includes the original point charges and the charge distribution (screening

potential) with opposite charges to the original point charges. Usually, the screening potential

is chosen as a Gaussian. An appropriate selection of the Gaussian potential will lead to an

electrostatic potential approaching zero at certain distance in the real space. Therefore, the

electrostatic potential generated by the screened point charges become short-ranged in the real

space (Vreal) and can be simply evaluated. While the reciprocal space includes the negative of

the above screening potential within the Coubomb cut-off distance and Gaussian

representations for all the other point charges beyond the Coubomb cut-off distance. The

electrostatic potential in the reciprocal space (Vrecip) can be computed efficiently by Fourier

transformation methods. In addition, the self-interaction (Vself) introduced by the screening

potential must be subtracted. Finally, the total electrostatic potential is composed by three

parts:

coul real recip selfV V V V (2.14)

Figure 2.2 Representation of Ewald summation


15

By suitably adjusting the Gaussian function, optimal convergence of Vreal and Vrecip may

be achieved. The original Ewald method scales as O(N3/2) and therefore is not efficient

enough for use in very large systems. The Particle Mesh Ewald (PME) method, [55, 56] in

which the atomic charges are interpolated on grid points and the fast Fourier transformation is

used for the calculation of the reciprocal sum, is even more efficient than the Ewald

summation and scales as O(NlogN).

2.5 Constraint Algorithms

In MD simulations, a proper choice of integration step is important in order to save

computational time without loss of accuracy. When Newton’s equation of motion is integrated,

the highest frequency that occurs in the simulation system is the limiting factor of the chosen

time step. Usually, the vibrations of bonds involving hydrogen atoms are of the highest

frequency. For example, the bond stretching frequency of an O-H bond is about 104 Hz, which

limits the time step to be around 0.5 fs (the integration time step should be about 1/20 of the

fastest period).

In order to increase the integration time step, constraint algorithms are often introduced in

MD to maintain certain degrees of freedom (such as bond lengths or bond angles), which

allows to increase the time step to a typical value of 2 fs. Constraints can be imposed by the

traditional SHAKE [57] algorithm and LINCS (Linear Constraint Solver) algorithm. [58, 59]

In the SHAKE algorithm, the equations of motion must fulfill Nc holonomic constraints:

2 2( ) ( ) 0,k a b abt t d r r

1,..., ck N (2.15)

where the distance between two atoms, a and b, is constrained to a prescribed distance of dab

in the kth constraint and Nc the total number of constraints. The equations of motion are

expanded by the introduction of a constraining force in the form of Lagrange multipliers.

While in the LINCS algorithm, a different set of constraint equations is used:

( ) ( ) 0,k a b abg t t d r r

1,..., ck N (2.16)

The LINCS algorithm is 3 to 4 times faster than the SHAKE algorithm and it can be easily

parallelized. [58]


16

2.6 Temperature Coupling and Pressure Coupling Methods

The Hamiltonian of an isolated system is time-independent, translationally invariant and

rotationally invariant. Integration of the equations of motion for such a system leads to a

trajectory mapping a microcanonical (NVE) ensemble. However, the NVE ensemble does not

correspond to the conditions under which most experiments are performed. In most cases, a

canonical ensemble (NVT) or an isobaric-isothermal ensemble (NPT) will be a more suitable

representation of the real system. To achieve this, the equations of motion need to be modified

with temperature coupling and pressure coupling methods.

Table 2.1 Different types of thermodynamics ensembles used in the MD simulations

ensemble type constant parameters

microcanonical ensemble (NVE) fixed number of particles N, volume V and energy E

canonical ensemble (NVT) fixed number of particles N, volume V and temperature T

isobaric-isothermal ensemble (NPT) fixed number of particles N, pressure P and temperature T

grand canonical ensemble (μVT) fixed chemical potential μ, volume V and temperature T

2.6.1 Temperature Coupling Method

The temperature of the system T is related to the kinetic energy K by:

2

1

| | /2 / 2N

i i df Bi

K m N k T

(2.17)

where Ndf is the total number of degrees of freedom in the system, vi the velocity of particle i,

and kB the Boltzmann’s constant. The temperature of the system T can thus be adjusted

through the regulation of the velocity of particles. This is achieved by either the weak

coupling scheme of Berendsen, [60] the extended ensemble Nosé-Hoover Scheme, [61, 62] or the

velocity rescaling scheme. [63]

Berendsen Temperature Coupling

The Berendsen coupling scheme was developed in 1984 by Berendsen. The system is

supposed to be coupled with first-order kinetics to an external heat bath with reference


17

temperature T0. The change in temperature is proportional to the temperature difference

between the heat bath and the system, with a coupling strength determined by the coupling

parameter τ, as defined by:

0T TdT

dt

(2.18)

The heat flow into or out of the system is achieved by scaling the velocity every step with

the scaling factor λ:

01 ( 1)Tt

T

(2.19)

The Berendsen coupling scheme is simple to implement and efficient to relax the system

to the target temperature from far-from-equilibrium conditions. However, it does not produce

a correct canonical ensemble due to the neglect of the stochastic contribution to the

temperature fluctuations on the microscopic scale. The Berendsen coupling scheme is usually

employed before the production run.

Nosé-Hoover Temperature Coupling

The weak Berendsen thermostat is extremely efficient for relaxing a system to the target

temperature, but it might be more important to probe a correct canonical ensemble once the

system has reached equilibrium which can be achieved by the extended-ensemble

Nosé-Hoover scheme. In the Nosé-Hoover scheme, an additional degree of freedom s is

introduced with conjugate momentum Pξ and effective mass parameter Q. The system

Hamiltonian is modified by a friction term in the equation of motion, as given by:

2

2

i i i

i

d d

dt m dt

r F r (2.20)

where ξ = Pξ/Q. The equation of motion for the friction parameter ξ is:

20

d 1( 1)

di i df B

i

m N k Tt Q

v (2.21)


18

2.6.2 Pressure Coupling Method

The pressure P of the system is related to the box volume Vbox, the kinetic energy K and the

virial by:

2 1

3 2ij ij

i jbox

P KV

r F (2.22)

where Fij is the force on particle i exerted by particle j, and rij is the vector going from particle

j to i: rij =ri – rj. Similar to the temperature coupling, the system can also be coupled to a

“pressure bath” via weak Berendsen pressure coupling [60] or extended ensemble

Parrinello-Rahman pressure coupling. [64, 65]

Berendsen Pressure Coupling

In the Berendsen algorithm, the system is coupled to an external pressure bath with reference

pressure P0. The coordinates and box vectors are rescaled every step. The system pressure is

regulated with first-order kinetics according to the equation:

0

P

P PdP

dt

(2.23)

where τP is a time constant. The volume of the box is scaled by a factor χ, and the coordinates

are scaled by χ1/3 assuming the system is isotropic and the simulation box cubic. χ is given by:

01 ( )P

tP P

(2.24)

Here, is the isothermal compressibility, the value of which only affects the time

constant of the pressure relaxation but not the average value of pressure itself, thus is usually

set to the value of water, 4.6×10−10 Pa−1. The Berendsen pressure coupling does not yield the

exact NPT ensemble.

Parrinello-Rahman Pressure Coupling

Within the Parrinello-Rahman barostat, the box matrix h, formed by arranging the box vectors


19

{a, b, c}, obeys the matrix equation of motion:

21 T 1

02( ) ( )box

dV

dt

hW h P P (2.25)

where W is a matrix parameter that determines the strength of the pressure coupling and how

the box can be deformed. The equation of motion for the particles becomes:

2

2;i i i

i

d d

dt m dt

r rM

F T

1 T T 1( )d d

dt dt

h hM h h h h (2.26)

The combined use of the Nosé-Hoover thermostat and Parrinello-Rahman barostat can

provide realistic fluctuations in temperature and pressure when the thermodynamic properties

of a system are requested.


20

21

Chapter 3

Force Field Models

As shown in eq. (2.1), the force on a particle is defined as the negative gradient of a potential

energy function U with respect to the position of the particle. A force field refers to the form

and parameters of mathematical functions used to describe the potential energy of a system of

particles. There are several assumptions in the force field models: (1) Molecules are

represented by a “ball and spring” model, in which atoms are of different size and “softness”

and bonds are of different length and “stiffness”; (2) The potential energy function is

described as a sum of individual functions for bond stretching, angle bending, torsional

energies, and non-bonded interactions; (3) The transferability of a force field enables the

parameters derived from either experimental or high-level computational data of the small set

of models to be applied to a much larger set of related models. [66]

3.1 The Potential Energy Function Form

The force field energy is expressed as a sum of bonded interactions, non-bonded interactions

and sometimes cross-terms. It is an effective pair-wise potential parameterized in such a way

that a significant proportion of the many-body effect can be implicitly incorporated through

parameterization. The energy of the bonded interactions is generally accounted for by bond

stretching (Vbond), angle bending (Vangle), dihedral angle torsion (Vtorsion), and improper

dihedral torsion (Vimproper) terms. The energy of the interactions between non-bonded atoms is

accounted for by van der Waals (VvdW) and electrostatic (Velec) terms. Cross-terms (Vcross) are

included in the second-generation force fields to achieve higher accuracy, describing the

coupling between the bonded terms.

Chapter 3 Force Field Models

22

FF bonded non-bonded crossV V V V

bonded bond angle torsion improperV V V V V

non-bonded vdW elecV V V

Figure 3.1 Schematic representations of the key components in the potential energy function. [66]

3.1.1 Bond Stretching

The simple harmonic potential is a reasonable approximation to the shape of the potential

energy curve near the minimum of the potential well for the bond stretching although it is less

accurate away from equilibrium. The functional form is:

2bond

1( ) ( )

2b

ij ij ij ijV r k r b (3.1)

The Morse potential is particularly useful for the description of potential energy curve of

a typical bond. It has an asymmetric potential well and a zero force at infinite distance. It is

capable of describing a wide range of behavior from equilibrium behavior to bond

dissociation. The functional form is:

2morse ( ) [1 ( ( ))]ij ij ij ij ijV r D exp r b (3.2)

with 2

ij

ij

k

D (3.3)

Here Dij is the well depth, bij is the reference bond length, and k is the stretching constant

of the bond. However, it is not so widely adopted in force fields since it is not so

computationally efficient to implement and for near equilibrium situations it is good enough

to use the harmonic approximations.


23

3.1.2 Angle Bending

The deviation of angles from their reference values can also be approximated by a harmonic

potential on the angle θijk:

0 2angle

1( ) ( )

2ijk ijk ijk ijkV k (3.4)

Sometimes the Urey-Bradley angle potential is used, where an additional harmonic

correction term on the distance between the end atoms i and k is added to the harmonic

potential on the angle θijk:

0 2 0 2angle

1 1( ) ( ) ( )

2 2UB

ijk ijk ijk ijk ijk ik ikV k k r r (3.5)

3.1.3 Proper Torsional Terms

The proper torsional angle potential function models the presence of steric barriers between

atoms separated by three covalent bonds. The motion associated with this term is a rotation

around the middle bond. This potential is periodic in a 360° rotation and is most commonly

expressed as a cosine series expansion. The torsional potential can take the periodic form of:

torsion ( ) [1 cos( )]2

nijkl ijkl n

n

VV n (3.6)

Here in each term Vn is the torsional rotation force constant, which indicates qualitatively

the relative barrier. n is the multiplicity. ϕn is the phase angle usually chosen so that terms

with positive Vn have minima at 180°.

3.1.4 Improper Torsional Terms

An improper torsion angle is a “virtual” torsion angle in which the four atoms are not bonded

in the sequence i-j-k-l. Improper torsional potentials are introduced to keep planar groups

planar or to maintain stereochemistry at chiral centers. A stiff harmonic improper term in the

angle χ or in the distance d (the equilibrium angle or distance for a planar structure is zero) is

usually employed:


24

2improper

1( )

2V k or 2

improper

1( )

2V d kd

(3.7)

3.1.5 Van der Waals Interaction

The essential feature of the van der Waals interaction is that the interaction energy is zero at

infinite inter-atomic distance rij; as rij is reduced, the energy decreases and passes a minimum;

then the energy rapidly goes up as rij decreases further. The van der Waals interaction can be

decomposed into two parts: the long-range dispersive interaction and the short-range

repulsive interaction. The dispersive force arises from the induced dipole-dipole interaction

due to the fluctuations in the electron clouds and the dispersion interaction approximately

varies as –1/rij6. The repulsive force is attributed to the repulsion between the electrons in the

internuclear region due to the Pauli principle. [66] There are different potential forms for the

description of the repulsion part.

The Lennard-Jones 12-6 function is the most widely used potential function for

describing van der Waals interaction.

12 6

LJ ( ) 4 ij ij

ij ij

ij ij

V rr r

(3.8)

where σ is the collision diameter (the separation for which the energy is zero) and ε the well

depth. The Lorentz-Berthelot mixing rules are applied for the evaluation of the cross-terms in

the non-bonded Lennard-Jones interactions:

1( )

2ij ii jj

ij ii jj

(3.9)

The twelfth power term is found to be quite reasonable for rare gases but is rather too

steep for other systems such us hydrocarbons. Nevertheless the 12-6 function is widely used

due to computational efficiency reason.

The Buckingham potential has a more flexible and realistic exponential function for the

repulsive term than the 12-6 potential.


25

bh 6( ) exp( ) ij

ij ij ij ij

ij

CV r A B r

r (3.10)

The combination rules for the Buckingham potential are:

1( )

2

ij ii jj

ij ii jj

ij ii jj

A A A

B B B

C C C

(3.11)

3.1.6 Electrostatic Interaction

The electrostatic interactions between atom pairs can be modeled by assigning partial atomic

charges to each atom or assigning a bond dipole moment to each bond. The atomic charge

model is easier to parameterize and is more commonly adopted in almost all force fields. The

interaction between point charges is given by Coulomb’s law:

elec

0

( )4

i j

ij

ij

q qV r

r (3.12)

3.2 Force Field Parameterization

Once the potential energy function form is determined, the parameters in the potential energy

function need to be derived. These parameters can be determined from fitting against

experimental data or quantum mechanics calculations. The use of quantum mechanics results

allows the determination of force field parameters for molecules where little or no

experimental data exist. The well-established macromolecule force fields include AMBER, [67,

68, 69] CHARMM, [70, 71] GROMOS [72, 73] and OPLS-AA.[74] The general AMBER force field [75] is compatible with AMBER and is designed for small organic molecules and drug

molecules. Each force field has its own protocol for parameter development.

Parameters for bond stretching and angle bending can be treated separately from the

others. [66] The reference bond distances and angles can be determined from crystallography.

The bond and angle force constants can be obtained from vibrational spectra or from quantum


26

mechanics calculations. These parameters can often be transferred from one force field to

another without modification with little effect on the results.

In contrast, the performance of a force field is quite sensitive to the parameters for

non-bonded and torsional contributions. These parameters are closely coupled and can

significantly affect each other. [66] Most force fields include non-bonded interactions for atom

pairs that are separated by three bonds or more. But the 1-4 interactions between atom pairs of

three bonds apart are in many cases scaled by a scaling factor between 0 and 1. For example,

in the AMBER force field, the scaling factors for the 1-4 electrostatic interaction and the 1-4

van der Waals interaction are 0.8333 and 0.5, respectively. In the GLYCAM force field, the

scaling factors are set to be unity. The use of scaling factors is for the purpose of

transferability of the force field parameters, especially the torsional terms. In the AMBER

force field, the torsional parameters depend solely on the atom types of the two atoms that

form the central bond, while in the GLYCAM force field, all atomic sequences have an

explicit defined set of torsional terms.

Van der Waals parameters are often obtained by fitting to experimental liquid properties

such as density, or fitting to the intermolecular interaction energy. The method of Monte Carlo

simulations on pure liquids to reproduce the experimental thermodynamic quantities proposed

by Jorgensen et al. [74] is used in both OPLS-AA and AMBER force field for deriving the van

der Waals parameters.

The atomic charges can be fitted to experimental liquid properties, X-ray crystallographic

electron density, or be fitted to electrostatic potential (ESP) calculated by quantum mechanics

methods. Within the ESP charge scheme, the ESP ϕesp arises from the nuclear charges and the

electron density in the form of:

esp

( ')( ) d '

| | '

Nnuca

a a

Z

rr r

R r | r - r | (3.13)

The fitting of the atomic charges Qa to the potential ϕesp is carried out by minimizing the

following error function under the constraint that the sum of the atomic charges Qa equals the

total molecular charge.

2points

esp

( )ErrF( ) ( )

| |

N Natomsa a

r a a

QQ

Rr

R r (3.14)

However, the ESP points used in fitting the charges must lie outside the van der Waals


27

surface of the molecule. [76] Hence “buried” charges (e.g. an sp3 carbon) tend to be poorly

determined because even the closest surface points at which the ESP is evaluated are

relatively far away. This problem has been resolved to some extent in the Restrained

ElectroStatic Potential (RESP) fitting scheme, [77] in which a penalty function is added in the

form of restraints on non-hydrogen atomic charges to target charges. The RESP scheme is

used in the AMBER force field. A more accurate modeling of the many-body effects requires

inclusion of polarizable effects. The average many-body effect is often approximately

accounted for by using a set of empirical charges which often correspond to a molecular

dipole moment 10–20% larger than that for the isolate molecule, or by fitting to e.g.

Hartree–Fock results with for instance 6-31G(d) basis set that are known to overestimate the

polarity.

Finally, the torsional angle parameters are usually fitted to the energy curve determined

from quantum chemical calculations for rotations around a bond, in combination with the van

der Waals parameters and partial charges. [66]

It is worth noting that each force field is devised under specific conditions and is

optimized for specific kind of molecules, such as biomolecules (nucleic acid, protein, lipid,

and carbohydrate), organic molecules, or inorganic molecules. It is crucial to choose an

appropriate force field for the studied system in order to get reasonable and reliable results

from MD simulations.

3.3 Force Fields for Carbohydrates

The inherent flexibility of carbohydrates, combined with the multitude of possibilities for

linkage and functionality, provides a unique challenging endeavor in the development of

carbohydrate force fields. [78] The growing interest in structural glycobiology has stimulated

the need to develop protein-consistent carbohydrate force fields. The above mentioned

macromolecular force fields AMBER, CHARMM, GROMOS and OPLS-AA usually can not

accurately reproduce the carbohydrate conformation, such as the experimental solution-phase

rotamer populations for the glycosidic linkages as well as the ω-angle. Therefore, special

carbohydrate parameter sets have been implemented into these macromolecular force fields,

including the GLYCAM series (complementary to AMBER), [79, 80, 81] and carbohydrate-tuned

versions of CHARMM, [82, 83, 84] GROMOS-45a4 [85] and OPLS-AA-ESI [86] force fields. The

current state of carbohydrate force fields has been reviewed by R. J. Woods. [87]


28

29

Chapter 4

Potential of Mean Force

4.1 Definition of Potential of Mean Force

Free energy is the underlying driving force behind a physical or chemical process of a system.

The free energy calculation is thus of supreme importance for our exploration of the

microscopic origin of the measurable macroscopic quantities. In statistic mechanics, for an

NPT ensemble, the Gibbs free energy G is related to the partition function ZN by:

1( , , ) ln ( , ) ln ( , )B N NG T P N k T Z T P Z T P

(4.1)

with β=1/kBT. The partition function of the system writes:

H( , )( , )N NN N

NZ T P d d e p xx p (4.2)

where H is the classical Hamiltonian for a system of N interacting particles. H is composed of

the kinetic energy and the potential energy. The kinetic energy depends on the momentum pi

of the particles, and the potential energy is a function of the positions of particles xN.

2

1

( , ) ( )2

NN N Ni

i i

Vm

p

p x x (4.3)

However, accurate calculation of the absolute free energy via eq. (4.1) is not possible for

a system consisting of thousands of atoms due to insufficient sampling during finite length

Chapter 4 Potential of Mean Force

30

simulations. The problem becomes much simplified if only the free energy difference is

required. For many practical applications, the change in free energy as a function of a reaction

coordinate, named potential of mean force (PMF), is of particular interest. The free energy of

the system for which a given observable ξ(x,p) is constrained to a generalized reaction

coordinate ξ is called constrained free energy. [88] A PMF is a constrained free energy. The

constrained free energy evaluates the impact of a subset of parameters on the free energy. The

constrained partition function Z(ξ) and constrained free energy G(ξ) are given by:

H( , )( ) ( ( , )- )N NN NZ d d e

p xx p x p (4.4)

1( ) ln ( )G Z

(4.5)

The probability (ρ(ξ)) to find the system in a state described by ξ is given by the ratio of

Z(ξ) and ZN. In this thesis, only when the reaction coordinate is a function of position, such as

the separation between two sets of particles, is considered. The ρ(ξ) thus takes the form of:

( )

( )

( ( )- )( )( )

N

N

N V

N VN

d eZ

Z d e

x

x

x x

x (4.6)

Then the free energy difference between two constrained states can be written as:

1 ( ) 1 ( )ln ln

( ') ( ')

ZG

Z

(4.7)

It can be seen that the relative probability of finding a particle changes exponentially with

the barrier height. In practice, when the barrier is as high as several multiples of kBT, the

distribution function ρ(ξ) is difficult to converge in a standard MD simulation within

reasonable computational time due to insufficient sampling. Several methods have been

proposed in order to overcome the insufficient sampling problem by guiding the system along

a predefined reaction coordinate, including the non-Boltzmann sampling techniques such as

umbrella sampling, [89] and steered MD (SMD) simulations. [90]


31

4.2 Methods for Potential of Mean Force Calculations

4.2.1 Umbrella Sampling

The method of umbrella sampling was proposed by Torrie and Valleau. [89] The essential idea

of this method is to overcome the sampling problem by modifying the potential function with

a restraining potential so that the unfavorable states in the high-energy regions can be sampled

adequately. The external restraining potential is designed such that its effect on the underlying

free energy can be removed.

The restraining potential in the modified potential function can be seen as a perturbation

to the original one:

'( ) ( ) ( )N NiV V x x (4.8)

Usually, we use the restraining potential that is a function of coordinates only. The

external restraining potential ωi(ξ) usually takes a quadratic form:

21( ) ( )

2i ik (4.9)

Figure 4.1 Schematic representation of umbrella sampling technique.

The basic principle of umbrella sampling is illustrated in Figure 4.1. The reaction

coordinate is divided into a number of simulation windows with reference positions at ξi. The

restraining potentials are then chosen in such a manner that the sampling distribution is


32

shifted along the reaction coordinate. The width of the simulation window is controlled by the

magnitude of the force constant k in the restraining potential. Sufficient overlap should be

guaranteed between the sampling distributions of adjacent simulation windows in order to

derive a continuous energy function from these simulations. The restraining potential should

not be too large either; otherwise the results may take too long time to converge.

Referring to eq. (4.6), the probability to find the particle at position ξ due to the

restraining potential ωi(ξ) in the ith window is given by:

( ( ) ( ))

( ( ) ( ))

( ( )- )( )

Ni

Ni

VN

biasi VN

d e

d e

x

x

x x

x

( )

( )( )

i

ii

e

e

(4.10)

Let Wbias(ξ) and W(ξ) denote the PMF associated with ξ when the simulation is

performed with or without the restraining potential, respectively, then we get:

( )

( )

( )1 1W( ) W ( ) ln ln

( )

i

i

bias ibiasi

e

e

(4.11)

Fi is named free energy constant associated with the biasing potential and is defined as:

( ) ( ) ( )i i iFe e d e (4.12)

where ρ(ξ) is the overall unbiased distribution function. Then we get:

W( ) W ( ) ( )biasi iF (4.13)

In order to get W(ξ), the unknown Fi needs to be resolved. To minimize the statistical

error, the Weighted Histogram Analysis Method (WHAM) [91] is widely used to determine Fi.

The WHAM equations express the optimal estimate for the unbiased distribution function ρ(ξ)

as a weighted sum over the individual unbiased distribution functions ρi(ξ). The following

WHAM equations are solved iteratively until self-consistency is achieved.


33

( ( ) )

( ( ) ) ( ( ) )1 1

1 1

( ) ( ) ( )i i

i i i i

FN Nbiasi i

i iN NF Fi ij jj j

n e n

n e n e

( ) ( )i iFe d e

(4.14)

where N is the number of biased window simulations and ni is the number of counts in the ith

window.

4.2.2 Steered Molecular Dynamics Simulations

Figure 4.2 The schematic figure of SMD principle. The green (left) atom represents the SMD atom and the

red (right) atom represents the dummy atom. They are connected by a virtual harmonic spring.

SMD simulations apply external steering forces in the desired direction to accelerate

processes that otherwise are too slow due to energy barriers. SMD simulation is a technique

mimicking the principle of the atomic force microscopy (AFM). There are two types of SMD

simulations. In the first protocol, named constant force SMD, a constant force is applied to

one or more atoms; displacement is recorded throughout dynamics. In the second protocol,

called constant velocity SMD, a harmonic potential (spring) is used to induce motion along a

reaction coordinate (see Figure 4.2). The free end of the spring is moved at constant velocity,

while the atoms attached to the other end of the spring (SMD atoms) are subject to the

steering force. The force is determined by the extension of the spring and is calculated as:


34

( ) ( ) ( )i i i

t U t U tx y z

F i j k

2

0

1( ) ( )

2U t k vt r r n

(4.15)

where r and r0 are the instantaneous position and the initial position of the SMD atoms, n is

the direction of pulling and k the force constant for the virtual spring.

The calculation of free energy from SMD simulations is based on multiple

non-equilibrium MD simulations. The Jarzynski’s equality [92] is typically applied to extract

equilibrium free energy difference from the non-equilibrium work distribution:

G We e (4.16)

where < > denotes an ensemble average of the work W from non-equilibrium simulations.

However, the average of exponential work appearing in Jarzynski’s equality is dominated by

the small work values which are very rare. Appropriate sampling of such rare trajectories is

thus required for an accurate estimate of free energy. Hence, Jarzynski’s equality is considered

practically applicable only for slow processes of which the fluctuation of work is comparable

to the temperature. Yet, in practice, a pulling speed of several orders of magnitude higher than

the quasi-static speed is employed in the simulation and only a small number of trajectories

can be sampled due to limited computing capability. To overcome this difficulty, approximate

formulas via the cumulant expansion of We were proved to be effective. [93, 94, 95]

2 32 32 3 2ln 3 2 ......

2! 3!We W W W W W W W

(4.17)

If the distribution of work W is Gaussian, the third and higher cumulants are identically

zero. [96] It has been shown that with a limited number of trajectories, the second order

cumulant expansion yields the most accurate estimate. [90]

22

2G W W W

2

2WW

(4.18)


35

where σW is the standard deviation of the work.

The proper choice of the force constant k in eq. (4.15) is important in SMD simulations.

The force constant should be chosen to be large enough to ensure that the reaction coordinate

closely follows the constraint position. But too large a k is also not allowed since PMF

calculated with a larger k shows larger fluctuation which scales as Bkk T .[97]


36

37

Chapter 5

Summary of Papers

5.1 Thermal Response of Crystalline Cellulose Iβ (Paper I)

Thermal behavior is one of the most fundamental properties of solid-state materials. Progress

in investigations of the thermal behavior of neat cellulose Iβ has been made by both

experimental [20-23] and theoretical [98] approaches. Atomic MD simulations of the thermal

response in crystalline cellulose Iβ was initiated by Bergenstråhle et al. [98] with the

GROMOS 45a4 force field. However, a relatively large discrepancy in the predicted unit cell

parameters a and γ compared to the crystallographic data was found, which might lead to a

controversial prediction on the thermal transition mechanism. The limited capability to

describe the van der Waals and intersheet coulombic interactions between the united CH

atoms and oxygen by the GROMOS 45a4 force field might be the reason.

Figure 5.1 Projection of the ab base plane of the crystal model for cellulose Iβ. Reprinted with permission

from Springer.

Chapter 5 Summary of Papers

38

The reported good performance of the GLYCAM06 force field [99]motivated us to apply

the force field to explore the atomic details regarding the thermal response of cellulose Iβ in

paper I. The crystal model for cellulose Iβ (Figure 5.1), built from the coordinates reported by

Nishiyama et al., [10] is composed of 6 × 6 chains. Each chain was composed of eight glucose

units and was covalently bonded to its own periodic image across the simulation boxes so that

infinite cellulose chains were modeled. Simulations were performed at seven different

temperatures (298, 350, 400, 450, 475, 500 and 550 K) to investigate the thermal response

behavior of crystalline cellulose Iβ.

5.1.1 Low Temperature Structure

At first, the structural properties for cellulose Iβ at 298 K were evaluated and compared to the

crystallographic data (Table 5.1, Figure 5.2). The predicted density was 1603.4 kg/m3 at room

temperature, which is about 2.0% larger than the experimental value (1636 kg/m3) [10]

determined from the diffraction data. The values for the unit cell axes of a (intersheet), b

(interchain) and c (chain direction) at 298 K were calculated to be 0.763, 0.823 and 1.080 nm,

respectively. The predicted unit cell angles for α, β and γ were 89.99, 89.99 and 97.17°,

respectively. The deviations of these calculated unit cell parameters from the crystallographic

data were quite small, with the largest deviation (4.0%) occurring for c.

Table 5.1 Predicted unit cell parameters and experimental data

Unit cell axial length (nm) Unit cell axial angle (°)

a b c α β γ

298 K (this work) 0.763 0.823 1.080 89.99 89.99 97.17

500 K (this work) 0.811 0.828 1.078 90.00 89.96 98.33

Exp. room temperature[10] 0.778 0.820 1.038 90.0 90.0 96.5

Exp. 523K [21] 0.819 0.818 1.037 90.0 90.0 96.4

Figure 5.2 Deviations for the calculated unit cell parameters with respect to crystallographic data.


39

An extensive comparison with the simulation data from other reports [98, 100, 101, 102] (see

Table 1 in paper I) allows an evaluation of different force fields. A remarkable advantage of

our results is manifested by the fact that the GLYCAM06 predicts the most accurate a and b

values among all the tested force fields. The a and b axes, corresponding to the inter-sheet and

inter-chain distances in cellulose Iβ, respectively, rely heavily on intermolecular non-bonded

interactions. The success of the GLYCAM06 in reproducing the a and b values should be

attributed to its partial atomic charge derivation scheme of fitting the partial atomic charges to

the quantum mechanical molecular ESP, especially the implementation of ensemble-averaged

charge sets, which has the advantage of reproducing intermolecular interaction energies. In

addition, the GLYCAM06 yields unit cell angles closest to the experimental values compared

to the other force fields. On the other hand, the slightly overestimated c axis by the

GLYCAM06 probably results from the bonded force field parameters, i.e. the reference values

for the bond stretching and angle bending parameters derived from gas phase quantum

mechanical calculations on small molecular fragments; the torsional terms derived from

density functional theory B3LYP/6-311++G(2d, 2p) calculations.

Another important structural characteristic of crystalline cellulose Iβ is its stabilizing

hydrogen bonding networks. The hydrogen bonds were analyzed based on geometric criteria.

At 298 K, two intrachain hydrogen bonds O2H2... O6, O3H3...O5 and one interchain (but

intrasheet) hydrogen bond O6H6...O3 were systematically detected on both the origin and

center chains (Figures 5.3, 5.4), which was consistent with the dominant hydrogen bonding

network determined experimentally. [10] On the whole, the structural properties of cellulose Iβ

at room temperature are satisfactorily reproduced by the GLYCAM06 force field.

5.1.2 Transition

The structure for cellulose Iβ was explored at different temperatures by the GLYCAM06 force

field. A sudden drop in density is an indication of a phase transition occurring between 475

and 500 K. In Figure 5.3, we can see that the a axis expanded slowly from 0.763 nm at 298 K

to 0.777 nm at 475 K, corresponding to a 1.9% expansion at 475 K from room temperature.

At 500 K, the a axis experienced a sudden increase of a 6.3% expansion from 298 to 500 K

(0.811 nm). On the other hand, the b axis showed anisotropic expansion by only about 0.6%

from 0.823 nm at 298 K to 0.828 nm at 500 K. On the contrary, the c axis showed slight

shrinkage at higher temperature. The monoclinic angle γ displayed a small variation in the

range 97.11–98.33° in our simulation. It is noteworthy that there were only small shifts

observed in the α and β angles during the whole heating process, indicating no chain sliding


40

effect during the transition, which is in strong contrast with the previous simulation result. [98]

Figure 5.3 Unit cell axes a, b, c, unit cell angles α, β, γ, and density of cellulose Iβ as a function of

temperature.

5.1.3 High Temperature Structure

It has been revealed by X-ray experiments [20] and temperature-dependent IR measurements [22], [23] that the cellulose crystal will change into a high-temperature phase upon heating.

Actually, Wada et al. [21] have recently confirmed by synchrotron X-ray fiber diffraction that

the high-temperature phase had a staggered half of a glucose residue along the chain axis

between the sheets, similar to the structure of cellulose Iβ, which indicates that no

longitudinal chain slippage is involved during the transition. The unit cell of the cellulose

high-temperature phase at 523 K was determined to have a two-chain monoclinic structure

with dimensions of a = 0.819 nm, b = 0.818 nm, c = 1.037 nm, and γ = 96.4°.[21]


41

Figure 5.4 (left) The low- and high-temperature structures for cellulose Iβ. Origin and center chains are

colored in red and blue, respectively (right). The corresponding hydrogen-bonding network for the low-

and high-temperature structures.

The high-temperature structure of cellulose Iβ predicted by the GLYCAM06 force field is

shown in Table 5.1 and Figure 5.4. The unit cell parameters calculated at 500 K are in

reasonable agreement with experiment. [21] At 500 K, the conformation of all hydroxymethyl

groups (ω O5-C5-C6-O6, defined in Figure 1.2) changed from tg (ω=180°) to either gt (ω=60°)

on origin chains or gg (ω=300°) on center chains. Meanwhile, change of the conformation of

hydroxyl groups also took place, especially for the hydroxyl groups at C2 and C6 positions. In

addition, in the high-temperature structure, all the origin chains rotated around the helix axis

by about 30°.

The change in the orientation of the hydroxymethyl and hydroxyl groups led to the

reorganization of the hydrogen-bonding network (Figure 5.4). At 500 K, the O2H2…O6

intrachain hydrogen bond completely disappeared and was substituted by the weaker

O2H2…O6’ interchain hydrogen bond. The number of the O3H3…O5 intrachain hydrogen

bonds in the center chains decreased by about 20%. The O6H6…O3’ interchain hydrogen

bonds collapsed completely at 500 K in the center chains and decreased to 28% of the original

population in the origin chains. Two new intersheet hydrogen bonds (Figure 5.5) emerged in

the high-temperature structure, namely O6H6…O2’, where only the primary OH groups at the

C6 position in the center chains act as donors, and O6H6…O4’, where 40% of the primary

OH groups at the C6 position in the origin chains act as donors.


42

Figure 5.5 A representative example of the intersheet H-bonds in cellulose I at 500 K. Reprinted with

permission from Springer.

In summary, compared to the previously employed GROMOS 45a4 force field,

GLYCAM06 yields data in much better agreement with experimental observations, which

reflects that a cautious parameterization of the non-bonded interaction terms in a force field is

critical for the correct prediction of the thermal response in cellulose crystals.

5.2 Adsorption of Xyloglucan on Cellulose (Paper II)

In paper II, we studied the adsorption of XGs on the surface of cellulose, in particular the

effects of explicit water and side chain variation. The adsorption properties were studied in

detail for three XGs on cellulose Iβ 1-10 surface in aqueous environment, namely

GXXXGXXXG, GXXLGXXXG and GXXFGXXXG, which differed in the length and

composition of one side chain.

5.2.1 Interactions Between XGs and Cellulose Iβ 1-10 Surface

The instantaneous interactions between XGs and cellulose Iβ 1-10 surface were analyzed after

the equilibration states were reached. In Figure 5.6, the total instantaneous interaction energy

(Eint) between XGs and cellulose Iβ 1-10 surface with different starting orientations, the

electrostatic (Eelec) and van der Waals (EvdW) contributions, along with the standard deviations,

are depicted. The results showed that each XG was able to adsorb onto the cellulose surface

regardless of the orientation; the adsorption of XGs on cellulose was stabilized by both Eelec

and EvdW; in addition, both the backbone and side chain residues interacted with the cellulose

surface. These characteristics are in agreement with the results of Hanus et al., [35] where the

XG-cellulose interaction was modeled in vacuo. In our model, when water molecules were

included, the interaction energy between XG and cellulose averaged over all the orientations

was -126.1, -123.0, and -117.8 kcal/mol for GXXXGXXXG, GXXLGXXXG and

GXXFGXXXG, respectively, indicating a small impact on the interaction energy by variation


43

in one side chain in our model. When the interaction energy per repeat unit was compared, it

was found that it was considerably lower in our solvated model than in Hanus et al.’s [35]

vacuo model. Moreover, a slightly prevailing role of EvdW in the total interaction energy

between cellulose and XG was revealed in our model, opposed to the previous conclusion [35]

on a 75% contribution of Eelec.

Figure 5.6 The electrostatic interaction energy Eelec, and the van der Waals interaction energy EvdW between

GXXXGXXXG (A), GXXLGXXXG (B) and GXXFGXXXG (C) and cellulose Iβ 1-10 surface with

different starting orientations, respectively, averaged over the last 5 ns for each run.

In order to investigate the effect of water, the water molecules in the solvated systems

with the largest interactions (GXXXGXXXG (A-[180°]), GXXLGXXXG (A-[0°]) and

GXXFGXXXG (A-[270°])) were removed and simulations were continued on these

desolvated systems. The interaction energies in vacuo were also found to be considerably

higher within the same force field, especially the electrostatic component Eelec. One factor that

contributes greatly to the higher Eint in vacuo is the larger number of hydrogen bonds formed

between XG and cellulose in vacuo than in water. By observation, in the solvated model, there

were several water molecules between XG and cellulose. Meanwhile, water mediated

hydrogen bonds were also present. Due to the absence of opportunities for hydrogen bonding

to water molecules, XG moved closer to the cellulose surface in vacuo accounting for the

larger interaction. Therefore, when interaction energies are considered, it is important to take

water molecules into account in the theoretical modeling of this system.

5.2.2 Structural Properties

The equilibrium structures of GXXXGXXXG (A-[180°]), GXXLGXXXG (A-[0°]) and

GXXFGXXXG (A-[270°]) adsorbed on the cellulose Iβ 1-10 surface, which are of the largest

interaction energies for each XG, are depicted in Figure 5.7. Two important structural

parameters are the conformation of the glycosidic bonds in the backbone and that of the side

chains. Analysis of the conformation around the glycosidic bonds revealed that for


44

GXXXGXXXG, all the glycosidic bonds took ‘twist’ conformations, while for

GXXLGXXXG and GXXFGXXXG, half of them took ‘flat’ conformations. Thus, extension

of XG side chains with galactosyl and fucosyl residues appeared to favor flattened

conformations of the XG backbone upon adsorption. Analysis of the branching side chains

orientation relative to the main chain showed that all the side chains in GXXXGXXXG were

closer to the gg conformation, while for GXXLGXXXG and GXXFGXXXG, the side chains

in the vicinity of the galactosyl and fucosyl residues tended to assume the tg or gt

conformations, facilitating extended conformations of the di- and trisaccharide side chains,

which leads to favorable interactions of both backbone and side chain residues with the

cellulose surface.

Figure 5.7 The equilibrium structures of GXXXGXXXG (A-[180°]), GXXLGXXXG (A-[0°]) and

GXXFGXXXG (A-[270°]) on cellulose Iβ 1-10 surface. Water molecules are omitted for clarity. Glucosyls

are indicated in magenta, xylosyls in cyan, galactosyl in yellow and fucosyl in green.

5.2.3 Binding Affinity Evaluated by SMD Simulations

To evaluate the binding affinity of the cellulose surface to XGs, free energy profiles for the

desorption of each XG from the surface were derived from the SMD simulations via

Jarzynski’s equality. SMD simulations were performed for each XG-cellulose system with the


45

largest interaction energy (GXXXGXXXG (A-[180°]), GXXLGXXXG (A-[0°]) and

GXXFGXXXG (A-[270°])) to generate representative results.

Figure 5.8 Free energy profiles for GXXXGXXXG (A), GXXLGXXXG (B) and GXXFGXXXG (C)

desorbing from the cellulose Iβ 1-10 surface over 25 ns of the SMD trajectory during which the pulling

distance of the pulled atom was increased by 2.5 nm at a constant rate of 0.1 nm/ns.

The calculated PMFs are depicted in Figure 5.8. The free energy difference ΔG is

calculated to be -31.4, -34.5 and -30.3 kcal/mol upon adsorption, corresponding to -3.5, -3.8,

and -3.4 kcal/mol/glucose, for GXXXGXXXG, GXXLGXXXG and GXXFGXXXG,

respectively. The calculated results fall well into the range of the experimental average

desorption energy of 1.82-5.12 kcal/mol/glucose determined from AFM measurements, [103]

which also to some extent justifies our SMD simulations. The free energy differences

demonstrate that the galactosyl and fucosyl substitutions on one side chain of XG does not

have significant effect on their binding affinity toward adsorption on cellulose. Nevertheless,

the characteristics of the three PMFs are different, indicating possibly different desorption

mechanisms for these XGs.

Therefore, our results addressed the importance of taking into account the water

environment in the theoretical model of the XG-cellulose assembly. In our model, variation in

one side chain did not have significant influence on the interaction energy between XGs and

cellulose and the binding affinity of XGs. However, the side chain variation did affect the

equilibrium structural properties of the adsorbed XGs.

5.3 Carbohydrate-Enzyme Interaction (Paper III)

In paper III, we have investigated by MD simulations tetradecasaccharide XXXGXXXG in

complex with hybrid aspen xyloglucan endo-transglycosylase PttXET16-34, with the

“nucleophile helper” residue, aspartate 87 (Asp87), either in protonated (conjugate acid) form

or deprotonated (conjugate basic) form. The “helper” residue, Asp87, lies in between and on

the same face of strand as the catalytic residues glutamate 85 (Glu85) and glutamate 89


46

(Glu89).

Two separate simulations were carried out. The sidechain of the Glu89, which functions

as a catalytic acid residue to deliver a proton to the nucleofuge in the first chemical step of the

canonical retaining mechanism employed by GH16 enzymes, was modeled in the protonated

form in both simulations. Meanwhile, Glu85, which functions as the catalytic nucleophile,

was modeled in the deprotonated form. The “helper” residue Asp87 was alternately modeled

in either deprotonated (Simulation 1) or protonated form (Simulation 2).

Figure 5.9 Schematic representation of interactions in the active site cleft of the hybrid aspen xyloglucan

endo-transglycosylase PttXET16-34 with XXXGXXXG. Subsites are numbered from the scissile

glycosidic bond according to Davies, Wilson, and Henrissat. Reprinted with permission from Taylar &

Francis.

By and large, similar global conformations of the relaxed XXXGXXXG in the active site

cleft were yielded in these two simulations. Both trajectories suggest that the PttXET16-34

active site cleft consists of seven primary backbone Glc-binding subsites. Three positive

subsites extend toward the reducing end of the substrate from the catalytic machinery on

strand β6, and four negative subsites extend toward the non-reducing terminus (Figure 5.9). It

is clearly shown that there are many hydrogen bonds and van der Waals interactions involved

in the substrate recognition. The multiple interactions in the active site encourage

XXXGXXXG to adopt a bent conformation. These interactions have been analyzed as a

function of the time in the MD trajectories (see Tables 2 and 3 in Paper III). In addition,

general trend was indicated that a correlation might exist between the number of interactions


47

and the low mobility of certain glycosyl residues with the contributions of these residues to

transition state binding and catalysis.

Careful analysis of the puckering angles in Glc(-1) revealed that in both simulations, the

undecorated Glc(-1) backbone glycosyl unit immediately flipped from the 4C1 chair starting

conformation into various boat and skew boat conformations (for example 1,4B, B2,5, B3,6, 1S3

and 1S5) after energy minimization and equilibration when all harmonic constraints were

removed. These boat-like conformations of Glc(-1) showed inter-conversions in both MD

trajectories, which are necessary pre-transition state conformations allowing access of the

catalytic nucleophile to C1 of the sugar ring Glc(-1).

Figure 5.10 Close-up of tetradecasaccharide XXXGXXXG in complex with PttXET16-34 from time steps

late in Simulations 1 and 2. Reprinted with permission from Taylar & Francis.

Although both simulations yielded similar global conformations of the relaxed

XXXGXXXG in the active site cleft, the charge state of Asp87 had a distinct influence on the

structure of the PttXET16-34 active site (Figure 5.10). To give a clear picture of the

difference in the active site, some critical distances at the catalytic site were monitored in the

last 10 ns in both simulations (Table 5.2).

When Asp87 was protonated in Simulation 2, a stable hydrogen bond was formed with

the catalytic nucleophile, Glu85. In addition, the 2-OH group of Glc(-1) was also found to be

hydrogen bonded to the nucleophile Glu85. Thereby, the Glu85 was precisely poised to

receive the anomeric carbon of Glc(-1) undergoing electrophilic migration from the departing

O4 atom of Glc(+1), leading to the formation of a covalent glycosyl-enzyme. However, in

Simulation 1, when Asp87 was deprotonated, the electrostatic repulsive interaction between

Asp87 and Glu85 pushed the catalytic nucleophile away from C1 of Glc(-1).


48

Table 5.2 Inter-atomic distances (nm) and hydrogen bonds averaged in the last 10 ns of the two trajectories

of XXXGXXXG in complex with PttXET16-34. Standard deviations are given in parentheses.

Simulation 1 Simulation 2

Glu85-Oε2 and C1-Glc(-1) 0.64(0.04) 0.36(0.02)

Glu85-Oε2 and Asp87-Oε1 0.63(0.04) 0.27(0.01) 99.8a

Glu89-Oε1 and O4-Glc(+1) 0.36(0.03) 0.38(0.04) 11.9a

Glu89-Oε1 and Asp87-Oε2 0.26(0.01) 98.0a 0.46(0.06)

aThe occupation time of the hydrogen bond (%) in the analyzed part of the trajectory.

On the other hand, the electrophilicity of the catalytic Glu89 was also affected by the

protonation state of Asp87. In Simulation 2, the OH group in Glu89 was found to occasionally

form hydrogen bond to the glycosidic oxygen atom of Glc(-1), which is catalytic productive.

On the contrary, in Simulation 1, the OH group in Glu89 was stably hydrogen bonded to the

carboxylate oxygen in Asp87, which is not favorable for the catalysis.

Our results thus suggest that catalysis in the glycoside hydrolase family 16 is optimal

when the “helper residue” Asp87 is protonated to align the nucleophile, the catalytic

nucleophile is deprotonated for nucleophilic attack on the substrate and general acid/base

residue is in the protonated form to deliver a proton to the departing sugar.

5.4 Working Mechanism of an α-cyclodextrin Based Rotaxane (Paper IV)

Molecular machines have recently attracted intense interest on account of their potential

application opportunities in the fields of nanoscience and materials science.[104 , 105, 106]

Redox-responsive bistable [2]rotaxanes and [2]catenanes are of considerable interest because

the redox-responsive switching process is usually rapid and precise and can be controlled

within solid-state devices, as well as on surfaces.

In paper IV, the binding preference and the working mechanism for a redox-responsive

bistable [2]rotaxane (Figure 5.11) incorporating an α-CD ring was investigated theoretically.

In this rotaxane, the α-CD ring can be induced to move between a tetrathiafulvalene (TTF)

recognition site and a triazole (TZ) unit on the dumbbell on account of the redox properties of

the TTF unit. When the TTF unit is in neutral state, the α-CD ring prefers to stay at the TTF

unit (denoted as the α-CD@TTF isomer); when the TTF unit is oxidized, the α-CD ring

leaves the TTF unit and binds with the TZ unit instead (denoted as the α-CD@TZ isomer).


49

Figure 5.11 Structure of the rotaxane. Reprinted with permission from American Chemical Society.

5.4.1 Potential of Mean Force

The binding preference of the α-CD ring is determined by the relative free energy between the

two isomers, i.e. the free energy difference between the α-CD@TTF isomer and the

α-CD@TZ isomer. The PMFs for the shuttling movement of the α-CD ring along the

dumbbell were calculated via the umbrella sampling approach. The reaction coordinate was

defined as the z-component difference between the center of mass (COM) of the α-CD ring

and the reference atom (the carbon atom in the TTF unit indicated with asterisk mark in

Figure 5.11) of the dumbbell.

The initial calculations on the PMFs for the shuttling motion of the α-CD ring along the

dumbbell for different states were carried out in vacuo. As shown in Figure 5.12 (left), the

α-CD ring was predicted to encircle (1) the TTF unit, (2) the tetraethyleneglycol chain, and (3)

the TTF unit, respectively, for the neutral, oxidized +1 and oxidized +2 states. Apparently,

these PMFs in vacuo are inadequate for the correct binding preference prediction for both

oxidized states, for which the α-CD ring is observed experimentally to encircle the TZ station.


50

Figure 5.12 The change of PMF as a function of the position of the α-CD ring along the dumbbell for

neutral, oxidized +1, and oxidized +2 states in vacuo (left) and water (right). Note: the hydrogen atoms are

omitted in the snapshots of energy minima for clarity. Reprinted with permission from American Chemical

Society.

Apparently, the water environment cannot be neglected in the theoretical model of this

rotaxane. Therefore we calculated PMFs (Figure 5.12 right) in explicit water for the shuttling

motion of the α-CD ring along the dumbbell for different states. For the neutral state, it is

obvious that the most stable isomer is α-CD@TTF (ring at 0.61 nm), which is energetically

more favorable by about 2.6 kJ/mol compared to the α-CD@TZ isomer (ring at 1.87 nm).

This free energy difference is comparable to the experimentally measured value [107] of 5.5 kJ/

mol. In contrast to the neutral state, the α-CD ring prefers to stay on the TZ station in both

oxidation states, in good agreement with experimental observations. In the oxidized +1 state,

the α-CD@TZ isomer is more stabilized than the corresponding α-CD@TTF+• isomer by

about 13.7 kJ/mol, while in the oxidized +2 state, the α-CD@TZ isomer is stabilized by about

48.0 kJ/mol. Accordingly, the free energy barrier for the ring shuttling from the TTF unit to

the tetraethyleneglycol chain decreases from 18.0 kJ/mol (neutral state) to 5.6 kJ/mol (+1


51

state) and this barrier completely vanishes for the oxidized +2 state. This means that the

response time for this rotaxane relies totally on the oxidization of the TTF unit and the

α-CD@TZ isomers in the oxidized states are very stable. While the PMFs in vacuo cannot

explain the experimental observations, the PMFs in water are in good agreement with the

observed binding preference, indicating that it is important to take into account the water

environment in theoretical calculations.

5.4.2 Energy Decomposition and Working Mechanism

Figure 5.13 Variation of (left) the intermolecular interaction ECD-dumbbell, ECD-H2O, and Edumbbell-H2O, and

(right) Eelec(CD-dumbbell) and EvdW(CD-dumbbell) with the position of the α-CD ring on the dumbbell for the neutral

(black), oxidized +1 (blue), oxidized +2 (red) states in water. The positions where the α-CD ring locates at

the TTF (0.2 nm < z < 0.6 nm) and TZ (1.4 nm < z < 1.90 nm) units are marked in cyan. Energies are

shown in unit of kJ/mol. Reprinted with permission from American Chemical Society.

The trajectories were further analyzed in order to elucidate the energy contributions

accounting for the working mechanism. The crucial components analyzed include

non-bonded interactions between the α-CD ring and the dumbbell, between the α-CD ring and

water, as well as between the dumbbell and water, denoted by ECD-dumbbell, ECD-H2O, and

Edumbbell-H2O, respectively. It is interesting to find that the non-bonded interactions between the


52

α-CD ring and the dumbbell are consistently attractive for all the three states. Additionally,

this attractive interaction ECD-dumbbell, in which electrostatic contribution dominates, becomes

stronger when the TTF unit is oxidized. Apparently, the working mechanism for this rotaxane

cannot be explained exclusively by ECD-dumbbell. Careful analysis on the non-bonded

interactions between the rotaxane subcomponents with water, ECD-H2O and Edumbbell-H2O,

reveals that both these interactions in the oxidized states favor the binding of the α-CD ring at

the TZ unit, which overwhelm the opposing effect of the attractive interaction ECD-dumbbell,

leading to the correct binding preference predicted for the oxidized states.

5.4.3 Quantum Chemical Calculation in Vacuo

Figure 5.14 (left) The total energy for the inclusion complexation of TTF-OCH3, [TTF-OCH3]+·,

[TTF-OCH3]2+ into the α-CD cavity calculated at the B3LYP/6-31G(d) level in vacuo. z denotes the

distance between center of mass of the guest and the α-CD ring. (right) Optimized geometries for the

inclusion complexes.

Quantum chemical calculations on the inclusion complexes between the α-CD ring and the

neutral/oxidized model guest molecules TTF-OCH3, [TTF-OCH3]+·, and [TTF-OCH3]

2+ were

also carried out. The fully optimized geometries for the complexes calculated at the


53

B3LYP/6-31G (d) level in vacuo show that both radical cation [TTF-OCH3]+· and dication

[TTF-OCH3]2+ can form stable inclusion complexes with the α-CD ring. Single point energy

calculations at the B3LYP/6-311++G(d, p) level for these fully optimized geometries further

demonstrate that the stabilization energy is increased from α-CD-TTF-OCH3 to

α-CD-[TTF-OCH3]+·, and α-CD-[TTF-OCH3]

2+ complexes, which is in disagreement with

experimental observations. These results suggest that quantum chemical calculation in vacuo

cannot correctly predict the interaction of the α-CD ring with the charged TTF units. It is also

reported that simple continuum solvation model is not sufficient to model the CD interaction

with some radical cations and anions, so that explicit solvent molecules should be taken into

account to describe the interaction properly. [108]

In summary, the free energy profiles calculated by MD simulations with explicit water

can correctly reproduce the experimentally observed binding preference for each state.

Analysis of the free energy components reveals that, for these α-CD based rotaxanes with

charged TTF units, the real driving force for the shuttling in the oxidized states is actually the

interactions between water and the rotaxane components, which overwhelm the opposing

effect of the attractive interactions between the α-CD ring and the charged dumbbell.


54

Chapter 6 Conclusions and Outlook

55

Chapter 6

Conclusions and Outlook

In this thesis, the atomic MD simulation technique has been applied for investigations of

structural and energetic properties of several carbohydrate related materials:

The atomic details of thermal induced transitions in crystal cellulose Iβ were studied by

MD simulations with the GLYCAM06 force field. The predicted crystal unit cell

parameters for the structures at low and high temperatures agreed well with the

experimental crystal structures of cellulose Iβ. The computed phase transition

temperature of 475-500 K also correlated well with the experimental one. The

reorganization of the hydrogen-bonding network along with the phase transition was in

qualitative agreement with temperature-dependent IR measurements.

The adsorption properties of XGs on the surface of cellulose Iβ, in particular the effects

of solvent molecules and side chain variation, were examined. The three XGs studied,

GXXXGXXXG, GXXLGXXXG and GXXFGXXXG, differ in length and composition

of one side chain. It is found that the interaction energies between XG and cellulose in

water were considerably lower than those in vacuo and the van der Waals interactions

played a prevailing role over the electrostatic interactions in the adsorption. In our model,

variation in one side chain did not have significant influence on the interaction energy

and the binding affinity. However, the side chain variation did affect the equilibrium

structural properties of the adsorbed XGs.

The interaction of tetradecasaccharide XXXGXXXG in complex with the hybrid aspen

xyloglucan endo-transglycosylase PttXET16-34 was analyzed. The effect of the charge

state of the nucleophile “helper residue” Asp87 on the PttXET16-34 active site structure

was emphasized. When Asp87 is protonated, the hydrogen bonding network formed led

to the correct positioning of the catalytic machinery of both the nucleophile Glu85 and


56

electrophile Glu89. Our results indicate that the catalysis in glycoside hydrolase family

16 is optimal when the catalytic nucleophile is deprotonated, while the “helper residue”

and general acid/base residue are both protonated.

The working mechanism for a redox-responsive bistable [2]rotaxane based on α-CD ring

was investigated. The free energy profiles were calculated employing an umbrella

sampling technique for the shuttling motion of the α-CD ring between a TTF recognition

site and a TZ unit along the dumbbell of the rotaxane for three oxidation states (0, +1, +2)

of the TTF unit. These calculated free energy profiles verified the experimentally

observed binding preference for each state. Analysis of the free energy components

revealed that, for the α-CD based rotaxanes with charged TTF units the real driving force

for the shuttling in the oxidized states is actually the interactions between water and the

rotaxane components, which overwhelm the attractive interactions between the α-CD ring

and the charged dumbbell.

The classical MD simulation method has proved to be a powerful tool to understand in

atomic detail the dynamics, structures and interactions of carbohydrate systems. It is a

powerful supplement to predict and interpret experimental observations. The quality of MD

simulations relies on two important aspects: the accuracy of the chosen force field and a

sufficient conformational sampling.

The development of force fields for carbohydrates has been a challenging endeavor.

Recent progress in carbohydrate force fields has promoted the computer simulations of

carbohydrates and protein-carbohydrate interactions to a new frontier. The cellulose crystal

structure can be predicted with an accuracy that can match, and even refine the experimental

precision. Improvements of carbohydrate force fields will be undertaken in order to further

increase the accuracy of simulations of cellulose based systems. With the current computer

technology, the simulation time and system size of atomic-scale simulations have been

practically limited to about 100 ns and a million of particles. However, the structural

flexibility and ruggedness of the free energy landscape for polysaccharides in condensed

phase pose a tough issue for conformational sampling. For these cases, enhanced sampling

techniques such as replica exchange molecular dynamics or its variants and metadynamics

simulations can offer a way forward. On the other hand, coarse-graining techniques, in which

the number of degrees of freedom is reduced and fine interaction details are eliminated, is also

a possible way to extend by orders of magnitude the time-scale and length-scale of molecular

modeling and enhance conformational sampling. The combined/hybrid quantum mechanics

and molecular mechanics calculations provides the opportunity to treat electronic excited


57

states, electron transfer processes and chemical reactions in large biomolecular systems with

high accuracy and efficiency. These are viable approaches for further studies of enzymatic

catalyses on carbohydrates. Multi-scale modeling is a key for further improved accuracy and

efficiency for simulations of carbohydrate and carbohydrate-protein interactions, and will,

together with dedicated experiments for validation and feedback, pave the way for future

breakthroughs in carbohydrate research.


58

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Molecular Dynamics Simulations of Biomimetic Carbohydrate ...416145/FULLTEXT01.pdfII Preface The work presented in this thesis has been carried out at the Department of Theoretical