UNIVERSITY OF CALIFORNIA, SAN DIEGO

Stretching and twisting chromatin

A dissertation submitted in partial satisfaction of the

requirements for the degree

Doctor of Philosophy

in

Engineering Science (Engineering Physics)

by

Irina V. Dobrovolskaia

Committee in charge:

Professor Gaurav Arya, Chair Professor Prabhakar Bandaru Professor Sergei Krasheninikov Professor Bo Li Professor Vlado Lubarda

2012 ©

Irina V. Dobrovolskaia, 2012

All rights reserved. The Dissertation of Irina V. Dobrovolskaia is approved, and it is acceptable in quality and form for publication on microfilm and electronically:

Chair

University of California, San Diego

2012

iii TABLE OF CONTENTS

Signature Page...... iii

List of Figures...... vii

List of Tables...... viii

Acknowledgments...... ix

Vita...... x

Publications...... x

Abstract of the Dissertation...... xi

1. Introduction...... 1

1.1. DNA structure and function...... 1

1.2. Chromatin as a DNA packager...... 3

1.3. Chromatin as a gene regulator...... 8

1.4. Forces on chromatin ...... 12

1.5. Torsional forces on chromatin...... 17

1.6. Thesis objective...... 21

1.7. References...... 23

2. Force-induced unraveling of nucleosomes...... 36

2.1. Introduction...... 36

2.2. Model Development...... 38

2.2.1 Modeling of DNA and histone octamer...... 39

2.2.2 Brownian dynamics simulations...... 43

iv 2.2.3 Parametrization of octamer groove charges...... 46

2.3. Results...... 51

2.3.1 Force-extension behavior...... 52

2.3.2 DNA unwrapping dynamics...... 54

2.3.3 Kinematics of nucleosome unraveling...... 57

2.3.4 Energetics of nucleosome unraveling...... 62

2.3.5 Role of non-uniform DNA/histone interactions...... 64

2.4. Discussion...... 70

Acknowledgments...... 74

2.5. References ...... 74

3. Torsional behavior of dinucleosome arrays...... 80

3.1. Introduction...... 80

3.2. Model and Simulation Methods...... 83

3.2.1 Linker and nucleosome model ...... 83

3.2.2 Dinucleosome mechanics and energetics...... 85

3.2.3 Monte Carlo twisting protocol...... 88

3.3. Results...... 90

3.3.1 Twist inversion...... 90

3.3.2 Nucleosome flipping...... 96

3.4. Discussion...... 101

Acknowledgments...... 105

3.5. References...... 106

v 4. Conclusions and future work...... 109

4.1. Summary...... 109

4.2. Future directions...... 111

4.3. References...... 113

Appendix 1...... 114

Appendix 2...... 115

vi LIST OF FIGURES

Figure 1.1: DNA double helix...... 2 Figure 1.2: Top and side view of the crystal structure of the nucleosome core...... 5 Figure 1.3: Major structures involved in DNA compaction...... 7 Figure 1.4: Single molecule experimental techniques and their natural force ranges.. 13 Figure 1.5: Forces and torques imposed by chromatin-acting enzyme...... 14 Figure 2.1: Coarse-grained model of the nucleosome...... 39 Figure 2.2: Normalized charge values assigned to the octamer groove beads as a function of their location along the wound DNA relative to the dyad...... 48 Figure 2.3: Mean unraveling force as a function of the normalized loading rate...... 51 Figure 2.4: Computed force-extension curves...... 53 Figure 2.5: Time evolution of extent of nucleosome wrapping...... 55 Figure 2.6: The octamer center of mass as a function of time for BD simulations...... 59 Figure 2.7: Time evolution of the nucleosome during simulation...... 61 Figure 2.8: Time evolution of the total energy...... 63 Figure 2.9: Representative force-extension plots...... 65 Figure 2.10: Comparison of the time evolution number of wrapped turns of DNA and octamer orientation for canonical and noncanonical nucleosomes...... 68 Figure 3.1: Dinucleosome array model and nucleosome and linker bead coordinate systems...... 84 Figure 3.2: Schematic of the applied and induced twist from the MC simulations...... 85 Figure 3.3: Twist inversion...... 91 Figure 3.4: Twist propagation simulation results...... 93 Figure 3.5: Effect of applied twist on the linker and nucleosome coordinate systems.95 Figure 3.6: Nucleosome flipping dynamics...... 97 Figure 3.7: Dinucleosome energy contributions...... 100 Figure 3.8: Total applied and induced twist...... 101

vii LIST OF TABLES

Table A.1: Physical parameters associated with the coarse-grained nucleosome and Brownian dynamics simulations...... 114 Table A.2: Physical parameters used in the Monte Carlo simulations...... 115

viii ACKNOWLEDGMENTS

I would like to thank Dr. G.Arya for being an inspiring scientific adviser and for his unresting enthusiasm. Its been a privilege to work under his guidance. All the members of Dr. G. Arya group, past and current, have been very supportive and helpful.

The text of Chapter 2 is partly based on paper I. V. Dobrovolskaia and G. Arya

“Dynamics of forced nucleosome unraveling and role of nonuniform histone-DNA interactions”, in press, Biophysical Journal, 2012.

The text of Chapter 3 is partly based on paper I. V. Dobrovolskaia, M.

Kenward, and G. Arya “Twist Propagation in Dinucleosome Arrays”, Biophysical

Journal, Volume 99, pp. 3355– 3364, November, 2010.

ix VITA

2005-2007 Master of Science, University of California San Diego

2007-2012 Research Assistant, University of California San Diego.

2012 Doctor of Philosophy, University of California San Diego

PUBLICATIONS

Irina V. Dobrovolskaia, Martin Kenward, and Gaurav Arya

Twist Propagation in Dinucleosome Arrays

Biophysical Journal, Volume 99, 2010, 3355–3364

Irina V. Dobrovolskaia and Gaurav Arya

Dynamics of Forced Nucleosome Unraveling and Role of Nonuniform

Histone DNA Interactions in press, Biophysical Journal, 2012

x ABSTRACT OF THE DISSERTATION

Stretching and twisting chromatin

by

Irina V. Dobrovolskaia

Doctor of Philosophy in

Engineering Science (Engineering Physics)

University of California San Diego, 2012

Professor Gaurav Arya, Chair

Eukaryotic DNA is present in the form of chromatin. The basic unit of chromatin is the nucleosome, composed of 147 bp of DNA wrapped around an octamer of histone proteins. The main function of chromatin is to package the DNA into the cell nucleus, while keeping it in an active, dynamic form. The DNA is subjected to various kinds of external forces and torques during processes like transcription and replication. While the mechanical properties of naked DNA are well understood, the dynamical response of nucleosomes and chromatin to such perturbations has not been studied in detail due to difficulties in probing chromatin

xi dynamics, both experimentally and theoretically. In this thesis, the dynamics of single nucleosomes subjected to extensional and torsional forces are investigated using mesoscale modeling and simulations.

The unraveling of nucleosomes subjected to extensional forces is studied. A mesoscale model of the nucleosome that accounts for the dynamics of the histone octamer and the DNA as well as their assembly and disassembly is developed. The model is parameterized to reproduce the histone/DNA binding free energies and the loading rate-dependent unraveling forces obtained from single-molecule experiments.

Brownian dynamics simulations under constant loading reveal that the nucleosome unravels in distinct steps. The detailed dynamics of the DNA and of the octamer during each step are elucidated and the role of the nonuniform histone/DNA interactions in unraveling is dissected.

The torsional response of a di-nucleosome array is investigated. A mesoscopic model of the array accounting for nucleosome geometry and linker mechanics is employed. Monte Carlo simulations are used to obtain the distribution and propagation of twist from one DNA linker to another while an external twist was applied in a stepwise manner to mimic quasistatic twisting of chromatin fibers. The magnitude and sign of the twist measured on linkers on opposite sides of the nucleosome are found to depend strongly on the relative orientation of two linkers. The simulations reveal rapid flipping of nucleosomes in response to continued twisting, which facilitates drastic changes in the entry/exit conformation of the DNA linker and rapid exchanges between DNA twist and writhe.

xii 1. Introduction

1.1. DNA structure and function

The instruction for the development, functioning, reproduction, and survival for the vast amount of life forms existing on Earth is encoded by a unique polymer, deoxyribonucleic acid or DNA1. The DNA single strand (ssDNA) is composed of subunits called deoxyribonucleotides. The nucleotide repeats contains one of 4 bases: adenine (A), cytosine (C), guanine (G), and thymine (T) that are connected to each other via a backbone made from phosphate-sugar residues (Figure 1.1) [1].

Most species have two DNA chains which run in opposite2 directions and are wound around a common axis, commonly referred to as double stranded DNA (dsDNA).

While dsDNA or simply DNA exists in several conformations, it exists as a family of B-

DNA3 structures in physiological conditions [2]. In B-DNA, the bases on one DNA strand interact with the bases on the other strand such that A is hydrogen bonded only to T, and

C only to G. Both strands form a helix4 of internalized bases with a pitch of 34 Å and a radius of 10 Å [3]. The spaces, or grooves, between the strands have a width of 22Å (a major groove) and 12 Å (a minor) [4].

1 With the exception of some group of viruses [1]. 2 Strands run in opposite direction with the 5' end having a terminal phosphate group and the 3' end a terminal hydroxyl group [1]. 3 There are several other forms in which dsDNA can be found: A-, Z-, P-DNA [1]. 4 The DNA double helix is stabilized primarily by two forces: hydrogen bonds between nucleotides and base-stacking interactions among the aromatic nucleobases [6].

1 2

Figure 1.1: DNA double helix. The atoms in the DNA structure are color coded such that white is for hydrogen, red for oxygen, blue for nitrogen, gray for carbon, and yellow for phosphorous. The hydrogen bonding between two base pairs is shown in the bottom, right corner. Adapted from [http://www.richardwheeler.net].

Due to this geometry, the bases in the major groove are more accessible to make a contact with DNA-binding proteins. In water, DNA obtains a negative charge and therefore cannot be tightly packed, but it can form dense toroidal clusters in multivalent counterions [5] and in the presence of counterions.

Even before the findings of DNA structure by Watson and Crick, it was understood that the DNA molecule was responsible for heredity [7,8]. It was later found by Jacob and Monod that the heredity information is encoded by sequences of four nucleobases is extracted by enzymes to form proteins in a process called 3 transcription5 [9]. Due to extensive research and novel techniques6 [10] that allow measurements of the total content of DNA (genome) in base pairs, scientists found that the genome varies from species to species. It’s length can be as short as 7,800 bp for

Commelina yellow mottle virus [11] to as long as 1.3×1011 bp for Protopterus aethiopicus (lungfish)7 [12,13]. Eukaryotes, referring to species with a defined, membrane-bound nucleus (fungi, plants and animals), possesses multiple DNA molecules and the average size of a eukaryotic genome is thousands times larger than that of a typical virus or bacterium.

1.2. Chromatin as a DNA packager

The typical length of DNA in eukaryotic organisms is meters long, which is hundred thousand times longer than the typical micron sized eukaryotic nucleus

[1,6,14,]. Such size mismatch creates consequently the problem of packing huge amounts of genetic material into tiny volumes that still allows the DNA to remain active and accessible.

Nature solves this problem by associating DNA with special proteins to create an organized and compact structure called chromatin8. For example, human DNA with linear size of ~1.8 m when associated with DNA-binding proteins shrinks to a linear

5 Transcription is a multiple step process at which the genetic code is copied from stretches of DNA into the related nucleic acid RNA [1]. 6 Enzymatic methods, for example, Chain-termination, Lynx Therapeutics' Massively Parallel Signature Sequencing [10]. 7 The largest genome is determined for Polychaos dubium ("Amoeba" dubia) and it is about 6.7×1011 bp long [13], though the size is under discussion [1]. 8 Chromatin is the complex of DNA and protein found in the eukaryotic nucleus, which packages chromosomes [1]. 4 size that is ~10000 times smaller to allow it to fit in the micrometer sized nucleus

[1,15]. Within the eukaryotic nucleus, chromatin can be visually observed during metaphase as the multiple classic four-arm structures, the tightly packed linear chromosomes9. Such densely packed chromatin was first described in the late 19th century as small colored objects that are presented in the nucleus and later, at the beginning of 20th century, recognized as the carrier of heredity [1,6]. However, it was only in the late 20th century that an understanding of chromatin complex dynamical organization and its multiple biological functions10 began to emerge [14,16-19].

The hierarchical process of DNA compaction begins with the binding of the negatively charged DNA around positively charged histone11 protein aggregates

(histone core) [20] to form the nucleosome core particle (NCP). The NCPs are separated by free DNA of length ranging from 5 to 30 nm [21] to yield a repeating motif called nucleosome. The nucleosome was first described by Roger Kornberg in

1974 and considered as the basic structural unit of chromatin [14, 22]. The X-ray crystallographic all-atom structure of the NCP [20, 24, 25] is shown in Figure 1.2.

The NCP is formed from eight histone proteins composed of two copies each of the H2A, H2B, H3, and H4 histones. About 146 bp of DNA is wrapped around this relatively rigid histone octamer. Usually, the shape of the NCP can be approximated by a cylinder of diameter 11 nm and height 5.5 nm. The pitch of the DNA superhelix is

9 A chromosome is a rodlike organized structure of DNA and protein found in cells. It is a single piece of coiled DNA containing many genes, regulatory elements and other nucleotide sequences plus DNA-bound proteins, which serve to package the DNA and control its functions [1,13]. 10 The functions of chromatin are: to package DNA into a smaller volume to fit in the cell, to strengthen the DNA to allow mitosis and meiosis and prevent DNA damage, and to control gene expression and DNA replication [1,13,14]. 11 Histones are highly alkaline proteins found in eukaryotic cell nuclei that package and order the DNA into structural units called nucleosomes [23]. 5 about 2.39 nm [26].

Figure 1.2: Top and side view of the crystal structure of the nucleosome core. Core histones: H2A, H2B, H3 and H4 colored correspondingly yellow, red, blue,green respectively. DNA is colored green and yellow. Adapted from [23].

The NCP is connected via a stretch of linker DNA to the next NCP. Such a linear chain of nucleosomes is known as a “bead on a string”. The nucleosomes can be randomly or uniformly positioned along DNA, on average, having 15–60 bp linker

DNA. The DNA linker length varies from species to species [18, 21].

The beads-on-the string array then folds into a chromatin fiber with a nominal diameter of 30 nm, which had been observed in several experiments [27,28] in the mid

1980s. The folding is regulated by salt concentration, positively charged histone tails, and positively charged linker histones H1 or H5 [14, 21, 29]. All mechanisms essentially shield the negatively charged phosphate groups on the linker DNAs, allowing the entry/exit linker DNAs to come closer leading to condensation. The internal structure of 30-nm chromatin has been a subject of investigation for several 6 decades [30, 31] due to the inability to visualize the internal structure of the fully compact 30-nm chromatin [32], leading researchers to propose various models of chromatin organization [14,30, 33]. The present models for the 30-nm fiber fall into two main classes: solenoid models [34,35] and zig-zag models [36-40]. Both models consider that the nucleosome faces orient perpendicular to the axis of the fiber with the linker histones placed internally holding together the linker DNA at the entry-exit site. Current experiments [41-43] and simulations [21] reveal that the fiber topology strongly depends on the linker length [30, 31, 43, 44]. Recent coarse-grained modeling of the 30-nm chromatin fiber has suggested that the fiber might exhibit a polymorphic structure, with simultaneous presence of solenoid and zigzag configurations [31, 35,

36, 45]. An all-atom model of the chromatin fiber containing linker histones also reveals a versatile structure tuned by the nucleosomal repeat length [46].

Thereafter, the 30-nm chromatin fiber undergoes several higher levels of folding, culminating in chromosomes (Figure 1.3) [47-49]. Mitotic chromosomes have a very condensed structure, which consists of tightly packed 100–300 nm-wide fibers

[55] and can be observed via optical microscope as the classic four-arm structure.

Interphase chromosomes, on the other hand, exhibit dual behavior, comprising of densely packed regions called heterochromatin12 as well as loosely packed regions called euchromatin13. Heterochromatin is especially prevalent in particular regions within chromosomes, such as the centromeres and telomeres [51,52] and in particular chromosomes, such as the human chromosomes 1, 9, 16, and Y [53]. At the same time,

12 Heterochromatin is the state of chromatin in which it stains intensely, indicating tighter packing. Heterochromatin is usually localized to the periphery of the nucleus. It is genetically inactive [1]. 13 Euchromatin is the state of chromatin in which it stains lightly. It is considered to be partially or fully uncoiled [1]. 7

Metaphase chromosome

Condensed scaffold- associated form

Chromosome scaffold Extended scaffold- associated form

30-nm chromatin fiber of packed nucleosomes

Beads on-a- string form of chromatin

Short region of DNA double-helix

Figure 1.3: Major structures involved in DNA compaction. dsDNA, nucleosome, 10nm "beads-on-a-string" fiber, 30-nm fiber, and metaphase chromosome. Adapted from [23]. other regions of chromatin are packaged as euchromatin [54], which contains loosely packed looped structures of 60–80 nm in width [55,56] with local regions of very loosely coiled 30-nm fibers [57] that can form structures resembling loops 8

[48,57,58,59]. Some regions can even exhibit local de-condensation over several microns containing tens to hundreds of kilobases of DNA [60-63].

It is acknowledged that overall chromatin packaging depends on the stage of the cell cycle14, but the higher order chromatin organization is poorly understood and remains under intensive investigation [14, 58]. At the same time, it is clear that such tight DNA compaction creates a problem in the cell, namely that of limited the access to DNA-binding proteins to DNA, which is necessary for the cell functioning.

1.3. Chromatin as a gene regulator

The normal growth, reproduction, and survival of an organism rely on processes by which information from particular DNA regions, the genes15, can be extracted, interpreted, and used. It is well known that these processes are highly regulated at the level of the DNA sequence via regulatory sequences such as enhancers, silencers, and promoters. For example, transcription is regulated by the binding of the proper set of transcription factors at the promoter region, which then recruit the RNA polymerase16, to synthesize RNA17 using the DNA sequence as its template.

It has now become clear that the nucleosome and chromatin fiber play an

14 The cell cycle is the series of events that take place in a cell leading to its division and duplication (replication).The sequence of events is divided into stages corresponding to the completion of one set of activities and the start of the next. These stages are interphase, prophase, prometaphase, metaphase, anaphase and telophase [1]. 15 A gene is a molecular unit of heredity of a living organism. It is a name given to some stretches of DNA and RNA (ribonucleic acid) that code for a polypeptide or for an RNA chain that has a function in the organism [1]. 16 RNA polymerase is an enzyme that produces RNA [1]. 17 RNA is one of the three major macromolecules which, like DNA, is made up of a long chain of components called nucleotides [1]. 9 equally important role in gene regulation. Specifically, the presence of nucleosomes and the folding of the 30-nm fiber hinder the access of DNA sequences to the RNA polymerase and transcription factors required for transcription. A number of studies

[64-66] have demonstrated that the presence of nucleosomes represses gene transcription by preventing the unzipping of DNA by the RNA polymerase [67].

Experiments have also showed that the RNA polymerase pauses for a minute when its encounters a nucleosome on its way [67]. To enhance the rate of transcription, the nucleosome structure and position needs to be perturbed [68]. This notion is supported by electron microscopy [69], which reveals extended U-shaped (partially unfolded) nucleosomes in actively transcribed regions of the chromatin [69], suggesting that nucleosomes are not static protein-DNA complexes but rather intrinsically dynamic macromolecular assemblies [16, 18, 35, 70] and that nucleosome conformational changes appear to be associated with gene activation [59, 71-74]. Thus, the nucleosome is an important player in gene regulation [67,75-77], and understanding how the nucleosomes and the chromatin fiber are regulated is an important component of modern biological research [78-82].

Several mechanisms have been identified that modulate the accessibility of wrapped DNA in nucleosomes and of linker DNA folded within the 30-nm chromatin fiber [132,58,81,83-87]:

a) histone posttranslational modifications (PTMs) and DNA methylation [88]

b) histone variants [89]

c) protein complexes [90]

d) thermal fluctuations 10

Post-translational modifications

A large number of post-translational modifications in the core histones have been identified over the last decade. For example, in histones H3 and H4, three arginine residues can be methylated, three serine residues can be phosphorylated, and six lysine side-chains can be acetylated. In the C-terminal domains of H2A and H2B, two residues can be ubiquitylated [91,92]. In the N-terminal tails of the core histone proteins, thirteen lysine residues can be reversibly acetylated. These histone modifications can operate either individually or through combinations to alter the functional state of chromatin [88]. Also, some histone modifications operate by altering the electrostatic charge of the histone tails, which leads to increased or reduced chromatin compaction, while others operate by recruiting specialized chromatin remodeling complexes. For instance, hyper-acetylation leads to more open, transcriptionally active chromatin [88,93]. On the other hand, methylation of H3 residues (Lys9, Lys27 and Lys35) results in more compact and transcriptionally silent chromatin [94]. Similarly, the coupling of DNA methylation, with the deacetylation and methylation of histone H3 Lys9 results in gene inactivation [95-102]. These post- translational modifications are carried out by highly specialized enzymes, which are often associated with sequence-specific DNA binding proteins.

Histone variants

In addition to canonical nucleosomes, in vivo chromatin also contains other types of nucleosomes in which canonical histones like H2A, H2B, H3, or H4 are 11 replaced by one or more histone variants. There exists a large diversity of histone variants for H2A and H3, but less for H4 and H2B. The most common histone variants

H2AZ and H3.3 are present in “active chromatin” while H2AX plays an important role in supporting DNA repair [103-106].

Protein complexes

There exist a number of non-histone proteins whose main purpose is to activate or repress transcription by either reducing or enhancing the shielding of the DNA backbone charge, which causes condensation or decompaction of local chromatin regions, respectively. As a result, some DNA sequences can become more transparent or more opaque to the transcriptional initiation proteins [107]. One example of such proteins is the high mobility group (HMG) proteins, such as HMG14 and HMG17, which bind to nucleosomes with high affinity and can disrupt higher order chromatin structure [109].

There also exist specialized protein complexes called ATP-dependent chromatin remodelers [108,138-140] that slide nucleosomes from one location to another along the DNA or evicting histone octamers from the DNA to enhance the accessibility of the DNA sequences occluded by the histone octamers. One such group of chromatin-remodelling proteins is the multi-subunit ATP-dependent SWI/SNF-like complex [90, 108]. Remodelling enzymes also actively participate in the unfolding of large chromatin domains [90]. In particular, to reorganize chromatin, regulatory proteins can bind directly to nucleosomal histones or influence its stability via posttranslational modification [75,76]. 12

Thermal fluctuations

Chromatin is a flexible polymer capable of fluctuating considerably far from its equilibrium state. Therefore, thermal fluctuations can also induce alterations in chromatin structure. In fact, the special organization of DNA into chromatin allows thermal fluctuations to make certain regions of DNA periodically available for interactions with various proteins (gene regulatory proteins, transcription factors, and

RNA polymerases) while keeping other parts of DNA hidden [81, 87], allowing suitable levels of transcription to be maintained in eukaryotes cells [75]. One such mechanism involves transient, thermally driven unwrapping and rewrapping dynamics of the DNA ends from the octamer surface [54,134]. Such “breathing” motions provide dynamic accessibility for wrapped DNA sequences close to the entry/exit region. Site exposure studies suggest that such dissociation/rebinding of the nucleosomal DNA allows proteins to bind to their cognate sites within these DNA sequences at physiologically relevant rates. In addition, ATP-dependent chromatin remodeling complexes (e.g., SWI/SNF) may exploit the intrinsic instability of nucleosomal DNA to dramatically enhance the rate of site exposure [135-137].

1.4. Forces on chromatin

Most alterations in chromatin structure such as those described above involve imposition of forces arising from different sources [110]. The forces can be exerted directly on the DNA or onto the chromatin fiber. The DNA-acting forces arise from 13

RNA and DNA polymerases that process the DNA template while the chromatin- acting forces arise from nucleosome-remodeling factors and structural proteins that hold chromosome together or govern their movements.

The mechanics of DNA interacting with DNA includes local bending, stretching, proteins twisting of the DNA at the interacting site as well as looping of the

DNA in the case of proteins that interact with two non-adjacent binding sites on the

DNA [111]. The response of DNA to external forces is governed by its mechanical properties, which have been investigated extensively through a variety of bulk [112] and single molecule techniques [110, 113-115]. In single molecule force spectroscopy, individual molecules are usually mechanically stretched or twisted and their elastic response is recorded. The range of the forces that can be measured or applied from these instruments ranges from 10-3 pN to 103 pN depending upon the technique, which is compatible with the typical forces exerted within the cell and the nucleus (Figure

1.4) [110].

Langevin Entropic Specific Covalent interaction

-3 -2 -1 0 1 2 3 Log, pN Magnetic tweezers AFM cantilevers Optical tweezers Figure 1.4: Single molecule experimental techniques and their natural force ranges. Adapted from [110].

However, in eukaryotes, all DNA-acting processes occur in the context of 14 chromatin (Figure 1.5) [115]. Even in the active gene regions, where the chromatin is relatively unfolded, the DNA still remains associated with histone proteins in a “beads on the string” structure [116].

(a) Transcription (b) Torques (c) Force (d) factory e

Cromatin loop u q r e

domain o c r T o F

(e)

Transcribing PolI, immobile

DNA screwing through polymerase Figure 1.5: Forces and torques imposed by chromatin-acting enzyme. Adapted from [115].

The forces imposed by DNA-based molecular motors [117], such as RNA and

DNA polymerases [118-121], and ATP dependent chromatin remodelers [122,123] typically fall in the range of tens of pN; in fact, the RNA polymerases have been shown to be capable of exerting transient forces as large to 40 pN. Clearly, these forces are sufficiently large to cause major structural changes in the chromatin structure. For example, these forces can stretch or compress the chromatin fiber, bend the fiber, under-twist or over-twist the fiber, and even unwind the underlying DNA

[124-126]. Such structural changes could in turn affect genetic processes. Thus, it is 15 important to understand the effects that these high forces can have on nucleosomes and chromatin [127-129].

Single-molecule pulling experiments on in vitro reconstituted chromatin fibers or nucleosome arrays [107,71] found two pulling regimes: a low force regime exhibiting a force plateau at ~5 pN and a higher force regime exhibiting saw-tooth type patterns. These experiments indicated that for forces between 6 and 20 pN, the chromatin fibers underwent stretching without any structural transition. Stretching of the fiber around and beyond 25pN resulted in the release of the histone octamers

[130,131]. The event of histone core eviction that follows nucleosome unwinding has significant consequences, as the obstacles for many DNA-binding proteins are now removed and the DNA becomes accessible.

The dynamical response of nucleosomes to forces critically depends on the intermolecular interactions between DNA and the histones. These interactions are fairly strong to overcome the bending stiffness of DNA and wrap it into the superhelical conformation found within nucleosomes. Calculations indicate that these interactions are almost twice as strong as the bending energy of the wrapped DNA to yield a net binding free energy of roughly 24 kcal/mol [141]. Such strong binding is provided by electrostatic, hydrogen bonding, and van der Waals interactions between histone residues and DNA phosphates and bases [20,24]. These interactions are concentrated within 14 distinct sites along the wound length of DNA at locations where the DNA minor groove makes contact with the histones [142]. The strongest of these sites are located at the nucleosome dyad and about 40 bp from the dyad

[131,142]. The presence of these three strong sites was recently confirmed by a novel 16 single-molecule assay involving the forced unzipping of wound DNA in a single nucleosome using optical tweezers [143].

While the molecular interactions between the histone/DNA have been well characterized, their role in dictating the dynamical response of nucleosomes to externally imposed forces on DNA remains less understood. Bennink et al. [130] were the first to observe the unwrapping of DNA from individual nucleosome within nucleosome arrays stretched via optical traps. Soon afterwards, Brower-Toland et al.

[131] conducted a detailed investigation of this phenomenon. They found that nucleosomes unraveled in two steps: the first (outer) DNA turn unwrapped spontaneously at low forces while the second (inner) turn unwrapped at high forces and gave rise to a prominent “rip” in the force-extension plot, which has now been confirmed by others [107,144-146].

Various physical mechanisms have been proposed to explain this difference in the unraveling of the two DNA turns. The original study [131] suggested that particularly strong histone/DNA interactions 40 bp from the dyad required large forces to dissociate, causing the outer turn to unravel irreversibly. Kulic and Schiessel [147] proposed that the intrinsic spool-like geometry of the nucleosome, in which the outer turn feels electrostatic repulsion from the inner turn, facilitates unwrapping of the outer turn; note that this repulsion is not reciprocated during the unraveling of the inner turn. Recently, Sudhanshu et al. [148] noted that the unraveling of a DNA turn is accompanied by significant deformation of the linkers, which contributes strongly to the net energy barrier of unraveling, and that this deformation energy increases with

DNA tension. Since the outer turn unravels at lower tension values, it was suggested 17 that its energy barrier must be smaller than that for the inner turn unraveling, which takes place at higher tensions.

Several studies have also explicitly simulated the dynamics of nucleosomes subjected to unraveling forces. Wocjan et al. [149] used a coarse-grained model of the nucleosome along with Brownian dynamics simulations to examine nucleosome unraveling under dynamic force loading. The study confirmed the experimentally observed reversible and irreversible release of the outer and inner turns, respectively, and also provided estimates of the location and height of the unraveling energy barrier.

However, since the study employed a uniform histone/DNA interaction strength across the wound length of DNA, the effects of the strong patches of interactions at the dyad and at the two off-dyad locations could not be determined. Ettig et al. [150] recently carried out steered molecular dynamics simulations of an all-atom model of the nucleosome in explicit water. Though these simulations were limited to short times

(100 ns) and large pulling rates (5 m/s), they yielded several new insights. Most notably, the histone tails were found to impede DNA unwrapping at different stages of the unraveling process.

1.5. Torsional forces on chromatin

In addition to pulling forces induced by enzymes that reshape chromatin structure, DNA is also subjected to torsional stresses. Such torsional stresses are typically generated during DNA transcription by the action of the polymerase enzyme, which is capable of exerting torques as large as ~1.25 kBT/rad, where kBT is the 18 thermal energy. In general, the advancing RNA polymerase results in positive supercoils downstream and negative supercoils in its wake [151,152].

The local and global torsional response of DNA can strongly impact its biological activity. Structural distortions of DNA can alter its binding affinity for proteins such as transcription factors, thus influencing gene transcription [153]. Twist- dependent protein/DNA binding has also been suggested as an indirect-readout mechanism for protein/DNA recognition [154]. DNA supercoiling can alter the accessibility of DNA to protein binding [153-155] and the juxtaposition probabilities of distant DNA sites, potentially affecting genetic recombination. Given its importance, it is not surprising that DNA supercoiling is tightly regulated inside cells.

In fact, an entire class of enzymes, the topoisomerases, is devoted to removing excess positive and negative supercoils from DNA [156,157]. Emerging evidence also suggests that supercoiling is not an entirely unfavorable byproduct of transcription; it may instead serve to signal protein binding over long genomic distances [151,158].

The topology of circular or closed DNA can be described by two variables: the twist Tw, which gives the number of helical turns around the local axis, and the writhe

Wr, which gives the number of DNA axis self-crossings averaged over all orientations.

The linking number Lk, which gives the number of times the two single stranded DNA molecules cross over each other, provides a relationship between Tw and Wr. For closed or end-constrained DNA, Lk is topologically invariant and equal to the sum of the twist and the writhe, Lk = Tw + Wr [159]. A key issue in the field has been understanding how torsionally stressed DNA, characterized by the deviation of Lk 19

from its relaxed value Lk0, distributes its stresses internally through changes in twist and writhe. Torsional stresses in open DNA can result from external twisting of its ends and in circular DNA through internal cutting, crossing, and rejoining of single strands by topoisomerases [156,157,160]. Extensive studies have examined the relationship between Tw and Wr during the relaxation of torsionally stressed DNA including the effects of torsional and bending rigidity and the presence of binding proteins [161]. A large body of work has also elucidated various forms of local structural changes associated with twisted DNA, by using both experimental [162,163] and theoretical approaches [159,164-166].

Though our knowledge of DNA supercoiling in response to torsional stresses is fairly advanced, very little is known about the behavior of chromatin fiber in response to torsional forces. To date, few studies have examined the torsional behavior of chromatin [167-169]. Bancaud et al. [167,169] used magnetic tweezers to twist individual reconstituted nucleosome arrays at fixed stretching forces from 0.9 up to

5pN and from -60 to 40 rotation turns. The arrays were found to accommodate large torsional stresses without significant changes in its length, in sharp contrast to DNA.

Moreover, the length variations were found to be highly asymmetric with respect to applied twist direction. A quantitative model that accounts for the torsional response of both the DNA and nucleosome was proposed. The latter response was modeled in terms of dynamic equilibrium between three conformations of the nucleosome: the nucleosome entry/exit DNAs cross negatively or positively or not crossed [170,171].

The authors proposed a model for these variations based on nucleosome flipping, 20 which modulated the entry/exit linker conformation [167,169]. Recent single-molecule twisting studies [167] have confirmed chromatin’s lower torsional rigidity compared to naked DNA. Other studies have examined the functional consequences of transcription-generated DNA supercoiling in eukaryotic organisms. For example,

Kouzine et al. [158] have shown that torsional stresses can propagate over kbp domains in chromatin, promoting formation of non-B-DNA structures that signal binding of specific proteins. 21

1.6. Thesis objective

The above discussion illustrates the critical role played by the nucleosome and the chromatin fiber in both the packaging and regulation of eukaryotic genomes. The discussion also highlights the ubiquity and importance of forces and torques acting on the DNA for enacting specific structural changes within chromatin. While the effect of these external perturbations has been extensively studied in naked DNA, much less is known about their effects on DNA wrapped in nucleosomes and subsequently folded into chromatin. The dynamics of nucleosomes in response to such forces and torques is even less studied.

The focus of this thesis is on elucidating the microscopic dynamics of this basic structural unit of the eukaryotic genome – the nucleosome – under pulling and twisting forces through computational modeling.

This thesis research is divided into two main parts. In the first part of the thesis, the dynamics of a single nucleosome subjected to external forces is studied, focusing on nucleosome unraveling, i.e., unwrapping of DNA from histone octamers.

This research is described in Chapter 2. To investigate the dynamics of force-induced unwrapping of DNA from histone octamers, a new coarse-grained model of the nucleosome has been developed in which the DNA and the histone octamers are treated as separate entities. The interactions between the DNA and the octamer in this model are parameterized to reproduce both the nonuniform histone/DNA interaction free energy profile obtained from single-molecule nucleosome-unwrapping experiments and the loading rate-dependent unwrapping forces, obtained from single- 22 nucleosome pulling experiments. Brownian dynamics (BD) simulations is then employed to determine the dynamics of the nucleosome under constant-loading conditions. The results from these simulations help to elucidate the force-extension behavior associated with nucleosome unraveling, the detailed dynamics of DNA unwrapping from octamers, the translation and rotational dynamics of the histone octamer, and the role of non-uniform histone/DNA interactions on unwrapping kinetics.

In the second part of the thesis, the propagation of DNA twisting forces within chromatin is studied, as discussed in Chapter 3. Since the focus is on twist propagation across individual nucleosomes, an array composed of two nucleosomes and linkers

(dinucleosome array) is considered as the model system. The dinucleosome array is modeled using a simplified version of a mesoscopic model of chromatin recently developed in the Arya group. The model was developed via a rigorous coarse-graining procedure that retains energetic interactions and structural features relevant to chromatin folding and dynamics [172-174]. The model reproduces many structural, thermodynamic, and dynamic features of experimental chromatin. Metropolis Monte

Carlo (MC) simulations are used then to generate an ensemble of conformations compatible with a dinucleosome array that is twisted in a step-wise manner, mimicking quasistatic twisting. The simulations help to elucidate the distribution of

DNA twist along the dinucleosome array, the dynamical exchange between the twist and writhe modes of the storage torsional stress, and the flipping dynamics of nucleosome during twist propagation.

Finally, Chapter 4 concludes the work by summarizing the main results, 23 discussing the implications of these results, and outlining future extensions of this thesis work.

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2.1. Introduction

Our aim is to study the unraveling dynamics of a single nucleosome subject to extensional forces acting via its linkers. A common approach for tacking the macromolecular dynamics is all-atom molecular dynamics (MD) [1,2]. In this technique, the atoms of the macromolecule are treated classically, where inter-atomic energetic interactions are described using semi-empirical potential energy functions (force field).

The dynamics of the macromolecule are then determined by integrating Newton’s equations of motion over millions of femtosecond-long time steps. [2]. However, all- atom MD simulations are limited to short length and time scales on the order of several nanometers and tens of nanoseconds [3] for systems as large as nucleosome, which comprises of a ~108 kilo-Dalton histone octamer and ~130 kilo-Dalton DNA

(corresponding to ~200 bp). Nonetheless, a number of studies have employed all-atom

MD simulations to study the dynamics of DNA [4-8] and of nucleosomes [21-26], though probing long time dynamics has been challenging. For example, an all-atom MD simulation of a nucleosome in explicit water being pulled by stretching forces could only probe the dynamics for 100 ns and was limited to abnormally high pulling rates (5 m/s)

[25].

36 37

To address this limitation of all-atom MD simulations, several coarse-grained

(CG) models of the nucleosome [27,28] and of the chromatin fiber [29-51] have been developed. The basic principle of CG modeling is to eliminate short length and time scale phenomena by treating subunits of the macromolecule as rigid bodies. The CG units interact through “effective” potentials that may be derived from inter-atomic force fields considering interactions among the subunits as well as interactions arising from the solvent molecules.

The simplest of these CG models of chromatin are the “two-angle” models

[29,30], which characterize the fiber structure using three parameters: the linker length, the entry-exit angle, and the twist angle. These models assume straight linker

DNAs, regular fiber architecture, and neglect the excluded volume of the nucleosomes. The two-angle model was used to explain the stretching behavior of chromatin fibers [35] and also demonstrated that the chromatin fiber is able to fluctuate between solenoid-like and irregular zigzag structures in response to salt condition and forces [36-38]. Several extensions of this model have been proposed to additionally account for the mechanics and electrostatics of the linker DNAs along with simple descriptions of inter-nucleosome interactions [31-34]. Such a model was recently used to examine nucleosome unraveling under dynamic force loading [39].

The most detailed models of the chromatin fiber have included more rigorous electrostatic descriptions of the nucleosome and also account for histone tail flexibility and linker histone binding [41-47], but model the histone octamer and wrapped DNA as a single nucleosome entity. A number of CG models have also been developed to explain chromatin looping and chromosome territorial behavior [48-51]. 38

In summary, all-atom simulations of nucleosomes are incapable of examining the long-time dynamics of nucleosome unraveling while most CG models cannot capture the independent dynamics of the DNA and the octamer as they model the nucleosome as a rigid entity. The only CG model that was used to study nucleosome unraveling did not examine the dynamics of the wound DNA and of the histone octamer in detail and did not accurately capture the histone/DNA interactions.

2.2. Model Development

In this study, we introduce a new coarse-grained model of the nucleosome that permits simulation of the unraveling dynamics of nucleosome over timescales in excess of hundreds of microseconds. Most importantly, the model is parameterized to reproduce the nonuniform histone/DNA interaction energy profile along the wound length of DNA, as obtained from a recent nucleosomal-DNA unzipping assay [52], and nucleosome unraveling forces obtained from dynamic force spectroscopy experiments on single nucleosomes [53]. Brownian dynamics simulations of this nucleosome model under constant-speed pulling conditions reveal the detailed dynamics of the wound DNA and the octamer during unraveling and also allows us to dissect the role played by nonuniform histone/DNA interactions in the unraveling process.

While we have used this model to study unraveling in this work, the model could potentially also be used to investigate octamer/DNA dynamics during other biological processes, such as twist-induced unwrapping of DNA from nucleosomes 39 and the effects of histone modifications on the stability of wrapped DNA.

2.2.1 Modeling of DNA and histone octamer

Our model nucleosome comprises of double-stranded DNA wrapped 1.7 times around a histone octamer such that the DNA ends, each of length 100 bp, overhang from the octamer as shown in Figure 2.1 A.

Figure 2.1: Coarse-grained model of the nucleosome. (A) Coarse-grained model of the nucleosome showing: the DNA (red) wrapped around histone octamer (gray), the directions in which the two linkers are pulled (thick arrows), and the coordinate system (thin arrows). (B) Nucleosome core showing the charged, groove beads (blue) and neutral beads flanking the groove (green) and present inside the core (orange). (C) Top view and (D) side view of the coordinates of groove (blue circles), flanking (green circles), and inner beads used to facilitate the wrapping of DNA in a helical path (black line) approximating the axis of wrapped DNA (red dots) obtained from the crystal structure. 40

The DNA and the octamer are treated as distinct entities using separate coarse-grained

(CG) models capable of assembling and disassembling.

The DNA is treated using the discretized worm-like bead chain model (Figure

2.1 A), where each “bead” represents a 3 nm segment of relaxed DNA [44, 54], yielding a total of Ndna = 39 DNA beads. Each bead is assigned a salt-dependent charge qdna according to Stigter [55]. The DNA bead-chain is also assigned an intramolecular force field comprising of stretching, bending, twisting terms that capture all aspects of

DNA mechanics.

The stretching energy Ustr of the DNA is given by

− Ndna 1 = h ∑  − 2 U str l i l 0 (1) 2 i=1 where h is the stretching constant of DNA, Ndna is the total number of DNA beads, li is the length of the segment connecting beads i and i + 1, and l0 is the equilibrium segment length. The bending energy Uben of DNA is given by

− N dna 2 = g ∑  2 U ben i (2) 2 i=1 where g is the bending constant of DNA derived from its persistence length and βi is the Euler angle describing the angle between the neighboring segments connected by beads i and i+1 and by beads i+1 and i+2. The torsional energy Utwi of DNA is given by

− N dna 2 = s ∑   2 U twi i i (3) = 2l 0 i 1 where s is the torsional energy constant and αi and γi are the Euler angles describing the twist between the local coordinate systems of adjacent DNA segments, 41

respectively. The total electrostatic repulsion Urep of the DNA is obtained as the sum of the Debye-Huckel potential between all interacting DNA beads, as given by

− 2 N dna 2 N dna −  = qdna ∑ ∑ exp rij U rep (4)    = =  4 0 i 1 j i 2 rij where κ is the salt-dependent inverse Debye screening length, rij is the separation distance between beads i and j, and ε and ε0 represent the dielectric constant of water and permitivity of vacuum. As the strong electrostatic repulsion prevents the DNA beads from overlapping with each other, no special excluded volume interactions are required. We use a monovalent salt concentration of 150 mM for parameterization and simulations.

The histone octamer is treated as a rigid body constructed from a collection of

CG beads (Figure 2.1 B). To promote superhelical wrapping of DNA around the octamer, as observed in the crystal structure of the nucleosome, we create a shallow spiral groove in the curved face of the octamer that is attractive to DNA beads. This groove is expected to follow a helical path with the same axis and pitch as that of the wrapped DNA but with a smaller helix diameter. We create such a groove from Ngr =

17 charged beads (blue, Figure 2.1 B) placed equidistant from each other along this helix and an additional Nflk = 34 neutral beads (green, Figure 2.1 B) flanking the two sides of the groove. These flanking beads are also placed equidistant from each other along helical paths offset axially from the groove helix. Their purpose is to ensure stable wrapping of DNA around the octamer. Further, we introduce Ncen = 14 neutral beads into the empty space at the center of the groove/flanking bead spirals (orange,

Figure 2.1 B) to ensure that the DNA does not enter the octamer during simulations. 42

Thus, we employ a total of Noct = 65 CG beads to describe the octamer.

To obtain the dimensions of our groove and flanking helices, we first determine the pitch and radius of the DNA superhelix. For this purpose, we fit a 3D helix to the axis of the wrapped DNA in the 1KX5 nucleosome crystal structure [56].

We define DNA axis in terms of midpoints of lines connecting pairs of phosphates on the two DNA strands involved in the Watson-Crick base-pairing interactions (red dots,

Figure 2.1 C,D). We have adapted the program PDBSUP [57], originally developed for determining the best fit superimposition of two sets of molecules, to obtain the best-fit helix passing through the axis points (black lines, Figure 2.1 C,D). The positions of the groove and flanking beads obtained from such fitting are shown by blue and green circles in Figure 2.1 C,D and their dimensions are tabulated in Table

A.1 (Appendix 1).

The charged octamer beads composing the spiral groove interact with the DNA beads using the Debye-Huckel attractive potential, with an energy given by:

N dna N gr −  = qdna ∑∑ qoct , j exp rij U att (5)    = = 4 0 i 1 j 1 r ij where qoct,j denotes the charge on groove bead j. The charge magnitudes are parameterized according to two types of experimental data, as detailed below. The groove, flanking, and interior beads interact with the DNA through an excluded volume potential treated using the Lennard-Jones (LJ) potential with energy given by:

N dna N oct  12  6 =  ∑∑[ ev  − ev  ] U ev 4 ev (6) = = i 1 j 1 rij r ij where εev and σev are the LJ energy and size parameters, which are chosen to ensure 43 that the DNA beads do not enter the interior of the histone octamer and that they do not overlap extensively with the oppositely charged groove beads.

The total potential energy of the octamer/DNA complex Utot shown in Figure

2.1 A is then given by the sum of the contributions from the DNA stretching, bending, torsion, electrostatics, and excluded volume interactions:

=      U tot U str U ben U twi U rep U att U ev (7) All force field parameters associated with the CG model are provided in Table A.1.

2.2.2 Brownian dynamics simulations

The Brownian dynamics (BD) simulations refers to an approach in which the explicit dynamics of the solvent molecules are replaced by random stochastic forces, mimicking the collisions of the solvent molecules with the macromolecule of interest, and by a friction term, mimicking the hydrodynamic drag force exerted by solvent molecules on the macromolecule [2]. The approach is based on the assumption that there exists a wide separation of time scales between the macromolecule and solvent motions. Furthermore, BD simulations assume that the velocity of the macromolecule relaxes much faster than the characteristic translational motion, allowing one to further neglect the inertial (acceleration) terms in the equation of motion of the macromolecule. Thus, BD simulations alleviate significant computational costs associated with simulating solvent molecules [2].

We employ a Brownian dynamics (BD) approach to simulate the dynamics of the octamer/DNA complex subjected to extensional forces through its linkers. We account for friction but not explicit hydrodynamic interactions, whose effects are 44 secondary to the primary phenomenon being investigated here. The second-order

Runge-Kutta algorithm of Iniesta and de la Torre [44,58] is employed for updating the translation ri(t) and rotation vectors Ωi(t) of each component i of the system (i = DNA beads or rigid octamer):

T    p   =   Di Fi t Fi   ri t t ri t t R i (8) 2 k B T

R    p     =   Di Ti t Ti   i t t i t t Wi (9) 2 k B T

T R where Di and Di are the translation and rotational diffusion coefficient, respectively; Fi(t) and Ti(t) are the forces and torques experienced by component i at

p p p time t; F i and T i are the predicted force and torque at time t =t+Δt; and Ri and Wi are

Gaussian-distributed random vectors satisfying the fluctuation-dissipation theorem

[59,60]:

〈 〉= 〈 〉= T   R i 0, Ri R j 2Di t ij I (10)

〈 〉= 〈 〉= R   Wi 0, Wi W j 2Di t ij I (11)

where I – unit matrix 3 Χ 3, δ – Kronecker's delta.

The above equations of motions apply for all components except the two terminal linker DNA beads, which are pulled at constant speeds Vpull/2 in opposite directions along the y-direction, as depicted in Figure 2.1 A.

While the octamer exhibits rotation and torque along its three axes, the DNA 45 beads only exhibit rotation and torque along their axis, defined by a vector pointing towards the next bead along the DNA bead-chain. The reader is referred to Ref.

[44,46] for a detailed description of the DNA model. The translational and rotational

T = /  diffusion coefficient of the octamer is given by Doct k B T 6 Roct and

R = /   3 Doct k B T 8 Roct , where Roct is the hydrodynamic radius of octamer and η is the solvent viscosity. Similarly, the translational and rotational diffusion coefficient of the

R = /  2 DNA beads along the DNA axis Ddna k B T 4 Rdna l0 , where Rdna is the hydrodynamic radius of each DNA bead and l0 = 3 nm is the segment length associated with each DNA bead. The force on each DNA and octamer bead is computed from the gradient of the total potential energy:

F =−∇ U i ri tot (12) where ri and Fi are the position vector and force acting on component i, respectively.  The torque on the ith DNA bead acting along its axis vector ai , arising from the twisting potential, is given by

=− s   − −   Tdna ,i i i i−1 i−1 ai (13) l 0 while the torque on the histone octamer is given by

N =∑oct  − × Toct ri rcm Fi (14) i=1 where ri - rcm denotes the position vector of octamer bead i relative to the octamer center of mass. All parameters associated with the BD algorithm are provide in Table

A.1. 46

2.2.3 Parametrization of octamer groove charges

We use two different types of experimental data to parameterize charges qoct;i on our octamer groove beads. First, we use the position-dependent DNA/octamer binding free energy profile recently obtained by Forties et al. [52] to determine the relative magnitudes of qoct;i . The free energy profile was derived by analyzing the single-molecule experiments of Hall et al. [61], where the DNA on single nucleosomes was unzipped at constant force and the time intervals, or “dwell times”, between the unzipping of successive basepairs were measured. Large dwell times signified strong local histone/DNA interactions that hinder unzipping. Second, we use the single- molecule experiments of Pope et al. [62] measuring the average force Funr at which the

DNA unravels completely from the octamer when nucleosome arrays are stretched apart at a constant speed. These experiments indirectly provide us the net strength of histone/DNA interactions, allowing us to derive the absolute values of qoct;i .

We first derive the relative magnitudes of qoct;i from the position-dependent free energy profile of histone/DNA interactions, provided in a cumulative fashion Gexp(j)

[52] (Figure 2.2 inset). However, this profile is of 5-bp resolution, which is incompatible with the resolution of x = 146 bp/Ngr = 8.59 bp/bead required by our model. Hence, we convert the discrete Gexp(j) into a continuous function through spline fitting, which is then converted back into a discrete G'exp(i) of resolution 8.59 bp, where i = 1;...;Ngr. The expected contribution from each groove bead is then given by:

  ≡   −    = G ' exp i G ' exp i 1 G ' exp i ,G ' exp 0 0 (15) 47 shown in a normalized fashion in Figure 2.2 (red circles).

Our aim is to obtain a set of qoct;i that would yield ΔG'exp(i) to within a multiplicative constant. For this purpose, we break down the total binding free energy

N  ≡∑gr    G ' exp G ' exp i into two contributions: i=1

N  =  ≡∑gr    G ' exp U ' do G ' rem U ' do i G ' rem (16) i =1

where ΔU'do denotes position-dependent attraction between the DNA and the octamer while ΔG'rem accounts for all remaining position-independent contributions arising from DNA bending and entropy. Because ΔG'rem is independent of position, we can

“smear” it equally over all groove beads, allowing us to write the local free energy as:

  =   / G' exp i U ' do i G ' rem N gr (17)

In our model, most of the contribution to ΔU'do(i) comes from screened electrostatic interactions via Eq. (5) whose strength is proportional to qoct;i, i.e.

  ≈− U ' do i Kq oct ,i , where K is a multiplicative factor greater than zero to enforce attraction.

We also note from our simulations that the total energy of the octamer/DNA complex remains roughly half to one-third of the total electrostatic attraction between the DNA and octamer. We tentatively assert that ΔG'exp drops down to a fraction α=

1/2 of the total attraction energy ΔU'do due to unfavorable contributions that resist 48

DNA wrapping, whereby

N  =  ≡∑gr    G ' exp U ' do U ' do i (18) i =1

Figure 2.2: Normalized charge values assigned to the octamer groove beads as a function of their location along the wound DNA relative to the dyad.

The charges qoct;i derived from the cumulative free energy profile of Forties et al. [29] (inset) are shown as black squares. For comparison, the charges obtained when ΔG'rem is neglected and when uniform histone/DNA interactions are assumed are shown in red circles and green triangles, respectively. The inner turn span is indicated by dotted blue lines.

Equations (13-16) provide us a way to obtain qoct;i from the available ΔG'exp (i) to within a multiplicative constant via:

N − gr =−  − 1 ∑    Kqoct , i G ' exp i G ' exp i (19)  = N gr i 1 49

Since the qoct;i thus obtained are relative, we can scale them for convenience to yield

N gr 0 ∑ 0 / = 0 q oct;i such that qoct ,i N gr 1 . Figure 2.2 plots q oct;i (black squares) along the i=1 wound length of DNA relative to the dyad. For comparison, we have also plotted in

Figure 2.2 the scaled charge profile if all groove beads were assigned equal charge

(green line) and the scaled charge profile if the ΔG'rem term was ignored in the charge parameterization (red circles); in this case the scaled charges can be shown to be equal to the scaled ΔG'exp (i).

0 We next determine the absolute magnitudes of q oct;i that would yield the same ˙ unraveling forces Funr at different loading rates F (rate at which the applied force increases) as those measured experimentally [53]. To this end, we perform Brownian dynamics simulations of our model nucleosome in which the ends of the linker DNA are pulled apart at a relative speed Vpull. We repeat these simulations for different

0 values of the scaling factor λ used to obtain qoct;i according to qoct;i = λq oct;i. Multiple pulling speeds within the range 0.025–0.25 cm/s are employed for each λ. Note that the speeds or loading rates typically imposed in simulations are orders of magnitude higher than the 10~100 nm/s rates typically employed in the single molecule experiments used for obtaining Funr. Hence, a direct comparison between computed and experimental Funr is not possible. To facilitate quantitative comparison between the two, we use a model based on Kramers’ theory [30–32] for extrapolating the computed

Funr to small loading rates. According to this model, the average force Funr at which a ˙ system subjected to a loading rate F crosses over a transition state separating two 50 stable states is given by:

  b  b k e G = G [ −{ 1 0 } ] (20) F unr b 1 b ln b ˙  x G  x F where ΔGb is the height of the activation energy barrier associated with unraveling, xb is the distance between the barrier and the initial fully wrapped state, k0 is the intrinsic unraveling rate at zero force, γ = 0.578, ν= 2/3, and β= 1/kBT. Through a trial and error procedure, we find that λ= 7.1 (≡λ0) provides a good fit between the computed and

b b -6 -1 experimental Funr, yielding values of ΔG ≈17.7kcal/mol, x ≈4.28 nm, k0 ≈2.53×10 s , similar to those obtained earlier [21]. Hence, we set our groove charges according to =

7.1 for all our simulations.

Figure 2.3 illustrates how the computed Funr for λ0 fit much better with the experimental data compared to those computed for slightly smaller and a larger λ.

Hence, for the rest of our simulations, we establish our groove charges according to

λ=7.1. 51

Figure 2.3: Mean unraveling force as a function of the normalized loading rate.  ˙ / ˙  Normalized loading rate ln F F 0 is computed from simulations for = 6.8 (red circle),7.1 (black square), and 7.4 (blue circle) compared to the experimentally ˙ measured values (green triangles), where F 0 = 1 pN/s. λ = 7.1 yields the best match between simulations and experiments. The dashed line represents the best fit of the simulated and experimental Frup via Eq. (19).

2.3. Results

We first investigate the dynamics associated with the forced unraveling of

“canonical” nucleosomes possessing nonuniform histone/DNA interactions, i.e., groove beads possessing charges described by the distribution shown in black squares

0 in Figure 2.2 multiplied by λ0, i.e. qoct,i = 7.1×q oct,i. We then investigate the role of nonuniform histone/DNA interactions by comparing the results from canonical 52 nucleosome with those obtained for a “non-canonical” nucleosome possessing uniform histone/DNA interactions. Each groove bead is assigned a fixed charges of +7.1e such that the net charge on the groove beads remains the same across the two types of nucleosomes. The unraveling simulations for both systems involve pulling the two linker ends apart with equal and opposite speeds, starting from the crossed-linker configuration of Figure 2.1 A, until the nucleosomes have unraveled and the linker ends are at least 120 nm apart. We employ five different pulling speeds vpull in the range 0.025–0.25 cm/s for the two systems spanning 20–400μs. For each pulling speed, we perform 36 simulations starting from different initial seeds for the random number generator.

2.3.1 Force-extension behavior

Figure 2.4 A presents the force-extension (F-x) behavior of the canonical nucleosome obtained from 36 separate BD simulations carried out at vpull = 0.05 cm/s.

The computed F-x behavior agrees well with those measured experimentally for mononucleosomes and nucleosome arrays using single-molecule pulling experiments

[53, 62-65]. 53

Figure 2.4: Computed force-extension curves. (A) Computed force-extension curves (36 in total) for pulling speed v = 0.25 cm/s are shown in gray, and one representative plot is shown in black. (B) Representative force- time curves for five different pulling velocities: v = 0.025 cm/s (black), 0.05 cm/s (red), 0.075 cm/s (green), 0.1 cm/s (blue), and 0.25 cm/s (magenta). Rough locations of the three unraveling regimes are also indicated.

The F-x behavior from other pulling speeds also follows the same qualitative behavior and are not shown here. All F-x curves exhibit a prolonged, gradual rise in force in the initial stages of pulling until an extension of 79–83 nm or a force of 10 pN

R has been reached .We denote this regime of slow force growth by 1. The gradual rise in force gives way to a steep, near-linear rise in the force culminating in a sudden drop or “rip” in the F-x plots at an extension of 100 nm. We denote the regime of sharp 54

R R force growth by 2. The sudden drop in force at the end of 2 signifies nucleosome unraveling, where upon most of the DNA dissociates from the octamer. After unraveling, the force begins to rise again in a sharp, non-linear manner with respect to extension, which resembles the stretching of naked DNA. We denote this regime of

R the sharp drop in force and its eventual rise by 3.

Figure 2.4 B presents representative force-time (F-t) plots for all five pulling speeds vpull investigated here. The average force at which the nucleosome unravels, termed rupture force Frup, increases with vpull and is plotted in Figure 2.3. This increasing trend in Frup–vpull is common to most microscopic transitions like protein unfolding, drug-ligand binding, and DNA unzipping that involve the thermal crossing of an activation energy barrier, where the application of a stretching force results in the tilting of the energy landscape and an effective lowering of the energy barrier. Fast pulling leads to higher dissipation and thereby a higher force is required to unravel nucleosomes. The observed increase in Frup with vpull is consistent with existing theoretical models, as discussed earlier in detail [66-68].

2.3.2 DNA unwrapping dynamics

We characterize DNA unwrapping dynamics in terms of the time evolution of the number of turns nwrap of DNA wrapped around the histone octamer. Hence, nwrap

≈1.7 represents a fully wrapped, canonical nucleosome while nwrap = 0 represents a completely unwrapped nucleosome. Since all pulling speeds investigated here yielded similar unwrapping dynamics, we present only results from one pulling speed. 55

Figure 2.5: Time evolution of extent of nucleosome wrapping.

(A) Time evolution of extent of nucleosome wrapping nwrap(t) for vpull = 0.05 cm/s. A representative trajectory is shown in black while the remaining 35 trajectories are shown in gray. (B) Frequency histograms of nwrap(t) obtained before reaching 50 nm extension (red),immediately before the rip (blue), immediately after the rip (violet), and well after the rip(green).. (C) Schematics showing the wrapped state of the nucleosome; the inner and outer turns are shown in gray and black, respectively.

Figure 2.5 A shows a representative nwrap(t) profile computed from a BD simulation performed with vpull = 0.05 cm/s. The corresponding force-time F(t) plot and nwrap(t) from 35 other simulations performed at the same pulling speed are shown alongside for reference. Figure 2.5 B plots the distributions in nwrap at four distinct

R points along the unraveling process: (i) initial portion of 1 within x < 50 nm; (ii) end

R R of 2 within a 6 μs time window preceding the rip in the F-x plot; (iii) beginning of 3

R within a 3.6μs window immediately following the rip; and (iv) later part of 3 for x >

115 nm when the DNA is taut again. The distributions have been obtained by averaging over multiple simulations at the specified pulling speed. Note that the 56

computed nwrap has a resolution of ~0.111 turns due to the discrete nature of our DNA.

Figure 2.5 A shows that the wrapped DNA close to the entry/exit site comes on

R and off the surface of the histone octamer within regime 1. In particular, nwrap fluctuates between a partially unwrapped state containing ≈1.22 DNA turns and a fully wrapped state with ≈1.67 turns (Figure 2.5 B, red distribution), which corresponds to

~20 bp of DNA coming on and off the octamer surface from each end. The unwrapping/rewrapping dynamics likely arises due to the relatively weak

DNA/histone interactions at the flanking portions of wound DNA (see Figure 2.2).

These dynamics are related to the nucleosome “breathing” observed experimentally, where the nucleosome fluctuates between fully and partially wrapped state to facilitate access to transcription factor binding sites buried in the DNA near the entry/exit site

[69, 70]. Apart from these breathing motions, we do not observe any further

R unwrapping of DNA in 1.

The flanking portions of wound DNA, which fluctuate between wrapped and

R R unwrapped states in 1 , become permanently unwrapped within regime 2 due to the increasing tension in the DNA, which suppresses rewrapping (Figure 2.5 A). We observe further unwrapping of DNA from both ends in this regime till nwrap ≈ 1.08

(Figure 2.5 B, blue distribution), i.e., a single turn of DNA remains unwrapped before the nucleosome unravels abruptly. This gradual unwrapping of ~0.6 turns of DNA

R (outer turn) within 2 is not associated with any visible rips in the F-x or F-t plots, suggesting that it occurs near-reversibly. Such reversible unwrapping of the outer turn of DNA is in agreement with experimental single molecule pulling studies [53, 62-65]. 57

R The abrupt unwrapping of the nucleosome at the onset of 3 releases about 20–

30 bp of the inner turn of DNA until nwrap ≈ 0.78 (Figure 2.5 B, violet distribution).

Interestingly, this unraveling coincides with the unwrapping of the strong off-dyad patch of histone/DNA interactions discussed in detail later. The DNA released from the octamer takes some time to become taut upon further pulling before the F-x curves

R begin to rise sharply. In this part of 3, additional DNA gets unwrapped from the octamer, but this unwrapping occurs gradually (reversibly). The DNA however does not dissociate completely from the octamer, even with extended pulling. Instead, nwrap

≈ 0.22–0.33 turns or 20– 30 bp of DNA remains attached the octamer (Figure 2.5 B, green distribution), likely as a result of the strong histone/DNA interactions at the dyad (Figure 2.2 A).

2.3.3 Kinematics of nucleosome unraveling

The translational motion of the histone octamer is characterized using the cartesian coordinates rcom ≡ (xcom, ycom, zcom) of its center of mass.

Figure 2.6 shows representative trajectories of the octamer from a simulation

R performed with vpull = 0.05 cm/s. The trajectories indicate that during 1 regime the nucleosome moves in the +x-direction towards the force axis (Figure 2.6 A). Recall ∓  that rcom = (0, 0, 0) at t = 0 and that the force is applied along the ez axis passing through (24,0, 0). This translational motion prevents a buildup of tension within the

DNA linkers, explaining why the force rises so gradually in this regime (Figure 2.4 A).

Since the linkers remain slack, we also note significant Brownian motion of the 58 octamer in the y and z directions (Figure 2.6 B,C). After sufficient pulling, the linkers

R become taut, which signals the onset of regime 2 . In this regime, the tension in the linkers no longer has a mechanism for relaxing it through translational motion, leading to a rapid build up of force (Figure 2.4 A) and a concomitant suppression in the

Brownian motion of the octamer. The positive and negative deflections in the y- position of the octamer (Figure 2.6 B) do not arise from Brownian motion but from an

R asymmetric unwrapping of DNA from its two ends. Within the 3 regime, the

Brownian fluctuations escalate upon nucleosome unraveling, as the DNA becomes slack, but decrease with continued pulling of the linkers as the DNA becomes taut again. The final spread in the y-position of the octamer is again indicative of the asymmetric unwrapping of DNA from the two ends.

We have analyzed the rotational motion of the nucleosome in terms of its  elevation angle Φele, defined as the angle made by the DNA superhelical axis c (see

Figure 2.1) and the pulling axis (+y-axis).

Figure 2.7 A shows representative plots of Φele as a function of pulling time for vpull = 0.05 cm/s and Figure 2.7 B–F shows snapshots of nucleosome orientation at different time points along the simulation. The azimuthal angle made by c along the x–z plane is not discussed, as it exhibits random behavior and does not reveal any meaningful trends. At time t = 0, the nucleosome is in its initial configuration shown in Figure 2.1 A and Φele = 90º. The strong electrostatic repulsion between the crossed linkers introduces a strong torque in the octamer, causing it to immediately “flip” until the superhelical axis is anti-parallel to the y-axis, i.e., Φele approaches 180º (Figure 2.7 59

B), which results in a more open conformation of the entering/exiting linkers. As the

Figure 2.6: The octamer center of mass as a function of time for BD simulations.

The xcom (A), ycom (B), and zcom- coordinates (C) of the octamer center of mass as a function of time for BD simulations with vpull = 0.05 cm/s. A representative trajectory is shown in black and the remaining 35 trajectories are shown in grey.

two linker ends are pulled apart, they impose a small torque to the octamer, gradually rotating it by another 90º to an orientation with Φele approaching 90º (Figure 2.7 C).

R Both these flipping events occur within the low-force 1 regime. Once the linkers have

R become taut in regime 2 , the octamer becomes tilted with respect to the force axis, with Φele close to 135º (Figure 2.7 D).

The taut linkers also importantly permit the nucleosome to swivel in the azimuthal direction about the pulling axis, rotating the nucleosome into orientations

R more amenable to unraveling. At the point of unraveling (beginning of 3), the 60 nucleosome orients itself such that its superhelical axis is anti-parallel to the y-axis with Φele approaching 180º, which provides the most favorable orientation for the

DNA to peel off from the surface of the octamer (Figure 2.7 E). After unraveling, the octamer quickly aligns parallel to the pulling axis with Φele ≈ 0 (Figure 2.7 E).

The “deterministic” rotations described above are also accompanied by significant fluctuations in Φele arising from Brownian motion. A rough timescale τrot of these fluctuations may be obtained from the rotational diffusivity Drot of the nucleosome, as calculated using the Stokes-Einstein-Debye relationship [71]:

 ~2 / ≈ /  3 rot D rot ; Drot k B T 8 R , where η is the solvent viscosity and R ~ 5 nm is the hydrodynamic radius of a nucleosome [44]. We obtain τrot ~10 μs, indicating that the nucleosome undergoes 10–100 of random rotations during the entire unraveling process spanning ~200 μs for vpull = 0.05 cm/s. Faster pulling speeds accommodate a lesser number of Brownian fluctuations making the deterministic rotations easier to distinguish from random rotations. 61

A

B

Figure 2.7: Time evolution of the nucleosome during simulation.

(A) Time evolution of the elevation angle elev(t) for vpull = 0.05 cm/s. A representative trajectory is shown in black while the remaining 35 trajectories are shown in gray. (B) Snapshots of the nucleosome captured at different stages (values indicated in s) during the simulation for the five different time-windows indicated in (A). In the snapshots, the DNA is shown as red cylinderial tubes spline fitted through the DNA bead centers and the nucleosome is shown as gray cylinder. 62

2.3.4 Energetics of nucleosome unraveling

We have plotted in Figure 2.8 the time evolution of the total energy Utot of the nucleosome during unraveling along with that of its five components arising from

DNA stretching Ustr, DNA bending Uben, DNA twisting Utwi, electrostatic repulsion between DNA Urep, and DNA/octamer interactions Uoct/dna, the last of which includes both electrostatic attraction and excluded volume interactions.

In regime R1, where the linkers are relaxed, Utot exhibits minimal changes. The only changes that occur within this regime are a gradual decrease in Uben due to straightening of the linkers and a small increase in Uoct/dna within the early stages of the regime due to the unwrapping/rewrapping of the flanking portions of wound DNA.

Note that the two energy changes do not include the loss in entropy incurred with linker straightening and the gain in entropy associated with DNA release.

In regime R2, where the linkers become taut, we note a sharp rise in Utot until the unraveling of the inner turn (force rip). The main contributors to this rise are increases in Uoct/dna, Uben, and Ustr even though Urep decreases. The increase in Uoct/dna and decrease in Urep both arise from the unwrapping of the “outer” turn of DNA from the octamer. The increase in Ustr arises from the DNA becoming taut while the increase in Uben arises from the bending of DNA at the entry/exit point in order to align with the linkers, which are oriented along the pulling direction. We also note that at the point of force rip, Uben and Ustr are both rising more sharply than Uoct/dna, with the former two rising almost quadratically with time (and extension) and the former almost linearly, suggesting that DNA stretching and bending might be responsible for the force rip. 63

Figure 2.8: Time evolution of the total energy.

Time evolution of the total energy Utot (black) of the nucleosome system (black line) and its four components from DNA stretching Ustr (magenta line), DNA bending Uben (brown line), DNA twisting Utwi (green line), and DNA/DNA repulsion Urep (blue line), and DNA/octamer attraction Uoct/dna (red line).

These results are in partial agreement with a model recently proposed by Sudhanshu et al. [72], where the largest contributor to the energy barrier related to the force rip was found to arise from the bending energy penalty imposed on the linkers as they bend to align parallel to the pulling direction. However, since their model assumed an inextensible worm-like chain model for the DNA/linkers, the importance of DNA stretching was overlooked.

Upon the unwrapping of the “inner” turn, at the beginning of regime R3, the

DNA becomes relaxed again and the DNA bending and stretching energies drop.

Continued pulling of the linkers results in the DNA becoming taut, leading to an increase in the DNA stretching energy. However, the bending energy term remains constant, as the small amount of DNA still attached to the octamer is already aligned 64

along the pulling direction and does not need to bend, as in regime R2. It is noted that the DNA twisting energy understandably remains nearly constant throughout the pulling, as the DNA ends are free to rotate during pulling.

2.3.5 Role of non-uniform DNA/histone interactions

To investigate the role of the nonuniform DNA/histone interactions exhibited in the canonical nucleosome—strong interactions at the dyad and at locations ±40 bp on either side of the dyad—we have repeated the pulling simulations for noncanonical nucleosomes in which all octamer groove beads carry the same fixed charge while the net charge carried by the groove remains the same as that in canonical nucleosome, i.e., qoct,i = +7.1e for all groove beads i. A comparison of the force-extension behavior and unraveling mechanism of canonical versus noncanonical nucleosomes reveals several important insights.

The force-extension (F-x) behavior of noncanonical nucleosomes yields features similar to those of the canonical nucleosomes (Figure 2.9 A). That is, the noncanonical nucleosomes also exhibit the three regimes in the F-x plots mentioned above: the slow rise in force with extension during which the nucleosome undergoes little unwrapping; the sharp rise in force during which the outer turn unravels reversibly; and the sharp drop and eventual rise in the force during which the inner turn unwraps abruptly and then gradually upon extended stretching of the linkers.

This similarity suggests that the strong histone/DNA interactions at the dyad and the two off-dyad locations are not responsible for the reversible and irreversible 65

Figure 2.9: Representative force-extension plots. (A) Representative force-extension plots of canonical (black lines) and noncanonical (red lines) nucleosomes for vpull = 0.05 cm/s.(B) Unraveling forces Funr computed for canonical (black squares) and noncanonical (red circles) nucleosomes as a function of normalized loading rate. The experimental data are shown as green triangles. The dashed lines represent model fits for the two data via Eq. (19)

unwrapping of the outer and inner DNA turns, respectively, as also suggested by others [73].

Even though the nonuniform DNA/histone interactions are not fully responsible for the rip in the F-x plot, they do significantly contribute to the stability 66 of the wound DNA against extensional forces. This effect can be gleaned from the results presented in Figure 2.9 B, which show that the unraveling forces Funr measured for noncanonical nucleosomes are consistently lower than those measured for the ˙ canonical nucleosomes for all loading rates F investigated here. A rough fitting of − ˙ the F unr F data with the model introduced in Eq. (14) yields an intrinsic activation energy barrier of ΔGb ≈17.4 kcal/mol, a barrier location of xb ≈ 5.20 nm, and

-6 -1 an intrinsic unraveling rate of k0 ≈ 3.78×10 s . It should be noted here that the computed Funr span only two orders of magnitude in loading rates; hence, a good fit between this data and the model is not possible. However, to obtain a rough fit, we also used the experimental Funr data for canonical nucleosomes at small loading rates

(green triangles) that were reduced by a fixed factor f = 0.83 obtained from the ratio of the computed Funr for noncanonical and canonical nucleosomes at high loading rates.

We also recall that the canonical nucleosome yielded ΔGb ≈ 17.7 kcal/mol, xb ≈ 4.28

-6 -1 nm, and k0 ≈ 2.53×10 s . Given that the canonical nucleosome possesses a larger activation barrier and a three-fold slower intrinsic unraveling rate indicates that the nonuniform DNA/histone interactions serve to stabilize the nucleosome under unraveling forces. Interestingly, the canonical nucleosome also exhibits a smaller distance to the activation barrier whose origin is discussed below.

The unwrapping of DNA with pulling time nwrap(t) (Figure 2.10 A) and distributions in nwrap (Figure 2.10 B) within distinct “windows” along the unraveling process reveal several critical differences in the unwrapping dynamics of the two types of nucleosomes. First, the noncanonical nucleosome maintains an almost fully- 67

wrapped state during R1, with nwrap ≈ 1.7 on average, and its flanking DNA exhibits minimal wrapping/rewrapping dynamics, as evidenced by the narrow nwrap distribution

(Figure 2.10 B). This behavior is in stark contrast to that of the canonical nucleosomes, which exhibit nwrap ≈ 1.45 during the same extension window and exhibits strong fluctuations in nwrap indicative of flanking DNA coming on and off the octamer surface (Figure 2.10 A,B). Thus, the weak DNA/histone interactions near the entry/exit regions of regular nucleosomes between 50 to 73 bp in Figure 2.2 facilitates breathing dynamics that is critical for protein/DNA binding. Second, the noncanonical nucleosome undergoes an “earlier” unraveling transition, at nwrap ≈ 1.23, compared to its canonical counterpart that undergoes unraveling at nwrap ≈ 1.0. As a result, the noncanonical nucleosome releases a larger length of DNA during this transition, which equals ≈32 bp corresponding to the 0.37 decrease in nwrap ; compare this to the ≈22 bp of DNA released by canonical nucleosomes (Figure 2.10 B). Clearly, the two strong patches of histone/DNA interactions ~30 bp inwards of the entry/exit site are responsible for the “delayed” unraveling of the canonical nucleosome.

Interestingly, the unraveling transition of the canonical nucleosomes at nwrap ≈ 1.11 coincides with the unwrapping of DNA off the two patches spanning nwrap = 0.79–

1.22. This observation suggests that DNA unwrapping pauses upon hitting the two strong off-dyad interaction patches and a sufficient buildup of force is required to rip the DNA off the octamer. Third, 10–15% of the pulling simulations resulted in the

DNA dissociating completely from octamers upon extended stretching beyond the unraveling transition. Such an effect was not observed in the case of canonical nucleosomes, suggesting that the strong interaction patches provide “sticky” spots for 68 the DNA to remain strongly bound to the octamer even after unraveling.

Figure 2.10: Comparison of the time evolution number of wrapped turns of DNA and octamer orientation for canonical and noncanonical nucleosomes.

(A) number ofwrapped turns of DNA nwrap(t), and (B) octamer orientation Φele(t) for canonical and noncanonical nucleosomes. The averages over results from 36 canonical and noncanonical nucleosome simulations are shown in red and black. The individual traces of nwrap(t) and Φele(t) for noncanonical nucleosomes are shown in grey. The inset in (A) shows the average nwrap(t) for the two types of nucleosomes when all nwrap(t) traces have been shifted relative to the rupture time. The x and y axes scale is identical to (A). The inset shows the orientation of the two nucleosome types at the onset of rupture. (C) Distribution in nwrap(t) over the same distinct time and extension-windows along the unraveling process as described in Figure 2.5 for canonical (black circles) and noncanonical (red bars) nucleosomes.

Thus, the weak DNA/histone interactions near the entry/exit regions of regular nucleosomes between 50 to 73 bp in Figure 2.2 facilitates breathing dynamics that is critical for protein/DNA binding. Second, the noncanonical nucleosome undergoes an

“earlier” unraveling transition, at nwrap ≈ 1.23, compared to its canonical counterpart 69

that undergoes unraveling at nwrap ≈ 1.0. As a result, the noncanonical nucleosome releases a larger length of DNA during this transition, which equals ≈35 bp corresponding to the 0.41 decrease in nwrap; compare this to the ≈26 bp of DNA released by canonical nucleosomes (Figure 2.10 B). Clearly, the two strong patches of histone/DNA interactions ~30 bp inwards of the entry/exit site are responsible for the

“delayed” unraveling of the canonical nucleosome. Interestingly, the unraveling transition of the canonical nucleosomes at nwrap ≈ 1.00 coincides with the unwrapping of DNA off the two patches spanning nwrap = 0.69–1.1. This observation suggests that

DNA unwrapping pauses upon hitting the two strong off-dyad interaction patches and a sufficient buildup of force is required to rip the DNA off the octamer. Third, 10–15% of the pulling simulations resulted in the DNA dissociating completely from octamers upon extended stretching beyond the unraveling transition. Such an effect was not observed in the case of canonical nucleosomes, suggesting that the strong interaction patches provide “sticky” spots for the DNA to remain strongly bound to the octamer even after unraveling.

Analyzing the rotational dynamics of the noncanonical nucleosomes reveals similar, albeit gentler, “flipping” motions as the canonical nucleosomes (Figure 2.10

C). In particular, the noncanonical nucleosomes unravel in a tilted orientation with respect to the pulling direction ( Φele ≈ 135º), as opposed to the canonical nucleosomes that orient with their superhelical axis antiparallel to the pulling axis ( Φele ≈ 180º) before unraveling (see inset of Figure 2.10 C). This difference arises due to the noncanonical nucleosome unraveling with 1.22 turns of wound DNA, which yields a large moment arm and thereby a large torque, causing the octamer to tilt. In contrast, 70 the canonical nucleosome unravels with one complete turn of wound DNA, which yields a negligible moment arm and thereby a negligible torque, causing the octamer to orient perpendicular to the pulling axis. The translational dynamics of the two types of nucleosomes are similar and are not discussed here.

2.4. Discussion

We have introduced a new coarse-grained model of the nucleosome in which the histone octamer and the doublestranded DNA are treated as discrete entities capable of assembling and disassembling. Such distinct treatment of DNA and the octamer thereby allows one to study the dynamics of DNA wrapping or unwrapping from the octamer, at equilibrium or in the presence of external forces and torques on the DNA. The DNA in our model is treated using a discretized worm-like bead-chin model and the octamer is modeled as a rigid cylinder carrying a positively charged, superhelical groove on its curved surface. The groove is designed to accommodate

~1.7 turns of DNA in a conformation similar to that observed in the crystallographic structure of the nucleosome. The most important and unique feature of this model is its parameterization of octamer/DNA interactions based on two dierent single-molecule measurements. The interactions between the groove and DNA have been parameterized to reproduce: 1) the free energy profile of histone/DNA interactions deduced from a nucleosomal- DNA unzipping assay and 2) the forces at which nucleosomes unravel in single-molecule pulling experiments conducted at constant loading. 71

We have next used this model to investigate the forced unraveling of nucleosomes resulting from pulling apart the free DNA linker ends at constant speeds, similar to single-molecule pulling experiments. The simulated force extension profiles, obtained via Brownian dynamics simulations, exhibit three distinct unraveling regimes that agree well with single-molecule pulling measurements on mononucleosomes at physiological conditions. However, the simulations go beyond the experiments by also providing the detailed mechanistic picture of each unraveling regime. Below we summarize our main findings.

Regime 1. At small extensions, the force rises gradually with extension due to the movement of the nucleosome towards the pulling axis, which keeps the linkers relaxed. Consequently, little DNA unwrapping occurs except for the spontaneous unwrapping and rewrapping dynamics of the DNA at the entry/exit and strong

Brownian fluctuations are prevalent in this regime. The octamer undergoes a fast 90º due to electrostatic repulsion between the crossing linkers and then a slow 90º flip due to the torque imposed by the two non-aligned linkers. At the end of this regime, the octamer is aligned with its superhelical axis perpendicular to the pulling direction. The octamer aligns with its superhelical axis perpendicular to the pulling axis.

Regime 2. At intermediate extensions, the force rises in a sharp, linear manner with extension. Since the two linkers are taut and aligned along the pulling axis, they permit swiveling motions of the octamer about the linkers and also suppress all

Brownian fluctuations that attempt to displace the nucleosome away from the pulling axis. The rapidly rising force causes a gradual, reversible release of the outer turn of

DNA until exactly one turn of DNA remains. The octamer also undergoes a slow 90º 72 flip until its superhelical axis is aligned parallel to the pulling direction.

Regime 3. An abrupt, irreversible release of a part of the inner turn of DNA, associated with a the rip in the force extension plots occurs at the beginning of this regime. Following unraveling, the linkers first slacken and then become taut again upon further pulling. Additional DNA unwrapping is observed, but ~10–20 bp of DNA always remains attached to the octamer, even after extended pulling. By the end of this regime, the octamer has flipped another 90º and assumes an orientation with its superhelical axis perpendicular to the pulling direction.

We have also used the model to dissect the role of non-uniform histone/DNA interactions present in canonical nucleosomes—stronger interactions at the dyad and

±40 bp from the dyad — by comparing the unraveling behavior of canonical versus

“noncanonical” nucleosomes, which possess uniform interactions across the wound length of DNA. That both nucleosome types exhibit reversible unwrapping of the outer turn and irreversible unwrapping of the inner turn unequivocally demonstrates that the nonuniform histone/ DNA interactions are not responsible for the observed differences in the unwrapping of the two turns. Instead, the nonuniform histone/DNA interactions serve to further stabilize the wrapping of the second DNA turn, as evidenced by the consistently lower unraveling forces of the noncanonical nucleosomes. A closer examination reveals that the noncanonical nucleosomes unravel with their superhelical axis tilted relative to the pulling axis and their octamers wrapped by 1 1/4 turns of DNA, as opposed to the superhelical axis being parallel to the pulling axis and the octamers wrapped by 1 turn of DNA for canonical nucleosomes. Intriguingly, the off-dyad sites of strong histone/DNA interactions are 73 located between 3/4 and 1 1/4 turns of wrapped DNA, suggesting that these two sites serve to delay the unraveling of the inner turn of DNA, which is clearly the rate limiting step in nucleosome unraveling. A rough estimation of the height and location of the unwrapping energy barriers for the two nucleosome types using a theoretical model reveals that the nonuniform histone/DNA interactions increase the barrier height and decrease its distance from the fully wrapped state, resulting in an overall lowering of the intrinsic unraveling rate of the nucleosome.

The nonuniform histone/DNA interactions in canonical nucleosomes also facilitate the unwrapping-rewrapping dynamics (breathing) of the flanking portions of the wound DNA due to the relatively weak histone/DNA interactions at the entry/exit sites. These breathing dynamics observed in canonical nucleosomes are well supported by FRET measurements on mononucleosomes [74, 75], which predict an equilibrium between partially open and closed states. On the other hand, noncanonical nucleosomes possessing stronger interactions at the entry/exit sites exhibit less breathing dynamics. It may then be speculated that the equilibrium between fully and partially wrapped states of the nucleosome could be modulated through histone modifications at the entry/exit site. The acetylation of Lys 16 on histone H3 has, in fact, recently been shown to facilitate the unwrapping of DNA [76]. The highly dynamic nature of the wound DNA at the entry/exit point also underscores the importance of the binding of linker histones at the nucleosome dyad for “stapling” together the entering and exiting DNA linkers to enable chromatin to adopt compact conformations [77]. Finally, our study reveals that the strong histone/DNA interactions at the dyad, and at the two off-dyad locations, in canonical nucleosomes prevent the 74 complete dissociation of the octamer from the DNA, even upon excessive stretching.

This effect might be useful for chromatin remodeling, where the octamer resists its detachment from the DNA during its partial dissociation and/or manipulation of its wound DNA by the remodeling machinery. Again, histone modifications, especially at the dyad region, could potentially modulate remodeling activities. Indeed, acetylation of Lys 115 and 122 of the H3 histone are known to modulate the stability of the octamer to complete detachment from the DNA [76].

ACKNOWLEDGMENTS

The text of Chapter 2 is partly based on paper I. V. Dobrovolskaia and G. Arya

“Dynamics of forced nucleosome unraveling and role of nonuniform histone-DNA interactions”, in press, Biophysical Journal, 2012.

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3.1. Introduction

As discussed in Chapter 1, the majority of studies focused on the torsional properties of naked DNA are not relevant to eukaryotic DNA, which is rarely present in a naked form. Eukaryotic DNA is instead wrapped around histone octamers to form nucleosomes [1]. Naked DNA is only present in short intervening sections between nucleosomes, called linkers, typically 20–70 basepairs (bp) in length. Under physiological conditions, the chain of nucleosomes folds into a compact, 30-nm thick chromatin fiber, which in turn compacts into higher-order structures [2]. We ask the all- important question, how are the torsional properties of DNA affected by its organization into nucleosomes and chromatin?

First, the strong electrostatic binding of DNA to the histone octamer is expected to critically hinder the propagation of torsional stresses between consecutive linkers along the wound DNA [3]. Second, as a result of hindered twist propagation, chromatin may instead relax its torsional stress through rigid-body-like nucleosome rotations or nucleosome flipping. The latter has been predicted to occur in the single-chromatin fiber twisting experiments of Bancaud et al. [4,5]. Third, the DNA within chromatin is subjected to significant steric constraints from nucleosomes and internucleosomal

80 81 interactions mediated by histone tails. Moreover, the strong twist/bend coupling in

DNA may be exacerbated in chromatin due to such steric constraints, thus modulating chromatin torsional stresses in an as-yet unknown manner.

The elastic properties of chromatin have been extensively studied over the last

10 years experimentally [6-10], theoretically [11-17] and by computer simulations

[18-20]. Single molecule pulling experiments have estimated the internucleosomal attractive energy to be in the range of 3.4 kBT to 16 kBT [6,21] depending on the compaction state of the fiber. The persistence length for a chromatin fiber remains controversial but depending on the experimental strategy by which it was measured, it has been found to vary between 30-200 nm [10]. A stretching modulus of 5-8pN was obtained by one set of single-moleclar manipulation experiments [5,6], while it was found to reach 150pN by another sets of experiments [7]. Computer simulations [22] have also predicted a wide range of elastic moduli, from 60 to 240 pN, depending on the salt concentration and the fiber geometry.

To date, few studies have examined the torsional behavior of chromatin

[4,5,23]. Bancaud et al. [4,5] used magnetic tweezers to twist individual reconstituted nucleosome arrays at fixed stretching forces. The arrays were found to accommodate large torsional stresses without significant changes in its length, in sharp contrast to

DNA. Moreover, the length variations were found to be highly asymmetric with respect to applied twist direction. The authors proposed a model for these variations based on nucleosome flipping, which modulated the entry/exit linker conformation

[4,5]. Recent single-molecule twisting studies [24] have confirmed chromatin’s lower torsional rigidity compared to naked DNA. Other studies have examined the 82 functional consequences of transcription-generated DNA supercoiling in eukaryotic organisms. For example, Kouzine et al. [25] have shown that torsional stresses can propagate over kbp domains in chromatin, promoting formation of non-B-DNA structures that signal binding of specific proteins. The above studies represent only the tip-of-the-iceberg, and undoubtedly, many more investigations will provide additional quantitative results on the microscopic torsional mechanics of chromatin and its regulatory roles in biology.

In this study, we show that simple computational models can be used to obtain new insights into the torsional behavior of chromatin. Specifically, we examine the propagation of torsional stress from one linker to the next, across a single nucleosome.

Our simulations yield two intriguing findings. First, the magnitude and sign of applied-versus-induced twist in the linkers is dictated by their relative orientation. A subset of these orientations leads to opposite twist direction in consecutive linkers, a phenomenon we refer to as ‘‘twist inversion’’. We propose a simple physical explanation of twist inversion based on the geometry of the linker DNA/nucleosome complex. Second, we observe a phenomenon analogous to buckling in twisted semirigid rods, whereby the nucleosome undergo sudden flipping in response to continued applied twist. Nucleosome flipping can induce drastic changes in the conformation of the dinucleosome and its overall writhe. We discuss the potential relevance of our findings to chromatin’s ability to absorb applied twist, higher-order folding of chromatin, and functional regulation of chromatin through mechanical stresses. 83

3.2. Model and Simulation Methods

3.2.1 Linker and nucleosome model

Our model system, shown schematically in Figure 3.1, contains two nucleosomes and two DNA linkers, which we refer to as a dinucleosome array or a dinucleosome. One end (the nucleosome) is held fixed and the other end (the linker) is twisted to examine how twist propagates from one linker to the next across the central nucleosome. The dinucleosome thus represents the most fundamental unit of chromatin for examining its torsional properties. It allows us to extract the essential physics governing propagation of twist across the chromatin fiber that is not affected by other more complex effects such as internucleosomal interactions and torsional forces arising from fluctuations in neighboring nucleosomes.

We model the dinucleosome based on previous work [19,26,27], but simplify some aspects of it. The linker is composed of six contiguous charged beads to mimic the 60-bp linkers of chicken erythrocyte chromatin [19,27]. Each linker bead (Figure

3.1) represents a 3-nm-long section of dsDNA. Linkers are ascribed interaction potentials that account for: salt-dependent electrostatics, stretching, bending, and twisting mechanics, and excluded volume interactions between other linker and nucleosomes. The linker connected to the first nucleosome (blue) is termed ‘‘entering’’ and the end linker is termed ‘‘exiting’’. The nucleosome is composed of two stacks of beads, each of radius 1.5 nm. Each stack contains 11 beads: nine beads are evenly spaced in a ring of radius 4.3 nm and two are placed diametrically opposite to each 84 other inside the ring (Figure 3.2).

A A

B

Figure 3.1: Dinucleosome array model and nucleosome and linker bead coordinate systems. (A) Dinucleosome array showing the orientation of the two linkers (red), as defined by the entry/exit angle θ and azimuthal angle φ. The rightmost nucleosome (blue), penultimate (orange), and last linker beads are spatially and rotationally constrained in our twisting simulations. (B) Nucleosome and linker bead coordinate systems.

All beads are fixed in space to model a rigid nucleosome. This simplified model allows us to clearly delineate effects of nucleosome-linker geometry and linker torsion in modulating the conformation of the dinucleosome without interference from more complicated effects arising from linker histones and histone tails. Our choice for a simpler model of the nucleosome is also motivated by observations that more- 85 detailed models exhibit essentially similar torsional behavior but are significantly more computationally demanding. Moreover, the simple dinucleosome system allows us to dissect twist propagation without the higher-order competing effects present in larger arrays.

Figure 3.2: Schematic of the applied and induced twist from the MC simulations. (Left panel) Twist is applied quasistatically by rotating the local coordinate system of the penultimate linker bead about its a axis by an angle Ω. (Right panel) Schematic of the applied and induced twist from the MC simulations.

3.2.2 Dinucleosome mechanics and energetics

To account for bending and torsion in the system, local coordinate systems are ascribed to each nucleosome and linker bead. Nucleosome orientations are prescribed

by a set of orthonormal unit vectors, Гc ≡{ac, bc, cc}, where ac and bc lie in the plane defined by the flat surface of the nucleosome. The values ac and bc point in directions tangential and normal (inwards) to the nucleosomal at the point of attachment of the exiting linker, respectively, and cc = acxbc (Figure 3.1). 86

The linker coordinate systems are denoted by Гi ≡{ai, bi, ci} where i is the linker bead index and ai points from bead i to i+1. Two additional coordinate systems are defined. The coordinate system of the exiting linker attachment is denoted

by Гex ≡{aex, bex, cex}. The two coordinate systems Гc and Гex are identical. The

coordinate system of the entering linker attachment is denoted by Гen ≡{aen, ben, cen}, where aen and ben are tangential and normal to the attachment, respectively, and cen is defined by cen = aen Χ ben.

Sets of Euler angles provide the transformation between consecutive

+ + + coordinate systems. Euler angles {αi , βi , γi } provide transformations from the nucleosome coordinate system to the exiting site coordinate system on the nucleosome. To ensure that the nucleosome-wound DNA with zero twist does not

+ + impart any twist to the exiting linker, αi +γi is set to zero [27]. Transformations between all other coordinate systems are given by {αi, βi, γi}, where i denotes the index of the linker bead, or the nucleosome, or the entering linker sites. More-detailed descriptions of coordinate systems and Euler angles are available elsewhere [19,27].

The total energy of the dinucleosome ET is given by the sum of five contributions from stretching Estr, bending Ebend, torsion ETw, electrostatic EC, and excluded volume energies EEV, i.e.,

=     ET Estr Ebend ETw EC E EV (21)

The stretching energy of linker segments is given by 87

N −1 = h ∑  − 2 E str l i l 0 (22) 2 i=1

where h, li, and l0 are the stretching constant, linker segment length, and equilibrium segment length, respectively. The summation in Eq. 22 is carried out over the N = 14 nucleosomes and linker beads. The bending energy is given by

N −2 = g ∑  2 E ben i (23) 2 i =1

where g and βi are the bending energy constant and the Euler angle describing the local bending of the linker, respectively. The torsional energy is given by

N −2 = s ∑   2 E twi i i (24) 2l 0 i =1

where s and αi+γi are the torsional energy constant and the Euler angles describing the twist between consecutive local coordinate systems, respectively.

The total linker-linker, linker-nucleosome, and internucleosome electrostatic energy, treated using the Debye-Huckel potential, is given by

− N bead 1 N bead = 1 ∑ ∑ qi q j −  EC exp r ij (25)   = =  4 0 i 1 j i 2 rij

where qi and qj are the charges on beads i and j and rij is the center-to-center distance between the beads. Nbead is the total number of linker and nucleosome beads. The other parameters in Eq. 25 are ε , the dielectric constant of the medium; κ, the inverse 88

Debye screening length [19,27]; and ε0 , the permittivity of free space.

Finally, the total linker-nucleosome, linker-linker, and internucleosome Lennard-

Jones excluded volume interactions are given by:

N N dna oct  12  6 =  ∑∑[  −  ] E EV 4 ev (26) = = i 1 j 1 rij rij

where εev is an energy parameter and σ is the effective diameter of the beads constituting the linker and nucleosomes.

The summations in Eqs. 25 and 26 are over all beads for simplicity of notation.

However, no excluded volume or electrostatic interactions between intranucleosome beads and contiguous linker beads exist.

3.2.3 Monte Carlo twisting protocol

We implement quasistatic twisting of the dinucleosome array using a Monte

Carlo (MC) approach. The initial linker and nucleosome positions are similar to that in

Figure 3.1, for a particular θ and φ. The entry/exit angle θ is defined as the angle between the two linker attachment points on the nucleosome and φ is the angle made by the exiting linker with the plane of the two nucleosomes. The rightmost nucleosome (blue) in Figure 3.1 a is torsionally and translationally constrained; the other (green) nucleosome is free to move and rotate. The penultimate and last linker beads labeled 13 and 14, respectively, in Figure 3.2 are also fixed in space.

Before twist application, the dinucleosome is first allowed to equilibrate

(relax), subject to the constraints mentioned above. All equilibration phases are carried 89 out using the Metropolis MC method at constant temperature. Two MC moves are used [19]: The first is rotation of a randomly chosen bead or nucleosome by a random angle in interval [-δr, δr] (δr = 0.3 rad). While linker bead rotations are only implemented about its ai axis, nucleosome rotations are chosen randomly about its ac, bc, or cc axis. The second is a translational move by a distance chosen randomly in the interval [-δt, δt] (δt = 0.3 rad) along a randomly chosen direction of a random bead or nucleosome.

Both types of moves are accepted or rejected using the standard Metropolis criterion based on changes in the total energy of the system ΔE. That is, if ΔE < 0, the move is accepted; and if ΔE > 0, the move is accepted with a probability exp(-

ΔE/kBT), where kB is the Boltzmann constant and T is the temperature. In all cases, we allow the system to equilibrate for Neq MC moves before the application of twist. We use 2 × 106 MC steps for each equilibration period.

0 The average twist in the exiting and entering linkers after equilibration is Tw ex

0 and Tw en, respectively. After equilibration, the penultimate linker bead (bead 13 in

Figure 3.2, left panel) coordinate system is rotated about its local a axis by an angle  = / 0 ± 4 . The system is then equilibrated again using the equilibration protocol.

Rotations are applied unidirectionally, in either the clockwise or counterclockwise

direction, which we denote by Ω+ and Ω- , respectively. Subsequent twist is applied using the same stepwise increments and equilibration phases. Figure 3.2 shows a plot of the typical applied and induced twist-versus-MC steps. Our twisting method ensures sufficient time for relaxation between each subsequent twisting event, thus 90 emulating the slow quasistatic twisting of chromatin in experiments.

We carry out a series of such twisting simulations at varying relative linker orientations, θ and φ, to probe their effects on the twisting response of the dinucleosome arrays. These angles are varied in increments of δθ =π/12 and δφ =π/6, in the range of θ ∈ (0, 2 π), φ ∈ (-π/3, π/3).

3.3. Results

3.3.1 Twist inversion

We define two parameters, ken and kex, to characterize the fraction of imposed twist stored per linker segment in the exiting and entering linkers,

 − 0  = 2 Tw Tw m k m  (27) N m n 0

where m = ex,en, and where Twex and Twen are the average twist in the exiting and entering linkers after n applications of twist, respectively. The twist in the two linkers is calculated as:

12 = 1 〈∑   〉 Twex  i i (28) 2 i=8

7 = 1 〈∑   〉 Twen  i i (29) 2 i=1

Note that the number of segments that absorb twist in the two linkers are 91

different (Nex = 5 and Nen = 7).

Figure 3.3: Twist inversion.

Plots of kex and ken for various φ as a function of θ for positive applied twist, Ω+, for a total applied twist of π/2. Twist inversion occurs when kex and ken are opposite in sign.

Figure 3.3 shows kex and ken versus θ for several φ for positive applied twist Ω+ during the second phase of twist application n =2. Both kex and ken are φ -invariant to first-order, but depend strongly on θ. Here, kex > 0 for all θ – and φ -values while, interestingly, ken is negative for a subset of θ. Regions where kex and ken are opposite in sign imply inverted twist. ken = 0 for a narrow range of θ corresponding to zero 92 twist propagation.

Twist propagation can be more clearly characterized by defining Λ=Sgn(kex ˟ ken ), where Sgn(x) is the sign of x and Sgn(0) = 0, and Λ > 0, Λ = 0, and Λ < 0 correspond to conserved twist direction; zero twist propagation; and twist inversion, respectively. Figure 3.4 shows twist in a dinucleosome with φ = -π/3 subjected to negative applied twist.

The upper panels of Figure 3.4 (left to right ) show representative plots of twist in the dinucleosome for θ = 3π/2, θ = π, and θ = π/6, respectively. Both cases for

Λ > 0 show that the magnitude of twist in the exiting and entering linkers are increasing as the applied total twist increases. The center panel corresponding to Λ < 0 shows that the magnitude of twist in the exiting and entering linker are increasing, although in opposite directions. This corresponds to twist inversion in the dinucleosome.

We now construct phase diagrams which show relative linker twist-direction as functions of θ and φ. These phase diagrams are shown in the two lower panels of

Figure 3.4 for positive and negative applied twist. We define three twist propagation zones. Zones I and III (Λ > 0) are both twist direction conserving. Zone II (Λ < 0), is twist-inverting. Figure 3.4 also shows schematic linker conformations in each zone.

Remarkably, the phase diagrams are invariant to f and direction of applied twist. Phase boundaries shown by dashed lines illustrate zero twist propagation, Λ = 0.

We provide a simple explanation of twist inversion that accounts for relative linker orientation and nucleosome geometry. Our Ansatz is that nucleosomes rotate as 93 rigid bodies, resulting from the strong DNA/nucleosome binding. Applied rotations on the exiting linker induce nucleosome rotations, which, in turn, induce rotations in the entering linker

Figure 3.4: Twist propagation simulation results. The top panel shows representative twist propagation results from our MC simulations for zones I (θ = 3π/2 ), II (θ = π), and III (θ = π/6) for counterclockwise applied twist Ω_ and φ=-π/3. Zone II is twist-inverting. (Solid blue and dashed green lines) Twist in the exiting and entering linker, respectively. (Black and red lines) Applied and induced total twist, respectively. (Bottom panel) Computed regimes of positive (red triangles) and negative (blue triangles) relative twists Λ for both directions of twist.

We define Гi ≡{ai, bi, ci} and Г'i ≡{a'i, b'i, c'i} to be the coordinate systems before and after rotation of the exiting linker, respectively. As shown in Figure 3.5, Г0 is the coordinate system of the entering linker bead, which we assume to be fixed for our argument; Г1 is the coordinate system of the entering linker attachment point; Г2 94

is the coordinate system of the exiting linker attachment point; and Г3 is the coordinate system of exiting bead being rotated anticlockwise. That is, the axes b'3, and c'3 are rotated while a'3 remains unchanged. A rotation is induced in Г2 about the a2 axis, thus rotating the nucleosome. The induced rotation about a2 is always smaller in magnitude than the applied rotation, resulting from the linker’s finite torsional rigidity.

Figure 3.5, A–C, shows the effect of applied rotation of Г3 for three entry/exit angles: θ = 180°, 360°, and 270°, respectively. For θ=180°, the nucleosome rotation causes rotation of Г1 about the a1 axis. The relative coordinate system rotations and directions required to go from Г0→Г'1 and Г'2→Г'3 are also shown. In fact, the above two rotations are by definition described by the angles (α0+γ0) and (α2+γ2), the twists in the entering and exiting linkers, respectively. Clearly, the two linker twists are opposite in sign, thus resulting in twist inversion across the central nucleosome.

Similar arguments apply for θ = 360° (and 0°) as demonstrated in Figure 3.5 B.

Here, the direction of rotations (α0+γ0) and (α2+γ2) required to transform

Г0→Г'1 and Г'2→Г'3, respectively, have the same sign. Thus, the twists in the two linkers are of the same sign. Finally, Figure 3.5 C shows the θ = 270° (and 90°) case.

By definition, the Euler angles α0 and γ0 can be calculated from the relationship [19].

⋅  ⋅   = b0 b' 0 c0 c' 1 cos 0 0  ⋅ (30) 1 a0 a ' 1

By examining the relative rotations of the coordinate systems, it can be deduced that (α0+γ0)=0, as a0·a'1 = c0·c'1 and b0·b'1 =1. Consequently, θ = 90° (and 95

270°) yields zero twist propagation.

A B

C D

Figure 3.5: Effect of applied twist on the linker and nucleosome coordinate systems. For three different values of θ: (A) twist sign inversion θ = π, (B) twist sign conservation θ = 2π, and (C) zero twist propagation example for θ = π/2. (D) Twist inversion zones obtained from a calculation of the dot product of aex·aen = cos (θ). Values of aex,aen = cos(θ) < 0 imply twist inversion.

To generalize the argument, we consider Гex and Гen to be the exiting and entering linker attachment coordinate systems for a nucleosome lying flat across the −   x y plane. We fix the exiting coordinate system aex = x . For the angle θ (and φ    = 0), aen is then given by cos(θ) x + sin (θ) y in terms of the unit vectors x  and y . Varying θ alters the relative orientations of aex and aen . The sign of the dot product, aex·aen = cos (θ), gives the relative twist direction in the linkers. A plot of aex·aen vs. θ is shown in Figure 3.5 D: cos (θ) =0 for θ = π/2, 3π/2, yields zero twist 96

∈ propagation; aex·aen < 0 for θ (π/2, 3π/2) corresponds to twist inversion; and the twist direction is conserved otherwise.

Our geometric argument illustrates the physical origin of twist inversion in the dinucleosome. It also provides a good first-order approximation of the dependence of the angle θ on twist inversion. Twist propagation in the dinucleosome is actually more complex. For instance, our explanation does not account for: linker twist/bend coupling, internucleosome interactions, steric effects, and effects arising from the size of the nucleosome. We also do not predict the magnitude of the twist propagated, only its sign. These additional effects may account for the observed differences in our predicted twist inversion zones and boundaries and those obtained by simulation.

3.3.2 Nucleosome flipping

Continued twisting of the dinucleosome leads to the second interesting finding of this study: the unconstrained nucleosome may undergo a sudden reorientation that we refer to as ‘‘nucleosome flipping’’. This effect is best illustrated by examining the continued twisting of a representative dinucleosome with θ = 225° and φ =0° for counterclockwise- applied twist. Figure 3.6A shows a series of snapshots of the dinucleosome conformation versus MC steps. Initially, the unconstrained nucleosome’s orientation remains unchanged. At ~1.5 × 107 MC steps, it rapidly inverts, and the linker bends sharply. Similar flipping events were observed in several other simulations. The finite twist storage capability of linkers and fast flipping of the nucleosomes is analogous to observed jump or buckling instabilities in end-clamped finite semirigid rods that are twisted at one end [28,29]. At a critical applied twist, the 97 rods buckle and form loops, thus converting their twist into writhe over very short times, which is qualitatively similar to observed buckling in our dinucleosome.

A

B C

Figure 3.6: Nucleosome flipping dynamics. Nucleosome flipping dynamics for dinucleosomes with θ=225° and φ=0°. (A) Series of snapshots from the MC simulations showing evolution of the dinucleosome conformations and associated nucleosome flipping and linker buckling. (B) Applied and induced twist in linkers on the exiting (red) and entering linkers (black), respectively. (C) Total applied and induced twist (solid black and red lines) and the total twist on exiting and entering linker beads adjacent to the nucleosome (dashed green and solid blue lines).

Three key features are associated with nucleosome flipping events as follow.

First, nucleosome flipping correlates with an exchange of the twist in the two linkers.

Figure 3.6B shows the average twist in the linker beads adjacent to the free nucleosome versus MC steps. The magnitude of twist in the exiting linker bead increases with larger applied twist. At ~1.5 × 107 MC steps, twist propagates from the exiting to the entering linkers. This results in a steep change in the exiting and entering linker twists, respectively. Clearly, this exchange is the largest for cases where the two 98 twists are of opposite sign with respect to each other. Indeed, we observe nucleosome flipping to be most abrupt for intermediate θ -values, corresponding to the observed range of twist inversion, and more gradual for small and large θ -values.

Second, an exchange of energy between the different modes accompanies nucleosome flipping. Figure 3.7 shows the evolution of stretching Estr, twisting ETw, bending Ebend, and Coulombic plus excluded volume EEVC energy contributions versus

7 MC steps for the above system. Before flipping (< 10 MC steps), ETw increases in the same steplike manner as the applied twist. The step height increases quadratically, as a result of ETw quadratic dependence on applied twist. We also observe slight increases in Ebend, Estr, and EEVC, likely due to the coupling between the bending and twisting modes, the stretching of bent linkers, and increased electrostatic repulsion between the nucleosomes, respectively. At the onset of flipping, Ebend rapidly increases while ETw stabilizes. Here, the twisting energy penalty begins to exceed that associated with the sharp bending of the linkers, thus relieving some of the linker torsional stress. At this point, linker buckling also induces the abrupt nucleosome flipping, followed by additional equilibration of the bending and twisting energies. Upon flipping, EEVC also decreases, possibly as a result of increased separation of the linkers and thereby reduced electrostatic repulsion. We further note that flipping occurs at ~ETw = 22 kcal/mol (~37 kBT ). Other simulations with different θ and φ , in which nucleosome flipping occur, yield similar energy barriers, perhaps suggesting a universality in the energy required for nucleosome flipping. Note that the above value is a very rough estimate of the free energy barrier, as it excludes various contributions including conformational entropy. It would be interesting to examine the energetics and kinetics 99 of nucleosome flipping in more detail.

Third, the imposed twist is not equal to the net observed twist, which can be interpreted as a change in the overall writhe of the system. Figure 3.6 C shows the average applied and induced twist in the system along with the total twist in both linkers. The difference between the induced and applied twist can be explained in terms of the linking number, Lk, given by the sum of the twist and writhe, Lk = Tw +

Wr: Although, strictly speaking, this relationship is only applicable to closed curves, we apply it to the dinucleosome under the assumption that the constrained ends are connected by a phantom curve. Similar approaches have been used by others to estimate writhe in open curves [30-32]. The change in linking number ΔLk is then equal to ΔTw + ΔWr - Twap; where Twap is the applied twist. Because ΔLk = 0 for closed curves, ΔWr = Twap - ΔTw and a numerical estimate of twist-to-writhe conversion is the difference in the applied and induced twist.

To better illustrate this point, we present an additional system with θ = 90° and

φ = 60° that undergoes a larger conformation change (Figure 3.8). As with the previous example, applied and induced twist are not equal. The spontaneous occurring

0 0 0 twist during the equilibration of the array is denoted Tw =(Tw ex + Tw en). At the end of the simulation Wr≃−0.64 ; illustrating the significant change in the overall writhe. The change in writhe is correlated with a conformational change of the dinucleosome. Figure 3.8 also shows the initial and final conformation of the dinucleosome. The dashed red line indicates that the dinucleosome is treated as a closed loop. 100

Figure 3.7: Dinucleosome energy contributions.

Dinucleosome energy contributions from twist ETw, bending Ebend, stretching Estr, and excluded volume plus Coulombic interactions EEVC for dinucleosome simulations with θ=225°and φ=0°.

We approximate the directional writhe as the sum of the signed crossings of the projection of the linker path onto a plane. The writhes for the initial and final conformations are Wr = 0 and Wr = -1, yielding ΔWr = -1. Thus, the directional writhe change computed from the above approximate approach exhibits the same sign as that computed from the difference in the measured and applied twists—further confirming twist-to-writhe conversion in nucleosome flipping. 101

Figure 3.8: Total applied and induced twist. Total applied and induced twist (solid black and red lines) and the total twist on exiting and entering linker beads adjacent to the nucleosome (dashed green and solid blue lines) for θ= 90° and φ=60°. Tw0 is the spontaneously occurring twist in the linker after equilibration and ΔWr is the change in writhe of the system at the end of the simulation. Also shown are the starting and final dinucleosome conformations and the projections of their linker paths onto a plane for counting the number of positive and negative crossings for directional writhe calculations. (Dashed red line) Phantom curve drawn to close the linker trajectory.

3.4. Discussion

Our simulations furnish two intriguing observations about twist propagation in dinucleosome arrays.

First, applied twist may lead to inverted twist in contiguous linkers. This 102 phenomenon is a result of strong DNA/octamer binding that inhibits twist propagation along the wound DNA; twist propagation instead occurs via rigid body- like rotations of the entire nucleosome. Hence, twist inversion is strongly dependent on the relative orientation of contiguous linkers at their point of entry and exit to the nucleosome.

Second, continued twisting in dinucleosome arrays may lead to nucleosome flipping and associated linker buckling. The latter occurs to relieve excessive stored twist in one linker; nucleosome flipping events facilitate quick transfer between contiguous linkers and also results in conversion of twist to writhe.

Our twist inversion finding leads to the obvious question: Can twist inversion occur in real chromatin fiber?

Our simulations suggest that a single parameter, the entry/exit angle q , dictates twist inversion, and that angles in the range 90° <≈θ <≈ 220° are required for twist inversion to occur. The entry/exit angles in the nucleosome crystal structure, θ0, are in the range 108° –126° [1], corresponding to 1.65–1.7 turns of wound DNA, well within the predicted range of twist inversion. However, θ is dependent on salt conditions, and even at fixed salt conditions, θ exhibits large variations due to spontaneous unwrapping of DNA [33]. At low salt concentrations, chromatin exhibits an unfolded conformation. The strong repulsion between the entering and exiting linkers causes them to diverge, resulting in a large increase in θ, potentially exceeding the upper angle limit of twist inversion. In contrast, at high salt conditions, chromatin is tightly folded and the wound DNA likely maintains its original entry/exit angle θ0. Hence, we expect unfolded chromatin at low salt concentrations to exhibit unidirectional twist and strongly folded chromatin at physiological salt conditions to exhibit twist 103 inversion (bimodal twist). Because the nucleosomes exhibit more restricted dynamics in chromatin fibers due to packing and histone tail interactions, it is difficult to extrapolate how far the unidirectional and inverted twists propagate along the fiber from our single-nucleosome results.

What are the possible implications of twist inversion? One possibility is that twist inversion could allow creation of alternate regions of overwound and underwound DNA, which could potentially have specific roles in chromatin function.

Some proteins are known to preferentially bind to either over- or underwound DNA

[34-37]. For example, the activity of the 434 repressor is dictated by the degree to which DNA is wound at its operator site [37]. Also, architectural proteins such as SRY prefer binding to overtwisted DNA while zinc finger proteins prefer undertwisted

DNA [36]. Opposite twists on contiguous linkers could therefore potentially enhance interactions between over- and underwound DNA-binding proteins. Twist inversion could also play a role in chromatin dynamics, especially under external twisting, as the direction of nucleosome rotation depends on the twists in the entering and exiting linkers. In addition, coupling between twist-inverting and non-twistinverting zones could modulate local nucleosome rotations, providing a method to modify local chromatin conformation and to locally over- or under-twist chromatin, depending on its topology.

The continued twisting of di-nucleosomes induces sharp bending of the linker.

Because some DNA sequences are more susceptible to bending than others, such torsion-driven buckling may provide a mechanism to modulate site specific bending in chromatin [37]. Localized buckling could synergize the binding of specific regulatory 104 proteins that preferentially bind to curved DNA. In particular, TATA sequences that play a key role in transcription by binding to various transcriptional factors are prone to bending. Hence, torsional forces from various cellular machinery could potentially instigate conformational changes near stress-sensitive sites, such as TATA boxes, to regulate the transcriptional activity of chromatin [15]. If these torques induce nucleosome flipping and buckling of linker, it may provide a method of governing local protein binding that could regulate protein binding and transcription. The conversion of twist to bending via local buckling of DNA may also serve as a transient reservoir or dynamic buffer to absorb twist in chromatin without causing drastic changes in its conformation [15]. The dependence of the abruptness of nucleosome flipping on linker orientation may also play an as-yet-unknown role in chromatin’s ability to transiently absorb torsional stress.

The analysis of energy of the dinucleosome system indicates that significant twist can be applied before inducing nucleosome flipping. If, for example, the applied twist originates from an advancing polymerase that exerts torques of ~1.25 kBT/rad, flipping would occur after 5-6 polymerase rotations, corresponding to the ~37 kBT energy barrier observed in our simulations. In our example, the ~180° nucleosome flipping transition occurs after ~270° of twist has been applied. The high energetic barrier of flipping indicates that dinucleosomes can transiently absorb applied twist and act as reservoirs for applied twist. Chromatin’s strong twist storing capacity contrasts sharply with that of DNA, testifying to chromatin’s structural stability.

Larger chromatin arrays will undoubtedly exhibit a more complex interplay among 105 twist, bending, and internucleosome interactions, and much remains to be understood about their fundamental physics, including how they propagate twist.

In summary, a systematic Monte Carlo study of the torsional response of end- constrained dinucleosome arrays is presented. The results suggest that the propagation of twist within a dinucleosome strongly depends on the relative orientation of contiguous DNA linkers, a subset of which leads to inverted twist. Twist inversion phenomenon illustrates how simple constraints on DNA imposed by the nucleosome lead to very unexpected consequences for twist propagation along DNA in nucleosome arrays. The results also demonstrate how dinucleosomes subjected to continued twisting can undergo dramatic conformational changes, nucleosome flipping. This flipping occurs at energy barriers that indicate chromatin’s significant ability to transiently store twist. Twist inversion and nucleosome flipping are expected to be real phenomena that could play important roles in chromatin function. It may be possible to verify their existence experimentally using, for instance, single molecule techniques

[5].

This study thus demonstrates how simple modeling and simulations provide an effective tool to probe nucleosome dynamics and to uncover previously unknown properties of chromatin.

ACKNOWLEDGMENTS

The text of Chapter 3 is partly based on paper I. V. Dobrovolskaia, M.

Kenward, and G. Arya “Twist Propagation in Dinucleosome Arrays”, Biophysical

Journal, Volume 99, pp. 3355– 3364, November, 2010. 106

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29–40. 14. Mozziconacci, J., C. Lavelle, M. Barbi, A. Lesne A, and J.-M. Victor (2006). A physical model for the condensation and decondensation of eukaryotic chromosomes. FEBS Lett. 580: 368–372. 15. Lavelle, C., J.-M. Victor, and J. Zlatanova (2010). Chromatin fiber dynamics under tension and torsion. Int. J. Mol. Sci. 11: 1557–1579. 16. Schiessel, H. (2003). The physics of chromatin. J. Phys.: Condens. Matter. 15: R699–R774. 17. Zlatanova, J. (2003). Forcing chromatin. J. Biol. Chem. 278(26): 23213– 23216. 18. Arya, G., and T. Schlick (2006). Role of histone tails in chromatin folding revealed by a mesoscopic oligonucleosome model. Proc. Natl. Acad. Sci. USA. 103: 16236–16241. 19. Arya, G., Q. Zhang, and T. Schlick (2006). Flexible histone tails in a new mesoscopic oligonucleosome model. Biophys. J. 91: 133–150. 20. Wedemann, G., and J. Langowski (2002). Computer simulation of the 30- nanometer chromatin fiber. Biophys. J. 82: 2847–2859. 21. Kruithof, M., F. Chien, M. de Jager, and J. van Noort (2008). Subpiconewton dynamic force spectroscopy using magnetic tweezers. Biophys J. 94(6): 2343– 2348. 22. Langowski, J. and D. W Heermann (2007). Computational modeling of the chromatin fiber. Seminars in Cell and Developmental Biology. 18(5): 659–667. 23. Gupta, P., J. Zlatanova, and M. Tomschik (2009). Nucleosome assembly depends on the torsion in the DNA molecule: a magnetic tweezers study. Biophys. J. 97: 3150–3157. 24. Celedon, A., I. M. Nodelman, and S. X. Sun (2009). Magnetic tweezers measurement of single molecule torque. Nano Lett. 9: 1720–1725. 25. Kouzine, F., S. Sanford, and D. Levens (2008). The functional response of upstream DNA to dynamic supercoiling in vivo. Nat. Struct. Mol. Biol. 15: 146–154. 26. Beard, D., and T. Schlick (2001). Modeling salt-mediated electrostatics of macromolecules: The discrete surface charge optimization algorithm and its application to the nucleosome. Biopolymers. 58: 106–115. 27. Beard, D. A. and T. Schlick (2001). Computational modeling predicts the structure and dynamics of chromatin fiber. Structure. 9: 105–114. 28. Neukirch, S., G.H.M. van der Heijden, and J.M.T. Thompson (2002). Writhing instabilities of twisted rods: from infinite to finite length. J. Mech. Phys. Solids. 50: 1175–1191. 108

29. Van der Heijden, G.H.M., S. Neukirch, and J.M.T. Thompson (2003). Instability and self-contact phenomena in the writhing of clamped rods. Int. J. Mech. Sci. 45: 161–196. 30. Rossetto, V. and A.C. Maggs (2003). Writhing geometry of open DNA. J. Chem. Phys. 21: 9864–9874. 31. Van der Heijden, G.H.M., M.A. Peletier, and R. Planque (2007). On end rotation for open rods undergoing large deformations. Q. Appl. Math. 65: 385– 402. 32. Starostin, E. L. (2005). On the writhe of non-closed curves. Physical and Numerical Models in Knot Theory. World Scientific, Singapore. 525–545. 33. Polach, K.J. and J. Widom (1995). Mechanism of protein access to specific DNA sequences in chromatin: a dynamic equilibrium model for gene regulation. J. Mol. Biol. 254: 130–149. 34. Hatfield, G. W., and C. J. Benham (2002). DNA topology-mediated control of global gene expression in Escherichia coli. Annu. Rev. Genet. 36: 175–203. 35. Remus, D., E. L. Beall, and M. R. Botchan (2004). DNA topology, not DNA sequence, is a critical determinant for Drosophila ORC-DNA binding. EMBO J. 23: 897–907. 36. Rohs, R., X. Jin, and R.S. Mann (2010). Origins of specificity in protein- DNA recognition. Annu. Rev. Biochem. 79: 233–269. 37. Koudelka, G. B. (1998). Recognition of DNA structure by 434 repressor. Nucleic Acids Res. 26: 669–675. 38. Kouzine, F., J. Liu, and D. Levens (2004). The dynamic response of upstream DNA to transcription-generated torsional stress. Nat. Struct. Mol. Biol. 11: 1092–1100. 4. Conclusions and future work

4.1. Summary

In this thesis, computational tools have been used to investigate the dynamics of single nucleosomes subjected to stretching and torsional loading. The nucleosome constitutes the basic unit of DNA packaging in eukaryotic organisms and greatly influences biochemical processes like transcription, replication, and repair by regulating the binding of proteins to DNA. The nucleosome achieves this in three ways. First, it allows for changes in the wrapped amount of DNA, sliding of the histone octamer along the DNA, and complete unraveling nucleosomes, which modulate the accessibility of the wound DNA sequences. Second, the nucleosome allows for changes in inter-nucleosome interactions, which alter the compaction of the chromatin fiber, thereby modulating the accessibility of the linker DNA. Third, by maintaining or releasing torsion in the linkers, the nucleosome could regulate the binding of proteins that exhibit differential binding affinity for over- and under-twisted DNA. Hence, the nucleosome is a highly dynamic system that is constantly being remodeled by external forces and torques, and a detailed understanding of the dynamical behavior of nucleosome to such external perturbations is required for fully understanding the regulation of DNA-related processes.

The dynamics of force-induced unwrapping of DNA from histone octamers has been investigated in the second part of the thesis. For this purpose, a new model of the nucleosome is developed. In the model the histone octamer and the DNA is treated as

109 110 separate entities capable of assembling and disassembling. The model is parameterized to reproduce the nonuniform histone/DNA interaction free energy profile and the loading rate-dependent unwrapping forces obtained from single molecule experiments.

Brownian dynamics simulations of this model at constant-loading conditions show three distinct regimes in nucleosome unraveling. At low forces, the flanking portions of wound DNA undergo fast unwrapping/rewrapping (breathing) dynamics. At medium forces, the outer half-turn of DNA unwraps gradually and the octamer flips

90o to orient its superhelical axis parallel to the pulling axis. At large forces, the inner turn unravels abruptly with a notable “rip” in the force-extension plot. The octamer flips another 90o after unraveling and remains attached to the DNA upon extended pulling. The simulations show that the nonuniform histone/DNA interactions serve to enhance the breathing dynamics, delay the unraveling of the inner turn, and inhibit the complete detachment of the octamer. The results suggest that modulation of the histone/DNA interaction profile of nucleosomes could constitute one possible mechanism by which nature regulates the accessibility of nucleosome-bound DNA sequences.

The torsional response of a di-nucleosome array has been studied in the third part of the thesis. A mesoscopic model that accounts for nucleosome geometry along with the bending and twisting mechanics of the linkers was employed to model the di- nucleosome array. Monte Carlo simulations were used to obtain the distribution and propagation of twist from one DNA linker to another while an external twist was applied in a stepwise manner to mimic quasistatic twisting of chromatin fibers. The 111 simulations revealed that the magnitude and sign of the twist measured on linkers on opposite sides of the nucleosome depends strongly on the relative orientation of two linkers. The relative sign of the twist in the two linkers has been characterized as a function of various geometrical parameters and a geometrical explanation for the observed behavior has been derived. Rapid “flipping” of nucleosomes in response to strong continued twisting has been observed, which leads to drastic changes in the entry/exit conformation of the DNA linker and to rapid exchanges between the twist and write modes has been observed. These findings shed light on the underlying mechanisms by which torsional stresses could impact chromatin organization and function.

In summary, the work demonstrates that mesoscale modeling and simulations provides a useful tool for probing nucleosomal dynamics to reveal previously unknown properties of nucleosomes and chromatin. Continued modeling efforts, at varying levels of complexity, will undoubtedly uncover more fundamental details of chromatin packaging and functioning.

4.2. Future directions

The development of the coarse-grained model of the nucleosome in Chapter 2 has opened up opportunities to investigate the behavior of multi-nucleosome array under stretching loads. Experimental studies on long nucleosomal arrays have shown that nucleosome unraveling requires larger rupture forces in arrays compared to those 112 in a mono-nucleosome (at least twice bigger) and the unraveling happens in a cascade- like manner, where the unraveling of one nucleosome triggers the unraveling of other nucleosomes. This observation leads to the hypothesis that the unraveling dynamics are influenced by nucleosome-nucleosome interactions as well as by the chromatin architecture. Both these effects are not accounted in the current nucleosome model; however, the model can be extended further to incorporate these effects and applied to the investigation of the stretching dynamics of long nucleosome arrays or chromatin fiber. It should also be mentioned that chromatin stretching in vivo occurs under conditions of changing linking number due to the DNA being processed by the RNA polymerase, which adds the complexity of stretching chromatin fibers.

New insights on the impact of a range of histone variants on the nucleosome unraveling dynamics can also be obtained by using the above single nucleosome model, where the histone/DNA interactions along the DNA path can be regulated. It is well established that different histone variants lead to different chromatin activities and structures. The future investigations can reveal as to what extent these effect are physical, i.e., due to regulation of electrostatic histone/histone and DNA/histone interactions, and to what extent they are biological, i.e. modulate the interactions with the chromatin-remodeling groups of proteins.

Finally, further improvements in the di-nucleosomal array model to account for more accurately for inter-nucleosome interactions and extension to longer arrays will allow one to study twist propagation in multi-nucleosomal arrays. The single-molecule twisting experiments on chromatin fiber conducted in [1] demonstrated significantly different behavior of chromatin versus naked DNA under torsion. It was shown that 113 chromatin fibers can accommodate significant amounts of supercoiling/torsional stress without large changes in chromatin end-to-end distance and that the maximum extension of chromatin fibers occurs at negative twists rather than zero twists as in

DNA. A quantitative model combining the torsional response of both the DNA and nucleosomal state was suggested, in which there existed a dynamic equilibrium between three conformations of the nucleosome in which the entry/exit linker DNAs either cross negatively, or positively, or do not cross at all. Repeating our twisting simulations for the longer arrays modeled with more details should shed light on the mechanisms responsible for these observed differences in the torsional response of chromatin versus DNA.

4.3. References

1. Bancaud, A., N. Conde de Silva, M. Barbi, G. Wagner, J.F. Allemand, J. Mozziconacci, C. Lavelle, V. Croquette, J.-M. Victor, A. Prunell, and J.L.Viovy (2006). Structural plasticity of single chromatin fibers revealed by torsional manipulation. Nat. Struct. Mol. Biol. 13: 444–450. Appendix 1

Table A.1: Physical parameters associated with the coarse-grained nucleosome and

Brownian dynamics simulations.

Parameter Description Value

l0 Equilibrium DNA segment length 3.0 nm 2 h DNA stretching constant 300kBT/l0

g DNA bending constant LpkBT/l0 s DNA torsional rigidity 300 pN nm2

Lp DNA persistence length 50 nm

σev Excluded volume size parameter 2.4 nm

εev Excluded volume energy parameter 0.025 kBT ε Dielectric constant of solvent 80

qdna Charge on DNA beads -24.1e

cs Salt concentration 150 mM κ Inverse Debye screening length 1.275 nm-1 T Temperature 300 K

Roct Hydrodynamic radius of octamer 4.0 nm

Rdna Hydrodynamic radius of linker bead 1.5 nm Δt BD simulation timestep 2 ps

Ngr Number of charged, groove beads 17

Nflk Number of flanking beads 34

Ncen Number of inner core beads 14

Noct Total number of octamer beads 65

Ndna Total number of DNA beads 39 λ Groove bead charge scaling factor 7.09e

wdna Width of wound DNA supercoil 4.5 nm

rdna Radius of wound DNA supercoil 4.18 nm

114 Appendix 2

Table A.2: Physical parameters used in the Monte Carlo simulations.

Parameter Description Value

l0 Equilibrium DNA segment length 3.0 nm 2 hstr DNA stretching constant 100kBT/l0

gbend DNA bending constant LpkBT/l0 2 sTw DNA torsional rigidity 300 pN nm

Lp DNA persistence length 50 nm κ Excluded volume distance 1.5 nm ε Dielectric constant of solvent 80 -1 κDH Inverse Debye screening length 0.3319 nm T Temperature 293 K

115