Schlick T. Molecular Modeling and Simulation: An Interdisciplinary Guide

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190 6. Topics in Nucleic Acids Structure: DNA Interactions and Folding

Figure 6.9. Solved crystal structures for the tetranucleosome and several nucleosomes:

tetranucleosome complex [1094]; nucleosome core particle containing a poly(dA•dT)

tract [88] (red); nucleosome core particle with H3 K79 dimethylated residues [786]. In

single nucleosome particles, the histone proteins are colored by type (i.e., H3, purple; H4,

silver; H2A, orange; H2B, blue). Arrows point to dimethylated H3 K79 residues (red).

ﬁbers with long linker DNAs and with linker histone and divalent ions produced

strong evidence for solenoid model [1055]. Studies also suggest that, depending

on the linker DNA length and presence of linker histone, different ﬁber dimen-

sions are produced; in particular, short linker DNAs cannot produce compact

ﬁbers [1055].

Modeling by Wong et al. [1388] also showed the dependence of ﬁber width on

the linker DNA length and the orientation of linker histones. Modeling of simpli-

ﬁed coarse-grained nucleosome models by the Rippe group reinforced the large

effect of the linker length and nucleosome twist angles on the extent of ﬁber com-

paction [1219]. Our mesoscale modeling (Fig. 6.11) combined with experimental

studies of EM and nucleosome interaction measurements [483](seeFig.6.12 and

Box 6.4) suggest a compact zigzag organization for the chromatin ﬁber at typical

linker DNA lengths with linker histones and a more heteromorphic architecture

6.5. Cellular Organization of DNA 191

0.2

0 2 4 6 8 10 12 0 2 4 6 8 10 12

024681012

Two-start zigzag One-start solenoid Interdigitated solenoid

0.2

0.4

0.6

0.8

0.4

0.6

0.8

0.2

0.4

0.6

0.8

I(k)

kk k

Figure 6.10. Various hypothetical models for the structure of the chromatin ﬁber. Three

hypothesized chromatin arrays are shown from the top and side with black nucleosome

cores and alternating colors for the wound DNA in consecutive nucleosomes. In the classic

two-start zigzag model [1094], nucleosomes i interact with i ± 2. In the classic one-start

solenoid model [851], nucleosomes i interact with i ± 1 and i ± 6. In the interdigitated

solenoid model [1055], nucleosomes i interact on the ﬂat sides with i ± 5, i ± 6, and on

the narrow edges at i ± 11.

of zigzag forms with straight linker DNA interspersed with bent DNA linkers at

divalent ion environments. Single-molecule force microscope studies [682] sub-

jecting 25-nucleosome arrays with two linker DNA lengths (167 and 197 bp for

the wrapped plus linker DNA) to forces up to 4 pN suggested a fundamental

solenoid organization, stiffer ﬁbers with short linkers, and only a stabilizing but

not structure-determining effect of the linker histone. While studies are ongoing

and no consensus has been reached regarding the structure of the 30-nm ﬁber,

both the zigzag and solenoid models appear to be viable and relevant at various

external and internal conditions (e.g., presence of linker histones, length of linker

DNA, monovalent and divalent salt concentrations). Indeed, chromatin structure

is likely heteromorphic [483,1388].

Still, the condensation at this polynucleosomal level does not achieve the 5

orders of spatial compaction realized by the chromatin ﬁber near the end of the

cell cycle (metaphase chromosomes).Various looping, scaffolding, wrapping, and

speciﬁc contacts with other proteins and possibly RNA have been suggested for

this higher folding to occur. Beyond the 30-nm ﬁber, a higher degree of coiling is

thought to lead to ﬁbers of dimension about 130 nm in diameter, and these forms

in turn are condensed to chromatids of dimension 200–400 nm ([844] and refs.

quoted therein).

192 6. Topics in Nucleic Acids Structure: DNA Interactions and Folding

Figure 6.11. The mesoscale oligonucleosome model in which the nucleosome core with

wrapped DNA (excluding histone tails) is treated as an electrostatically charged object

with Debye-H¨uckel charges optimized to approximate the electric ﬁeld using the Dis-

crete Surface Charge Optimization (DiSCO) algorithm [108,109,1240,1441]; linker DNA

are treated as beads using the wormlike chain model for supercoiled DNA; and the histone

tails and linker histones are coarse grained to approximate atomistic models [64,65, 483].

Undoubtedly, in the next decade, we will begin to understand better how this

nucleoprotein chromosomalmaterial is stabilized and packed in the cells, and how

packaging works with biological processes, such as replication and transcription,

that require full access to the DNA. Much work is in progress on understanding

the biochemical mechanisms by which the electrostatic charge density of polynu-

cleosomes is modulated (e.g., by acetylation and phosphorylation mechanisms) to

regulate transcription, and the sequence-dependent “code” in the DNA that binds

to nucleosomes.

6.5. Cellular Organization of DNA 193

Box 6.4: Simulations of Polynucleosome Folding

As mentioned in the text, little is known about the detailed organization of the chromatin

ﬁber — the DNA/protein complex in eukaryotes made of DNA wound around histone pro-

teins (see Figure 6.7) — both in terms of the linker DNA geometry and the nucleosome

packing arrangement in the ﬁber. However, the crystallographic triumphs that produced

an atomic-level view of the basic building block of the chromatin — the nucleosome core

[55, 287, 790, 1094] — provide ﬁrm foundations for complementary modeling work.

Over the past several years, our group has developed a mesoscale model of chromatin

(see recent overview [1114]) in which the nucleosome, excluding the histone tails, with

wrapped DNA is treated as an electrostatically charged object with Debye-H¨uckel charges

to approximate the atomistic electric ﬁeld computed by the Poisson-Boltzmann equation

using the DiSCO (Discrete Surface Charge Optimization) algorithm [108,109,1240,1441].

In this model, each nucleosome unit is represented by several hundred charges, as shown

in the Figure 6.11, optimized so that the effective Debye-H¨uckel electrostatic ﬁeld matches

that predicted by the nonlinear Poisson-Boltzmann equation [108]. This electrostatic rep-

resentation is important, since properties of chromatin are sensitively dependent on the

internal charges as well as on the ionic concentration of the medium.

Connecting these charged bodies of nucleosomes is the linker DNA, which we treat

as beads in the wormlike chain model used for supercoiled DNA. The histone tails

[62, 65] and linker histones [64] are coarse grained by beads from united-atom protein

model (Figure 6.11). Following detailed model validation against available experimental

data (e.g., translational-diffusion constants, radii of gyration, sedimentation coefﬁcients,

etc.), as recently summarized [64], such a model simulated using Monte Carlo [63]and

Brownian dynamics permits detailed analysis of many structural, energetic, and dynamic

questions, such as the folding and unfolding of dinucleosomes and trinucleosomes as a

function of salt [109], the dynamics and energetics of nucleosome arrays as a function of

salt [1240], the roles of histone tails in stabilizing ﬁber architecture [64, 65], the inﬂuence

of linker histones and divalent ions in compacting the ﬁber structure [483], and the inﬂu-

ence of linker DNA length on ﬁber organization [989].

Recent studies show that without linker histones, the ﬁber structure has a loose zigzag

conformation [483]. Linker histones further compact the ﬁber by forming stems in the

linker DNA where they closely interact with the linker histone, thereby promoting an or-

dered zigzag ﬁber organization. When divalent ions are introduced, bending in some linker

DNAs results to minimize steric clashes along the ﬁber axis; this produces a mostly zigzag

ﬁber accented with some bent linker DNA, resembling the solenoid form. These studies

thus lend support for the both solenoid and zigzag models; moreover, they underscore the

heterogeneous and polymorphic nature of the chromatin ﬁber. These dynamic ﬁber struc-

tures and their dependence on the ionic strength and linker histones can be seen from

Figure 6.12, which shows snapshots at low and high salt conditions (including divalent

ions), with and without linker histones. A condensation trend is apparent as the ionic con-

centration increases and as linker histones provide additional screening of the negatively

charged linker DNA.

194 6. Topics in Nucleic Acids Structure: DNA Interactions and Folding

6.6. Mathematical Characterization of DNA Supercoiling 195

6.6 Mathematical Characterization of DNA

Supercoiling

Now that we have covered many local features of nucleic acids as well as basic

global characteristics, I will introduce quantitative tools to describe how DNA is

condensed in the cell.

6.6.1 DNA Topology and Geometry

Topological methods are important for analyzing some reactions of supercoiled

DNA. For example, successful collaborations between biologists and mathemati-

cians have led to techniques that establish mechanisms for various reactions

that produce knots and catenanes (linked DNA rings) from supercoiled DNA

substrates by recombination [1239].

Basic Topological Identity

The topological method, as well as the many computational methods used to study

DNA supercoiling, rely on the fundamental identity attributed to Grigore Calu-

gareanu, Jim White, and F. Brock Fuller [279, 431, 1369]. This equation relates

the topological invariant Lk,orlinking number, to the geometric quantities twist,

Tw,andwrithe, Wr as:

Lk = Tw+ Wr. (6.3)

Linking Number

Essentially, Lk characterizes the order of linkage of two closed-oriented curves in

space (Figure 6.13). It is unchanged by continuous deformations but altered when



Figure 6.12. Conﬁgurational snapshots of 24-core oligonucleosomes simulated at various

conditions [989]. The top shows three different orientations of the zigzag starting conﬁg-

uration used for all series. The four simulated conditions from left to right involve: low

(0.01 M) monovalent salt without linker histone, physiological monovalent salt (0.15 M)

without linker histone, physiological monovalent salt (0.15 M) with linker histone, and

physiological monovalent salt (0.15 M) with moderate levels of divalent ions (as modeled

to a ﬁrst order approximation [483]) and linker histones. The conﬁgurations are simu-

lated for 10 million steps using Monte Carlo simulations [63]. The nucleosomes are colored

white and navy in alternating order, and linker DNA segments are rendered red. The linker

histones (each represented by 3 beads) are colored in turquoise. The ﬁnal conﬁgurations at

bottom also show the other tail beads colored as: H2A—yellow, H2B—red, H3—blue, and

H4—green (see mesoscale model in Fig. 6.11). Overall, we see the increased compaction

trend from left to right. At low salt, the negatively charged linker DNAs (red segments)

repel one another and expand the array, producing the beads-on-a-string form shown also

in Figure 6.7. At higher salt concentrations, the chromatin ﬁber condenses. Note the com-

paction as enhanced by linker histone as well as divalent ions. With divalent ions, the

mostly irregular zigzag conformation is accented by some linker DNA bending [483].

196 6. Topics in Nucleic Acids Structure: DNA Interactions and Folding

(+)

CCW (+)

CW (−)

global

helix

axis

shown

Relaxed Circle

Lk = +4 = Lk

Tw = +4 = Tw

Wr = 0

Underwound Circle

Left-handed supercoil

Lk = +3 = Lk

Tw = +3 = Tw

Wr = 0

Lk = +3

Tw = +4

Wr =−1

Figure 6.13. Descriptors of supercoiling topology and geometry. For the relaxed and un-

derwound circles of DNA, each line represents one polynucleotide strand of the double

helix. For the supercoil, the curve represents the double helical axis of the DNA.

strand breaks occur. By convention, for right-handed DNA, the linking number of

the relaxed state, Lk

, is the number of primary turns: Lk

= n/n

where n is

the number of bps and n

is the number of bps/turn (e.g., 10.5).

Lk can be rigorously computed from any planar projection of these curves onto

the plane by summing all signed crossings (ignoring self interaction) and dividing

the result by two, as shown in Figure 6.13. By convention, a crossover index is

considered positive if the tangent vector of the top vector is rotated counterclock-

wise to coincide with the tangent of the bottom curve; the sign is negative if this

rotation is clockwise. For precise mathematical deﬁnitions, refer to [431, 1370].

Twist

The variable Tw reﬂects the familiar concept of the helical twist angle intro-

duced earlier, namely the winding angle per residue. For the case of a helix

spinning about a global helical axis, Tw measures the number of times the he-

6.7. Computational Treatments of DNA Supercoiling 197

lix revolves about that axis. As conventional for Lk, Tw is positive if the helix is

right-handed and negative if it is left-handed (see Figure 6.13).

When the global helix axis is a closed circle and the DNA winds n times around

this planar axis, Tw = Lk ≡ Lk

= n/n

.Fromeq.(6.3), this implies that

Wr =0. This case is shown for the relaxed circle in Figure 6.13.

A nonzero writhing number is introduced by the freeing of the two ends of

the closed ring and overwinding or underwinding the duplex before resealing the

ends. Topoisomerase enzymes perform this magical act in the cell. The unwrithed

(planar) state is unfavored because there is substantial local disruption to the natu-

ral twist of the DNA (Figure 6.13). The induced torsional stress can be alleviated

through nonplanar writhing, or supercoiling, namely the twisting of the global

helix axis itself. This is illustrated in Figure 6.13 where a ﬁgure-8 interwound

forms the path of the double helical axis.

Writhe

The writhing number describes the geometry of this global helix axis in space.

Essentially, it is the average number of crossings in space of this curve. Like

the linking number, Wr can be obtained for each planar projection and averaged

over all these projections. The same sign convention used for Lk is used for Wr

(Figure 6.13).

Rigorously, the writhe can be computed by the Gauss double integral [1370]

(mathematically speaking, a map that associates each pair of curve points with

the unit sphere). For example, the simplest interwound — a ﬁgure-8 form — has

Wr = ± 1; the interwound with two self-crossings, separating the chain into

three regions, has Wr = ±2. The ﬁgure-8 interwound in Figure 6.13 (bottom)

has Wr = −1, and the interwound supercoil in Figure 6.6 (left) has Wr ≈−13.

See [1127, for example] for computational determination of Wr.

Typically, DNA is subjected to a linking number deﬁcit,orunderwinding, with

respect to its relaxed state: ΔLk = Lk − Lk

< 0. This state is often described

by the normalized quantity σ =ΔLk/Lk

.Thevalue−0.06 is typical of DNA

in vivo. In the absence of loop-constraining agents like histone proteins, DNA

tends to absorb most of this stress into writhing, by forming left-handed supercoils

(negative Wr).

6.7 Computational Treatments of DNA Supercoiling

While the mathematical concepts of Lk, Tw,andWr have been invaluable for

interpreting supercoiling, the energetic and geometric details of the process re-

main unknown and require further computational treatments. The processes of

interest involve fundamental reactions such as replication, recombination, and

transcription, that often require, or are facilitated by, a supercoiled DNA substrate

[631,1307]. Experimental techniques such as gel electrophoresis and electron mi-

croscopy (atomic, scanning, and cryo) have been crucial for studying properties

198 6. Topics in Nucleic Acids Structure: DNA Interactions and Folding

associated with large supercoiled DNA molecules. The former allows separation

of DNA molecules into topological isomers, and the latter offers views of over-

all shapes of supercoiled DNA. However, the experimental resolution is limited

due to the large size (thousands of bps) of the DNA subjects and their extreme

ﬂoppiness in solution. Hence, theoretical tools based on numerical analysis and

computational modeling have been useful.

Computational treatments of the energetics and dynamics of supercoiled DNA

have largely relied on the theories of polymer statistical mechanics and elas-

ticity in the framework of elastic rod mechanics and dynamics. For realistic

results, non-elastic contributions — electrostatics and hydrodynamics — must

be included. These can be easily incorporated into computational models and in-

vestigated using various conﬁgurational sampling techniques (Metropolis/Monte

Carlo) [1308] and dynamic techniques (by molecular, Langevin, and Brownian

dynamics) [1105].

6.7.1 DNA as a Flexible Polymer

Very long DNA can be described by concepts from polymer statistical mechan-

ics [410]. The DNA polymer is characterized by two key quantities: a contour

length L, and a bending rigidity A. This view of a random coil slithering

through space is reasonable only for naturally occurring DNA of mixed sequences

that are not intrinsically bent at lengths much greater than DNA’s persistence

length, p

The persistence length is essentially the length scale on which the polymer

directionality is maintained. It can be computed as the average projection angle

between the end-to-end distance vector on the ﬁrst bond vector of a polymer, in

the inﬁnite-length limit [410]. The main result from polymer statistical mechanics

is that the length of this vector is proportional to the square root of the contour

length; that is, the mean square displacement, R

, satisﬁes:

R

 =2p

L (6.4)

with 2p

as the proportionality constant.

Hence, for lengths  p

, the DNA can be considered straight, but for lengths

 p

, a better description is a bent random coil undergoing Brownian motion.

This is evident from the views of DNA on different length scales, as shown in

Figure 6.7. For DNA, the persistence length in vivo is around 500

A or about

150 bps at physiological monovalent salt concentrations [491]. A salt dependence

of the persistence length is recognized for DNA [104], but the magnitude of this

effect is not currently understood.

The persistence length is also related to the bending force constant of a long

ﬂexible polymer of length L and bending rigidity A via

A = p

T (6.5)

6.7. Computational Treatments of DNA Supercoiling 199

is Boltzmann’s constant and T is the temperature). In this description, the

ﬂoppy polymer writhes through space as a worm-like chain. The bending rigid-

ity — which tries to keep the DNA straight — is balanced by thermal forces —

whichtendtobenditinalldirections.

A decomposition of the persistence length into static and dynamic components

has been developed [1097,1272].

6.7.2 Elasticity Theory Framework

Since the pioneering applications of Fuller [431, 432], the elastic energy ap-

proximation has proven valuable for studying global (i.e., long range and time

ﬂexibility) features of superhelical DNA. In this approximation, the DNA poly-

mer is idealized as a long, thin and naturally straight (i.e., no intrinsic curvature)

elastic isotropic rod with a circular cross section. Homogeneous bending (i.e.,

equal ﬂexibility in all directions) is often assumed as a ﬁrst approximation,

though current computational models follow twist locally, for example by using

body-centered coordinate frames along the helical axis with associated Euler

transformations.

The elastic deformation energy can then be written as a sum of bending

and twisting potentials, with bending and torsional-rigidity constants (A and C,

respectively) deduced from experimental measurements of DNA bending and

twisting [491]. The bending term is proportional to the square of the curvature

κ, and the twisting energy is proportional to the twist deformation:

E = E

+ E



(s) ds +



(ω − ω

)

ds. (6.6)

In these equations, s denotes arclength, and the integrals are computed over the

entire closed DNA curve of length L

. The intrinsic twist rate of the DNA is

(e.g., 2π/10.5 radians between successive bps). The parameters A and C,

denoting the bending rigidity and the torsional rigidity constants, respectively,

can be estimated from various experimental measurements of DNA, such as the

persistence length. These force constants are key characteristics of an elastic ma-

terial.

See Box 6.5 for the relationship between the force constants A and C

and the bending and twisting persistence lengths, and Box 6.6 for the relation-

ship between A and measured variations in roll and tilt angles in solved DNA

structures.

For example, a rubber band, for which bending is facile, has a small ratio r = A/C, while

a stiff material with strong bending resistance has large r. This bending to torsional-rigidity ratio is

a key descriptor of an elastic material. It is related to both Poisson’s ratio (σ

) — a characteristic

of a homogeneous isotropic material — and the geometry of the rod cross section. Speciﬁcally, for

a homogeneous elastic rod of circular cross section, r =1+σ

. Note: the ratio A/C is usually

designated by the symbol ρ, but here we reserve ρ for the roll angle.