Computational Modelling of the Chromatin Fiber

Jorg Langowski1 and Dieter W. Heermann2

1 Division Biophysics of Macromolecules,German Cancer Research Center,

Im Neuenheimer Feld 580,D-69120 Heidelberg, Germany,

and2 Theoretical Biophysics Group,

University of Heidelberg, Philosophenweg 19,D-69120 Heidelberg, Germany,

The packing of the genomic DNA in the living cell is essential for its biological function. Whileindividual aspects of the genome architecture, such as DNA and nucleosome structure or the ar-rangement of chromosome territories are well studied, much information is missing for a unifieddescription of cellular DNA at all its structural levels. Computer modeling can contribute to sucha description. We present here two typical approaches to models of the chromatin fiber, includingdifferent amounts of detail in the description of the local nucleosome structure.

PACS numbers:

I. INTRODUCTION

DNA organization in the cell can only be adequatelydescribed on many length and time scales. A model en-compassing all the different scales is completely out ofrange both conceptually as well as technically. It is thusimperative to single out a reasonable scale and to takeinto account the missing degrees of freedom and smallerscales in an approximate manner. Such a multi-scalemodeling problem must thus be approached by someappropriate approximation, in which one has to definesubunits of the macromolecule that behave for examplelike rigid objects on the size and time scale consideredand thus eliminating short scales. These objects inter-act through effective potentials that may in principle bederived from the inter-atomic force fields considering in-teractions among the subunits as well as mediated inter-actions through proteins for example which can introducea long range interaction. Here we briefly describe modelsfor chromatin on the 30nm fiber scale.

Models on this length scale can take into account dif-ferent levels of details. As prototypical models we takethe worm-like chain model for the DNA and the two-angle model for the nucleosome chain geometry. Bothexhibit typical approaches to the prediction of quantitiesthat can be measured in experiments.

II. WORMLIKE CHAIN MODELS

The motif of the ’linear elastic filament’ in genomeorganization is repeated on many length scales: DNA,as well as the chromatin fiber and to some extent itshigher order structures may be approximated by a flexi-ble wormlike chain (WLC). Thus, we may develop mod-els of DNA and the chromatin fiber based on a coarse-grained description using a linear segmented chain. Seg-ments are assumed to behave like rigid cylinders on the

time and length scale considered; they are connected byelastic joints, with bending, torsional and stretching po-tentials approximated by Hookean springs with springconstants that are known independently. [39]

A. Segmented wormlike chain

Fig. 1 schematizes a segmented chain geometry. A vec-tor si defines the direction and length of each segment i,f i is a unit vector normal to the segment and gi is anauxiliary vector that is used to take into account per-manent bending of the DNA. The details of this chaingeometry are given in [4].

FIG. 1: Section of a segmented polymer chain as used in theDNA and chromatin models described here.

B. Intersegment interactions

Adjacent segments (i and i+1 in Fig. 1) interact witheach other through bending, twisting and stretching po-tentials. Independent of the form of the local interseg-ment potential, the WLC approximation always holds

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for sufficiently long chains, as has recently been shownby Wiggins and Nelson [5]. Thus, if one does not con-sider tight local bends, the potentials between adjacentsegments can be approximated by Hookean springs. Fur-thermore, potentials must be defined for long-range in-teractions between non-neighboring segments: in the caseof ’naked’ DNA, this interaction is the electrostatic re-pulsion between the negatively charged sugar-phosphatebackbones and can be described by a screened Coulombpotential [4]. If the DNA is associated with proteins as inthe case of chromatin, the geometry of these complexesand their specific interaction must be taken into account.In the chromatin chain model developed by Wedemannet al. [6], nucleosomes are approximated by rigid ellip-soids and their interaction by a Gay-Berne potential (ananisotropic Lennard-Jones potential[7]).

1. Bending rigidity

In the WLC model the length of the segments shouldbe chosen well below the persistence length Lp, which is ameasure of the bending flexibility of the chain molecule.It is defined as the correlation length of the direction ofthe chain measured along its contour:

〈~u (s) ~u (s + s′)〉 = e−s/Lp (1)

Here ~u (s) is a unit vector in the direction of the chain(e i in Fig. 2) and s resp. s’ is the position along thechain contour, the angular brackets indicating the aver-age over all positions and chain conformations. Moleculesshorter than Lp behave approximately like a rigid rod,while longer chains show significant internal flexibility.

The bending elasticity A - the energy required to benda polymer segment of unit length through an angle of1 radian - is related to the persistence length by Lp =A/kBT, kB being Boltzmann’s constant and T the ab-solute temperature. Thus, the energy required to bendtwo segments of the chain of length l by an angle θ withrespect to one another is:

Eb =kBT

2Lp

lθ2 (2)

For DNA, Lp has been determined in a number of ex-periments (for a compilation, see [2]). The existing dataagree on a consensus value of Lp = 45-50 nm (132-147 bp)at intermediate ionic strengths (10-100 mM NaCl and/or0.1-10 µM Mg2+). For high values of θ, the potential maydeviate from the simple harmonic form (see footnote [39]and ref. [3]).

2. Torsional rigidity

The torsional rigidity C, defined as the energy requiredto twist a polymer segment of unit length through an an-gle of 1 radian, may be related in an analogous way to a

torsional persistence length LT . The torsional rigidity Chas been measured by various techniques, including flu-orescence polarization anisotropy decay[8–10] and DNAcyclization[11–13], and the published values converge ona value of LT = 65 nm (191 bp).

3. Stretching rigidity

The stretching elasticity of DNA has been measuredby single molecule experiments[14, 15] and also calcu-lated by molecular dynamics simulations[16, 17]. Thestretching modulus σ of DNA is about 1500 pN, whereσ = F · L0/∆L (∆L being the extension of a chain oflength L0 by the force F ). The stretching energy of asegment of length l that is stretched by ∆l is:

Estr =12

σ

l∆l2 (3)

DNA stretching does not play a significant role in chro-matin structural transitions, since much smaller forcesare already causing large distortions of the 30 nm fiber(see below).

4. Intrachain interactions

The average DNA helix diameter includes the diame-ter of the atomic-scale B-DNA structure and – approx-imately – the thickness of the hydration shell and ionlayer closest to the double helix. For the calculation ofthe electrostatic potential and the hydrodynamic prop-erties of DNA, a helix diameter of 2.4 nm describes thechain best [4, 18–20]. The choice of this parameter issupported by the results of chain knotting [21] or cate-nation [22], as well as light scattering [23] and neutronscattering [19] experiments.

As pointed out in [4, 24] DNA intrachain electrostaticrepulsion can be adequately described by a Debye-Huckelelectrostatic potential between two uniformly chargednon-adjacent segments (i, j ) in a 1-1 salt solution:

E(e)ij =

ν2

ε

∫∫dλidλj

e−κrij

rij(4)

Here,κ is the Debye screening parameter, proportionalto the square root of the ionic strength, ν the linearcharge density of DNA and ε the dielectric constant ofwater. More details as to the normalization of the linearcharge density etc. have been given in our earlier paper[4].

III. MONTE-CARLO MODEL OF THECHROMATIN CHAIN

As an example of the application of a polymer chainmodel to genome structure, we describe the simulation

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of nanomechanical properties of the chromatin fiber bya Monte-Carlo model. The flexibility of the chromatinfiber has been measured in a set of experiments, eitherby relating the spatial distance of markers on the DNA totheir genomic distance [25–27] or by direct measurementsof cyclization probabilities [28, 29]. The persistence ob-tained cover a large range from unrealistically low valuesof about 30 nm [28, 29] to values of up to 200 nm [26]. Inour recent work [30], we show that depending on the localstructure of the DNA on the nucleosome, the nucleosomerepeat and the presence or absence of linker histone H1,this wide range of persistence lengths may be reproduced.

FIG. 2: Example of a Monte-Carlo equilibrated structure ofa chromatin fiber consisting of 100 nucleosomes (red), linkersegments (blue) of repeat length l = 205 nm.

In the model the chromatin fiber is approximated as aflexible polymer chain consisting of rigid ellipsoidal disks,11 nm in diameter and 5.5 nm in height. These disks areconnected by linker DNA, represented by two cylindricalsegments. Incoming and outgoing linker DNA are set 3.1nm apart of each other. This geometry used is essentiallythe “two angle” model developed earlier by Woodcock etal. [31], in which an attractive internucleosome interac-tion is introduced via an anisotropic Lennard-Jones typepotential, the Gay-Berne potential [7].

A typical conformation of a 100 nucleosome chain after3.106 MC steps is shown in Fig. 2. Simulations were donewith either a condensed fiber as a starting conformationor an initial conformation where all segments are orderedin a straight line.

The simulation results show that the bending and thestretching stiffness of the chromatin fiber strongly de-pend on the local geometry of the nucleosome. Both thepersistence length Lp and the stretching modulus ε de-crease if either the linker lengths or the opening angle areincreased, or the twisting angle is reduced. This behav-ior is independent of the presence of the linker histoneH1. The latter decreases the opening angle α betweenthe entry and exit of the linker DNA and as a resultleads to a more condensed fiber structure for high saltconcentrations [32]. This is in agreement with our sim-ulations, since the presence of the linker histone-inducedstem motif yields higher persistence lengths thus stifferfibers (Fig. 3).

0

50

100

150

200

250

300

350

400

191 195 200 205 210 215 220

repeat length [bp]

per

sist

ence

len

gth

[n

m]

0

100

200

300

400

500

600

195 196 197 198 199 200 201 202 203 204 205

repeat length [bp]

per

sist

ence

len

gth

[n

m]

0

50

100

150

200

250

300

350

400

191 195 200 205 210 215 220

repeat length [bp]

per

sist

ence

len

gth

[n

m]

0

100

200

300

400

500

600

195 196 197 198 199 200 201 202 203 204 205

repeat length [bp]

per

sist

ence

len

gth

[n

m]

FIG. 3: Persistence length of modelled 30 nm chromatin fiberswith different nucleosomal repeats in the presence and absenceof linker histone H1. The twisting angle between adjacentnucleosomes is adjusted to the canonical value of 360◦ per10.5 bp. The persistence lengths of fibers with linker histone(closed symbols, dashed lines) are higher than for fibers with-out linker histone (open symbols, solid lines). This effect isstronger for short repeats and weakens with increasing repeatlength. The peaks show that the twisting angle strongly influ-ences the stiffness of the fiber, leading to a non-monotonousvariation of Lp with nucleosome repeat.

The other major result of the simulation comes fromcomparing the persistence length of the modelled fibersto that of a hypothetical rod from a isotropic elastic ma-terial having the same stretching rigidity as the chro-matin fiber. Such a rod would have a bending rigidity4-10 times higher than that actually measured, or simu-lated here. Thus, the chromatin fiber is less resistant tobending than to stretching. This property of the chro-matin fiber is important for its ability to condense anddecondense, for example to prevent or allow transcrip-tional access. Chromatin fibers thus seem to be packedmore easily via dense loops than by a linear compression.The formation of such dense loops of hairpin structuresof interdigitated chromatin arrays has been recently sug-gested [33], and some hairpin conformations could alsobe seen in related simulations [34].

IV. TWO-ANGLE MODELS

Alternative to the ’linear elastic filament’ paradigmone may reduce the complexity without sacrificing the

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FIG. 4: The figure shows the basic parameters of the E2Amodel: The entry-exit-angle αi , the rotational angle βi , thelinker length bi and the vertical distance di between in andoutgoing linker. We chose a large entry-exit-angle here tohave better visualization.

relevant physics by considering fixed linker lengths andfixed angles as depicted in Fig. 4. The two-angle modelwas introduced by Woodcock et al. [31] to describe the ge-ometry of the 30nm chromatin fiber. It has been shownthat the excluded volume of the histone complex playsa very important role for the stiffness of the chromatinfiber [34] and for the topological constraints during con-densation/decondensation processes [35]. In [36] a roughapproximation of the forbidden surface in the chromatinphase diagram was given. A more detailed analysis [37]we answered questions concerning the fine structure ofthe excluded volume borderline which separates the al-lowed and forbidden states in the phase diagram with thebasic assumption of spherical nucleosomes and no verti-cal shift between in and outgoing strand.

A further refinement was introduced [38], the ex-tended two-angle model (E2A-model) which takes into

account the cylindrical shape of the nucleosome as wellas explicit H1 as shown Fig 4. Here H1 plays a doublerole: It is responsible for the pitch as well as for thestability of the complex. This can be clearly seen if onedepletes H1, i.e., if one assumes H1 to present only witha probability p. In Fig 5 is shown the radius of gyrationas a function of the chromatin length. Already a smallnumber of missing H1 leads to a drastic compactionof the chromatin fiber, indicating a much more flexiblestructure in agreement with the Monte-Carlo simulationsshown above.

0 5 10 15 20 25 30 35 40 45 50 55 60 650

0.02

0.04

0.06

0.08

0.11

0.13

Squared radius of gyration for chromatin fibers with H1 defects

genomic distance [kbp]

<R

G2>

[µ

m2]

p = 0.00

p = 0.01

p = 0.05

p = 0.10

p = 0.30

FIG. 5: The squared radius of gyration for chromatin fiberswith different defect probabilities p. With increasing numberof H1 defects the fiber becomes much more compact whichcould be an important mechanism to compact the chromatinfiber.

V. CONCLUSION

Simple geometric models of the chromatin fiber, inconnection with techniques that take into account thefree energy of the chain, can give surprisingly deep in-sight into the nanomechanical properties of chromatinand therefore its packing properties, accessibility to tran-scription machinery etc. In particular, the role of changesin the local geometry to the global structure and flexi-bility may be studied, helping to consolidate many ex-perimental observations on chromatin, such as the roleof the linker histone, histone modifications, nucleosomepositioning sequences or the occurrence of certain nucle-osomal repeats with high probability.

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[39] For DNA, the approximation of Hookean bending elas-ticity has recently been challenged by the finding thatshort DNA fragments have a cyclization probability much

higher than expected for a homogeneously elastic worm-like chain [2], and atomic force microscopy observationsthat sharp bends occur more frequently than expectedfor a Gaussian distribution of bending angles [3].