REVIEW Open Access
Hierarchies in eukaryotic genome organization:
Insights from polymer theory and simulations
Balaji VS Iyer, Martin Kenward and Gaurav Arya*
Abstract
Eukaryotic genomes possess an elaborate and dynamic higher-order structure within the limiting confines of the
cell nucleus. Knowledge of the physical principles and the molecular machinery that govern the 3D organization of
this structure and its regulation are key to understanding the relationship between genome structure and function.
Elegant microscopy and chromosome conformation capture techniques supported by analysis based on polymer
models are important steps in this direction. Here, we review results from these efforts and provide some
additional insights that elucidate the relationship between structure and function at different hierarchical levels of
genome organization.
Introduction
The term “genome” refers to the complete linear DNA
sequence containing all of the hereditary material possessed
by an organism. One of the goals of the human
genome project was to determine the sequence of the 3
billion base pairs that constitute human DNA. Despite
the wealth of information this tour-de-force of scientific
enterprise has generated, it is increasingly becoming clear
that the cellular function of the human genome is not
merely determined by the linear ordering of its DNA
base pairs. In fact, many of the functional aspects of the
genome are governed by its three dimensional (3D)
structure, which involves meters long DNA packaged
into the limiting space of a micrometer sized cell nucleus.
The DNA, thus packaged, occupies a significant portion
of the nucleus volume while cellular factors that
read, copy, modify, and maintain the genome, occupy
the remaining. Ultimately, sophisticated patterns in cellular
function arise due to a coupling between the accessibility
of genetic information in the packaged DNA, and
the organization and activity of cellular factors within
the cell nucleus. For instance, nuclear processes like
transcription, translation, repair and recombination do
not occur ubiquitously in the nucleus, but are spatially
compartmentalized in transcription, replication and
recombination factories [1-3]. Clearly, how the 3D organization
of the genome modulates these nuclear
processes and how the nuclear processes in turn modify
genome structure are important questions in modern
cell biology. A critical step in addressing these questions
requires a fundamental understanding of the genome
3D structure and the physical principles governing its
organization, as articulated concisely yet powerfully in
the cartoon of Figure 1.
Increasingly, ideas from polymer theory and simulations
coupled with state-of-the-art microscopy and chromosome
conformation capture techniques are being
used to determine the 3D structure of the genome and
the physical principles governing its folding. In this article,
we present an overview of the key aspects and
insights gained from these studies at the different hierarchical
levels of organization shown in Figure 2. We
begin with a discussion of mesoscale models and simulation
methods used to decipher the secondary structure
of the genome, the folded chromatin fiber, on the scale
of 1-10 kbp. Next, we discuss coarse-grained models
and simulations of the genome tertiary structure. At the
tertiary structure level genome compaction varies all the
way from ~50 to 100,000 fold, therefore, we have chosen
to divide it into two sub-structures: tertiary-a structure
at the gene locus level (10-2000 kbp) and tertiary-b
structure at the chromosomal level (1-200 Mbp).
Primary and Secondary Structure: Chromatin
Organization
The classic image of double stranded DNA (dsDNA) is
that of a naked double helix. However, in eukaryotes,
* Correspondence: garya@ucsd.edu
Department of NanoEngineering, University of California San Diego, 9500
Gilman Drive, La Jolla, CA 92093-0448, USA
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© 2011 Iyer et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
dsDNA seldom occurs in its naked form. Instead, it is
packaged through a ubiquitous hierarchical process
involving specialized proteins called histones. This
packaging serves two purposes. First, it compacts the
DNA allowing it to fit into the confines of the cell
nucleus, and second, it controls the accessibility of DNA
to cellular machinery for transcription, regulation, repair
and recombination [4].
The primary structure of the eukaryotic genome consists
of DNA wrapping ~1.7 times around histone octamers
comprising of two copies of the four histone
proteins H2A, H2B, H3, and H4 [4-7]. The combined histone
octamer-DNA complex is called the nucleosome.
This nucleosomal organization of DNA is considered to
be the primary determinant to accessibility of genetic
information. Although the atomic structure of the
nucleosome has been resolved through X-ray crystallography
[6,7], the dynamics of nucleosomes remains far
from fully understood. However, theoretical modeling
and simulations are beginning to provide new insights
into nucleosomal dynamics by answering questions
related to: (1) how histone post-translational modifications
and histone variants affect nucleosome structure
and intra/inter-nucleosome interactions and (2) how
nucleosomes undergo spontaneous conformation transitions
between fully- and partially-wrapped states.
Recently, we have [8] employed all-atom molecular
dynamics (MD) simulations and molecular docking to
examine the structure of the H4 histone tail and its
interactions with the acidic patch of nucleosomes. The
H4 tail was found to exhibit a propensity for an a-helical
conformation within a stretch of six residues encompassing
the lysine 16 (K16) residue. Interestingly, this ahelical
region interacted very strongly with the acidic
patch; K16, in particular, mediated strong electrostatic
interactions with the negatively-charged residues of the
acidic patch. Acetylation of K16 diminished these binding
interactions, suggesting a plausible mechanism by
which post-translational acetylation of K16 could trigger
chromatin unfolding [9]. To examine nucleosomal
dynamics beyond the time scales accessible to all-atom
simulations, Sharma et al. [10] and Voltz et al. [11]
have developed coarse-grained models of the nucleosome.
The former has helped in the identification of
important histone residues, termed “cold sites”, that
maintain the stability of the histone octamer and the
latter has been used to compute long wavelength fluctuations
of the nucleosome. Some effort has also been
devoted to elucidating the energy barriers and the
kinetic rate constants associated with the accessibility of
DNA in the nucleosome [12] and explaining specific
features of force-induced nucleosome unraveling
observed in single-molecule experiments [12-14].
Beyond the single nucleosome level, contiguous
nucleosomes separated by short sections of naked DNA
called linker DNAs yield the classic beads-on-a-string
structure depicted in Figure 2. The secondary structure
of chromatin involves folding of this beads-on-a-string
motif into a ~30-nm thick fiber called chromatin.
While, in vivo imaging of cell nuclei has yielded little
information on the secondary structure of chromatin,
and has even brought into question the very existence
of a 30-nm fiber [15,16], electron microscopy of isolated
nucleosomal arrays have been successful in imaging the
transition from 10-nm beads-on-a-string structures to
30-nm condensed structures with increasing salt concentrations
[15,17]. Regardless of the gaps in our
Figure 1 Illustration of the Genome Folding Problem. Illustration
of the important question on genome organization. (Adapted with
permission from cartoonist John Chase-http://www.chasetoons.com)
Figure 2 Hierarchies of Genome Organization. The hierarchical
process by which eukaryotic double-stranded DNA (two meters
long, in the case of humans) is packaged within the confines of a
micrometers-sized cell. As shown schematically in the figure, this
process encompasses three main organization levels classified as
primary, secondary and tertiary [115,116].
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knowledge of the secondary structure of chromatin
in vivo, this structure is expected to dictate the accessibility
of DNA sequences for interactions with nuclear
machinery and is also likely to play a role in the recruitment
of histone modifying and remodeling factors to
specific regions of the genome. Local packing of nucleosomes
within the chromatin fiber could also potentially
affect its interactions with distant portions of the fiber
or other chromatin fibers through processes like interdigitation
[18].
Despite decades of research, the internal structure of
the 30-nm chromatin fiber remains controversial. This
controversy arises because of two main reasons: (1)
in vivo chromatin is too “messy” to be visualized with
even the most advanced microscopy techniques [15] and
(2) in vitro reconstituted nucleosome arrays are too
large and flexible to be crystallized, and too compact at
physiological conditions for their linkers to be fully
resolved through microscopy. These limitations have led
researchers to adopt indirect ways of deducing the internal
structure of chromatin, using chemical cross-linking
and single-molecule pulling techniques, with varying
degrees of success. Based on the data, two types of models
for the chromatin structure have been proposed that
differ mainly in the location and configuration of the
linkers. In the one-start solenoid model, linkers exhibit
a strongly bent configuration and reside at the fiber
interior [17,19,20]. In the two-start helix, linkers exhibit
a straight or gently bent configuration, and they could
reside at the periphery of the fiber in one version of the
model [21] or inside the fiber (close to its axis) in a zigzag
manner in another version [22-24].
Computational modeling has played an integral role in
providing new insights into chromatin architecture. The
main challenge in modeling chromatin is the vast
degrees of freedom possessed by even small segments of
the fiber (e.g., an array of 50 nucleosomes contains >1
million atoms). Hence, all-atom approaches demand
prohibitive amounts of computational resources to converge
to equilibrium structures. Further, the larger and
the more flexible the system, the less adequate a
description by one single equilibrium conformation
because of the large variation in possible structures
around thermal equilibrium. In other words, the systems
become “fuzzy” and can only be described in terms of
statistical averages. The advantage of this situation is
that atomic-detail resolution may easily be abandoned
for vast gains in computational speed in a coarsegrained
model. To this end, a major focus of computational
modeling has been on developing lower-resolution
models or “force fields” of nucleosome arrays that still
account for the energetic interactions and constraints
between the different components of the arrays. A large
effort in computational modeling also lies in developing
efficient procedures for “sampling” low-energy conformations
of the nucleosome array subject to appropriate
force fields.
A range of computational models that include widely
differing amounts of detail have been developed. The
simplest of these models are the so-called “two-angle”
models [25,26], where the nucleosome arrays are modeled
using two physical parameters: the entry-exit angle
a of the linker DNA and the angle b describing the relative
rotation between consecutive nucleosomes, the latter
being determined by linker length L (Figure 3a).
Woodcock et al. [25] illustrated how a and b (or L) play
a key role in dictating the 3D structure of the resulting
nucleosome array. At fixed a, variations in L lead to
stark changes in chromatin structure and variations in
the nucleosome packing ratios. The authors also showed
that small variations in L introduce sharp bends in the
fiber axis similar to those observed in cryo-electron
microscopy of moderately-folded arrays, possibly due to
overwrapping/underwrapping of nucleosomal DNA or
displacement of nucleosomes along the DNA. Twoangle
models with consideration for the excluded
volume of nucleosomes have been extremely useful in
providing sterically permissible conformations of nucleosomal
arrays for specified constraints on nucleosome
packing ratio, fiber diameter and linker length [27-31].
The E2A model is an extension of the two angle
model [25] that also accounts for the cylindrical shape
of the nucleosomes [32]. Simulations based on the E2A
model are ideally suited for studying long nucleosomal
arrays as they overcome the difficulty of lack of knowledge
of actual interaction potentials between nucleosomes
by using model parameters obtained from
experimental data. In particular, the E2A model has
been used for studying the effects of variability in linker
lengths and variability in linker histone occupancy [32].
In combination with high-resolution light microscopy
experiments these simulations promise a way to capture
local chromatin structure in the range of 10 - 40 nm
resolution [33].
The next class of models, in addition to including the
features of the two-angle model, include the mechanics
and electrostatics of the linker DNAs along with a simple
treatment of inter-nucleosome interactions [34-36].
The linker DNA is treated as a discretized wormlike
chain, possessing energy terms for stretching and bending
of the bead-chain with an additional energy term for
the relative twist angle between adjacent beads to
account for the twisting rigidity of DNA [37] (Figure
3b). The linker beads are also assigned suitable effective
charges that interact with each other via the DebyeHückel
potential [38]. The inter-nucleosome interactions
have thus far been treated as hard spheres, interacting
with a hard-core attraction [34] or as cylinders
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interacting with Gay-Berne [35,36] and Zewdie potentials
[36]. The nucleosome array conformations and
packing ratios obtained using Monte Carlo simulations
[35,36] based on these models generally resemble those
obtained by electron microscopy for moderately folded
chromatin. These models [34] have also provided valuable
insights into the stretching behavior of chromatin
and yielded first estimates of the strength of internucleosome
interactions by matching simulated forceextension
curves to those obtained experimentally [39].
Importantly, the models have revealed the sensitivity of
the chromatin structure to the internucleosome interaction
strength and linker lengths [36]. Furthermore, it
was observed that nucleosome arrays exhibit a zigzag
(a) (b)
(c)
Figure 3 Chromatin Modeling. Coarse grained models of 10 nm chromatin fiber with different level of details. (a) The simple two angle model
has two parameters, the angles a and b [25]. (b) The detailed two angle model includes energy terms for stretching and bending of the beadchain
with an additional energy term for the relative twist angle between adjacent beads to account for the twisting rigidity of DNA [37]. (c)
Representation of the detailed mesoscale model of Arya et al [44,45].
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structure without the linker histone and a more solenoidal
structure for linker histone-bound arrays [40].
Very detailed models of local interactions within oligonucleosomes
have been developed by Schlick and coworkers
[41-49] (Figure 3c). The earliest model [41,42]
treated the nucleosome as a rigid cylinder (along with a
small protrusion to represent the H3 histone tail) with
hundreds of “pseudo” charges scattered uniformly on
the surface. The magnitude of the charges were optimized
to reproduce as closely as possible the electric
field in the vicinity of the nucleosome. This approach
allowed one to account for the salt-dependence in the
inter-nucleosome interactions. The model was later
refined by employing an irregular-shaped representation
of the nucleosome [43] that was based on a more recent
nucleosome crystal structure with all histone tails fully
resolved [50]. The model reproduced the experimentally
observed [15,17] compaction of the arrays with increasing
salt concentration and indicated that the arrays
maintain a zigzag morphology under monovalent salt
conditions. Further, the simulations demonstrated that
reduced electrostatic repulsion between the linkers is
the main mechanism responsible for the folding of
arrays at high salt. These models are also being used to
study the dynamics of chromatin arrays, especially
under different kinds of forces including torsional stresses
[51].
Recently, this model was further improved by accounting
for histone tail flexibility [44], linker histone binding
[47], and effects of divalent ions [47]. The tails were
treated as coarse-grained bead-chains, where each bead
represented five amino acid residues. The stretching,
bending, and the electrostatic terms in the bead-chain
were parametrized using an iterative procedure. The linker
histone was coarse-grained as three charged beads
rigidly bound at the nucleosome dyad with the magnitude
of the charges optimized to reproduce the electric
field of the atomic linker histone. Divalent ions were
treated phenomenologically in terms of their effect on
flexibility and electrostatic screening of the linker
DNAs. A configurational-bias Monte Carlo approach
was used to sample the tail configurations and translation,
rotation and pivot moves were used to sample the
global array configurations [46]. This model helped elucidate
the role of each histone tail, the linker histone
and physiological salt condition in chromatin folding
[45,49]. Specifically, the H4 tails were found to mediate
the strongest internucleosome interactions, the H3 tails
mediated strong internucleosome interactions and
screened electrostatic repulsion between the entering/
exiting linkers, and the H2A and H2B tails mediated
inter-fiber interactions [45]. The linker histones constricted
the linker entry/exit angle to bring alternate
nucleosomes together. Divalent ions were also found to
facilitate tight packing of nucleosomes by allowing a
fraction of the linkers to bend and by strongly screening
the linker repulsion at the fiber axis. Moreover, the
model, in conjunction with sophisticated cross-linking
experiments, confirmed the existence of a heteromorphic
fiber containing both zigzag and solenoid conformations
in the presence of additional divalent cations
and linker histones [49]. This model has also recently
been used to reproduce the linker length dependence in
the observed chromatin structures [48].
In summary, mesoscale models at varying levels of
sophistication such as those discussed above are proving
to be valuable tools for examining chromatin structure.
The choice of which model to use is dictated by the
amount of detail required and the amount of computational
resources available. For instance, the detailed
models accounting for histone tail flexibility and nucleosome
geometry may provide the most accurate representation
of short nucleosome arrays. For long arrays
containing hundreds of nucleosomes, these models
rapidly become computationally intractable and the
intermediate-resolution models like the E2A model [32]
become more suitable.
Tertiary-a Structure: Gene Locus Organization
In the previous section, we discussed the secondary
structure of chromatin from a static perspective. In reality,
the chromatin fiber within the cell nucleus is present
in a dynamic state [52]-it is flexible over lengths
much larger than the fiber diameter [53] and it is constantly
being subjected to various kinds of remodeling
activities, including histone modifications [54-56], sliding
and depletion of nucleosomes [57-64], and incorporation
of histone variants [65]. There is strong
evidence from light microscopy studies indicating that
at the gene locus level the chromatin fiber is organized
into loops [66]. Studies on several multigene clusters
conceive such loops as instrumental in bringing together
distant enhancer and promoter regions crucial for gene
activation, regulation and recombination [67-69].
Although the detailed mechanism of looping is not fully
understood, there is no question that the “intrinsic”
bending rigidity of the chromatin fiber dictates to a considerable
degree the loop size-dependent statistical probability
of two distant regions of the fiber coming into
close proximity to form a loop.
Emerging evidence suggests that the relationship
between flexibility and looping probability may be utilized
by cells for gene regulation. Specifically, the idea
that modulation of flexibility through remodeling processes
like acetylation of histones alters the probability
of interactions between a remote enhancer and cognate
gene by means of looping has been successfully applied
to explain gene regulation of the Hoxd gene cluster and
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the b-globin locus [70]. Strong support for this idea also
comes from recent experiments in the Murre lab that
show large-scale conformational changes in the IgHlocus
during B-Cell development accompanying genome-wide
and locus-specific histone modifications and
nucleosome depletion events [66,71,72]. Thus, it seems
that chromatin remodeling events, apart from modulating
the local structure of chromatin and DNA accessibility,
could lead to changes in the higher-order folding
of chromatin through its effects on macroscopic properties
of the fiber such as its flexibility.
So far, little effort has been devoted to investigating
the intrinsic flexibility of chromatin and the associated
looping probability as a function of loop size; especially
how the flexibility correlates with external conditions
like monovalent/divalent salt concentration and system
parameters like nucleosome repeat length, DNA wrapping
angle, histone variants, histone modifications and
presence/absence of linker histones. Chromatin flexibility
as characterized by its persistence length, Lp, seems
to exhibit large variations, depending on the experimental
method used for its determination and the conditions
and type of chromatin investigated. For example, singlemolecule
pulling of nucleosome arrays at low salt [39]
and in vivo looping driven cross-linking/recombination
assays [53,73] measure Lp as low as 30-50 nm while analysis
of fluorescent markers in the genome of erythrocytes
measure Lp of 100-200 nm [74,75]. However, it is
important to review these results in light of a recent
fundamental study [76] indicating that standard definitions
of persistence length as used in these studies may
not describe the local intrinsic flexibility of chromatin.
Aumann et al. [77] recently examined Lp of nucleosome
arrays using Monte Carlo simulations of a mesoscale
model of chromatin. The persistence length was
obtained from the decay in the correlation of the tangent
vector representing the local fiber axis. It was
found that Lp decreased strongly with increasing nucleosome
repeat length and increasing entry/exit angle, and
that binding of the linker histone led to an increase in
Lp, consistent with experimental observations of linker
histone deficient and inclusive chromatin. A comparison
between the magnitude of the bending and elastic rigidity
suggested that chromatin is much easier to bend
than stretch, leading to the interesting hypothesis that it
may be easier for the cells to pack chromatin via tight
loops rather than by linear compression of the fiber.
As discussed earlier, chromatin exists in a highly
dynamic state within the cell nucleus. In fact, histone
octamers are constantly being dissolved and rebound
with the average genome-wide nucleosome occupancies
being less than 75% [32]. Heermann and coworkers
[78,79] have recently examined the effects of such depletion
events on the persistence length and conformation
of nucleosome arrays (Figure 4). They employed an
extended two-angle (E2A) model [25,32], which allows
examination of very long nucleosome arrays containing
>1000 nucleosomes. An adaptation of the E2A model
with experimental distribution of nucleosome repeat
lengths yields a quantitative estimate for persistence
length modification. Recent Monte Carlo (MC) simulations
based on this adaptation indicate a decrease of Lp
from an initial value of 280 nm to 140 nm with ≈ 20%
increase in the nucleosome skip probability [78]. This
leads to sharp bends in the fiber allowing for formation
of loops in the kilo base pair range [79], an important
feature of genome organization visualized in experiments
[69].
We have recently begun to examine the mechanisms
behind the conformational collapse in the IgH - locus
observed by Murre and coworkers during B-cell development
[66]. We hypothesize that chromatin remodeling
events (nucleosome depletion and histone
modifications [71,72]) introduce flexible “hinges” within
the chromatin fiber causing it to collapse (Figure 4). By
treating the chromatin fiber as a worm-like chain
(WLC) with fixed contour length Lc and variable persistence
length Lp, we showed that the compaction, as
characterized by the ratio of the final to initial mean
square end-to-end distances, is given by the ratio of the
final to initial persistence length in the ideal chain limit
(see Appendix for complete derivation). Using the initial
and final Lp observed by Heermann and coworkers [78],
this ratio becomes 1/2, indicating that the gene locus
compacts by ≈30%. Interestingly, 3D distance measurements
in the IgH-locus during progressive stages of Bcell
development show a similar ~30% decrease in locus
dimensions in response to remodeling events [66]. To
examine the functional implications of this collapse, we
Figure 4 Nucleosome Depletion. Schematic of nucleosomal
depletion associated persistence length modification and
consequent local conformation change in the cell nucleus. Li
p and
L
f
p refer to the initial and final persistence length, respectively.
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used a simple Flory type argument to show that the collapse
leads to an 8-fold increase in the number of binary
interactions nb within the gene locus. Assuming that the
frequency of promoter-enhancer (P-E) interactions is
proportional to nb, we conclude that the conformational
collapse facilitates transcriptional regulation and/or
recombination by allowing for higher probability of P-E
interactions. Thus, the chromatin remodeling induced
changes in the persistence length offers one possible
mechanism for governing the conformational state of
the gene locus and consequent modification of the functional
state of the cell (Figure 4).
Tertiary-b Structure: Chromosomal Organization
Interestingly, the condensed higher-order structures of
chromatin, namely chromosomes, were observed as distinct
entities during mitosis as early as the 19th
century,
much before the primary and secondary structures of
chromatin were known to exist. However, disappearance
of this condensed structure during interphase and the
underlying chromatin organization remained a mystery
till the late 20th
century [80]. In the beginning two different
models were proposed for the interphase chromatin
organization: (1) random organization akin to a bowl
of spaghetti without any apparent structure and (2)
organization in territories that later condense during
mitosis to form distinct chromosomes [81]. The earliest
clues to deciding between the two models came from
experiments shining a laser light onto a specific volume
of the cell nucleus and observing the effects of the consequent
damage on replication [81]. It was found that
only a few chromosomes were affected by the laser
light, indicating existence of distinct chromosome territories
within the cell nucleus. This conjectural evidence
has now been confirmed beyond doubt by advances in
imaging techniques like fluorescent in situ hybridization
(FISH) [82-84] and chromosome painting [82,83].
While there is concrete evidence for the existence of
territories, the internal architecture of chromatin within
chromosome territories, the interaction between territories,
and the organizational principles governing their
formation remain poorly understood [85]. Results from
MC simulations of confined polymer chains suggest that
territory formation could arise from simple, non-specific
entropic forces and from segregation of long chains
attempting to conserve their topological state while
undergoing confined Brownian motion [86,87]. Increasingly,
insights from such coarse-grained polymer models
coupled with experiments are being used to examine the
internal architecture of chromosome territories.
Two predominant themes recur during discussions of
higher-order structures of interphase chromosomes: (1)
formation of loop structures and (2) confined fractal
organization. Interestingly, each of these themes derive
their support from different kind of experiments. The
loop structure theme is predominantly supported by
light microscopy experiments [75,88-90] and the fractal
theme by chromosome conformation capture [91], small
angle neutron scattering [92] and tracer diffusion [93]
experiments. Although the two themes are not mutually
exclusive, it is often possible to classify the polymer
models based on them. Here, we follow this classification
in discussing the polymer models and their central
features.
The earliest polymer model [94] considered the organization
as a classical fractal where the recurring motif
is that of the confined Gaussian chain. The confinement
idea was introduced to explain the observation that the
geometric separation between two points in the structure
follows a Gaussian polymer model between the 0.1-
1.5 Mbp genomic separation scale [88] while beyond
this scale the geometric separation tends to become
independent of genomic separation. However, an alternative
proposition explaining this leveling off of geometric
separation through incorporation of loops was
found to be consistent with the observation of loop
structures in different light microscopy experiments
[95]. The loop proposition consequently led to the formulation
of the random walk giant loop (RW-GL)
model [75] followed by increasingly refined models like
the multiloop subcompartment (MLS) [96] and the random
loop (RL) [97] (Figure 5).
The basic feature of the RW-GL model is the existence
of 1-3 Mbp size loops along a randomly oriented
backbone [75]. This analytical model explained the
observed leveling off of 2D mean geometric distance in
the 10-200 Mbp genomic distance range fairly well.
However, it had limitations in capturing the compartmentalized
chromosome territory structure and spatial
chromatin distribution observed in experiments [96].
These limitations were attributed to the use of phantom
chains in the model [96]. The MLS model based simulations,
with excluded-volume interactions and subcompartments
consisting of 120 kbp loops that can be
opened up to accommodate giant loops, overcame the
limitations of the RW-GL model and correctly predicted
the 2D mean geometric distances [96]. The RW-GL and
the MLS models demonstrated beyond doubt the usefulness
of simulations based on polymer models for study
of higher-order structures of the genome. However,
these models were unable to explain the drastic leveling
off of 3D geometric distances above the 10 Mbp range
that were observed by recent 3D-FISH experiments [90].
The random loop model that proposed existence of
loops at all scales >150 kbp could successfully explain
this leveling off beyond the 10 Mbp scale (Figure 6). A
key feature of this model is that it accounts for observed
experimental features of existence of loops at several
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scales along with a means to account for cell-to-cell variation
of measurements through averaging over different
configurations of loops [97]. Further, the presence of
random loops is expected to lead to formation of segregated
chromosome territories as they repel each other
more strongly than the linear structures [98].
The proposed loop models show a power-law behavior
for geometric distances versus contour length at scales
<10 Mbp, with scaling exponents ranging from ν = 1/3-1/
2. This suggests two different internal structures at work:
(1) equilibrium/globular state at short scales and (2) random
looped structure at large scales. The transition
between these structures is not clearly understood. In
contrast, the confined fractals do not encounter these difficulties
as they propose a single size exponent at all
scales. Recent advances in chromosome conformation
capture [91] have enabled genome-wide measurements
of probability of formation of loops as a function of the
size of the loops in the human genome (Figure 6). Comparison
of the probability measurements with confined
freely jointed chain (FJC) simulations indicate that the
Figure 5 Loop Models. Illustration of the different loop models; (a)
Random Loop model indicating loops at all scales > 150 kbp [90],
(b) multi-loop subcompartment model with 120 kbp rosette
structures [96] and (c) random-walk-giant-loop model with giant
loops organized along a random backbone [75].
Figure 6 Random Loop and Fractal Globule. (a) Predictions of the random loop model for mean-square displacement (adapted from MateosLangerak
et al [90]). (b) Predictions of the fractal globule model for probability of contact (adapted from the article of Lieberman-Aiden et al
with permission from AAAS [91]).
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human genome may be organized as a fractal globule
with size scaling exponent ν = 1/3 [91]. However, we
note that the agreement of the probability measurements
is observed only in the 1-10 Mbp range (Figure 6).
The advantage of this simplified picture of confined
fractal is that the universal exponent ν observed in these
experiments can be readily used to glean information on
the higher-order structure by using simple ideas from
the FJC model. For instance, we can define the “folding”
index F of chromatin as the ratio of fully extended contour
length Lc to the folded size Cs. The persistence
length Lp can now be computed as a function of F at
different levels of human genome organization and is
given by: F = (Cs,0 /2Lp)2
, where Cs,0 ≈ 25 μm is the
size of a typical human cell nucleus (see Appendix for
complete derivation) (Figure 7). Although this relationship
is valid for the human genome, it has to be appropriately
modified for the yeast genome, which is
expected to have an organization different from that of
higher eukaryotes. In particular, experiments on yeast S.
cerevisiae indicate a linear increase in inter-chromosomal
contact with the genomic size s of the chromosome
[99]. Such an increase is expected from an equilibrium
globule organization where the average size R of a chromosome
scales as s1/2
and hence the surface area of
contact R2
scales as s.
The simple arguments we used for the persistence
length calculations inherently assume that the scaling
exponent ν = 1/3 is universal within the human genome.
However, simulations investigating properties of compact
polymers have pointed out that the chromatin
fiber, although in a state that resembles the fractal globular
state of the compact polymers with ν = 1/3, shows
deviations from these type of polymers [100]. Specifically,
the moment ratios calculated from intrachain distance
distributions for chromatin are observed to be
different from that observed for globular state [100].
Instead, the intrachain distributions are observed to be
reasonably approximated by the random loop model
[100]. However, the random loop model also shows that
the distances become independent of genomic separations,
i.e., ν ® 0. The anomaly in these results can possibly
be explained by the argument that in the presence
of loops it is inappropriate to use the intrachain distances
to determine the scaling exponent ν. Instead, the
radius of gyration, a measure of size for closed structures
like rings, may be a more appropriate quantity for
determining the exponent.
We end this section with a perspective on combined
use of the information at the gene locus level and the
chromosomal level to infer qualitative aspects governing
the functional state of the cell. In the previous section,
we have shown that the decrease in persistence length
Lp causes a collapse and consequent increase in the concentration
(density) of segments in a gene locus, via Eqs.
(2) and (3). In contrast, in this section, we have shown
using Figure 7 that a decrease in Lp is accompanied by a
decrease in the linear mass density c (since c and F are
inversely related). The paradoxical nature of this result
indicates that chromatin remodeling events, which lower
the effective persistence length of chromatin, could have
opposing effects on gene-related activities like transcription.
On one hand, locus-wide remodeling could open
up regions for transcription and lead to increased frequency
of interactions between promoters and enhancers;
while on the other hand, genome-wide remodeling
events could cause genome folding leading to formation
of inaccessible, dense regions and potential suppression
of transcription. Thus, nucleosome depletion, in addition
to its established role in gene activation, may also
lead to repression. Such opposing effects are in fact
observed when the nucleosomal content is reduced
through H4 depletion in the yeast genome, resulting in
increased expression in 15% and reduced expression in
10% of genes [101].
Future Challenges
Understanding genome organization within the cell
nucleus and its functional implications are fundamental
issues in modern cell biology. In this article, we have
reviewed the incisive role played by polymer theory and
simulations in providing new insights into the structure
of the genome at various levels of organization, ranging
from packing of nucleosomes into the 30-nm chromatin
fiber to the higher-order folding of chromatin via
Figure 7 Persistence Length Vs Folding Index. Plot illustrating
persistence length as a function of folding index for ν = 1/3. The
folding index is analogous to the resolution of a lens through
which the genome is viewed and the persistence length
corresponds to the length scale of observable correlations at this
specific resolution.
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looping into chromosomes. Despite enormous progress
in both experimental and computational fronts, a comprehensive
model of genome organization at the chromatin
and chromosome level is still lacking. Below we
list four main challenges that computational models
would have to overcome to tackle this highly intriguing
and important problem.
First, genome organization is heterogeneous. At the
chromatin level, this heterogeneity arises from variations
in the nucleosome repeat length, nucleosome composition
(histone modifications, histone variants, and linker
histones), and potential entrapment of chromatin in
metastable states. Most models of chromatin, including
those developed in our group, tend to assume uniform
nucleosome positioning and composition. Only recently
have studies begun to examine the effects of chromatin
heterogeneity; preliminary work in this direction has
already yielded some very interesting results. At the
chromosome level, the size, shape and location of chromatin
loops is not fixed. Thus, a key aspect towards
examining heterogeneity in chromosome models would
be through more realistic modeling of the interactions
across chromatin fibers, e.g., allowing loops to dynamically
form and break according to the associated energetics
of loop formation in chromosome models, as
opposed to “fixing” looping points and loop sizes.
Second, the chromatin fiber and chromosomes are
highly dynamic. Nucleosomes are constantly being displaced,
modified, dissolved and reformed through various
mechanisms. At the higher scale, chromatin loops
are continuously being broken and reformed. In addition,
a range of nuclear proteins including transcription
factors and architectural proteins dynamically bind and
dissociate from chromatin [2]. Such binding/dissociation
events could affect chromatin structure and in turn may
be affected by chromatin structure. Most models currently
look at the genome from a static perspective.
Incorporation of the above mentioned kinetic features
into polymer models coupled with structural heterogeneity
would be a crucial step in building more comprehensive
models of the genome.
Third, development of accurate, coarse-grained models
of chromosomes poses a grand challenge. One
potential route is via “multi-scale” approaches, where
each level represents coarse-graining of the previous,
higher-resolution level. Such an approach generally
involves the use of potential of mean forces (PMFs) for
treating the interactions between coarse-grained subunits.
The PMFs account for effects of the degrees of
freedom that are “averaged out” during each coarsegraining
step. The challenge is to maintain self-consistency
from one level to another, as the PMFs are valid
only when there is a clear separation of length and
time scales across the levels [102]. When there is no
clear separation of time/length scales, one might need
to resort to “multi-resolution” approaches. In this
approach, different portions of the system are treated
at different resolutions, depending on their relative
importance to the phenomena being investigated (in
the same spirit as the QM/MM method [103]
employed in enzyme catalysis). The challenge here is
to identify the portions of the system (or degrees of
freedom) important enough to be treated at higher
resolution as opposed to those regions that can be
treated at lower resolution. Another challenge is developing
suitable potentials for linking low- and highresolution
regions.
Fourth, little modeling effort has been invested in
determining structure-function relationships in genome
organization. A definitive connection between structure
and function comes from the observation that gene density
is high in largely decondensed euchromatin and low
in highly condensed heterochromatin [104]. Further,
gene-rich and gene-poor regions are found to be physically
separated from each other [105,106]. Specifically,
in human lymphocytes gene-rich regions are positioned
in the nuclear interior while the gene-poor regions are
positioned towards its periphery [107]. Apart from gene
density patterns, the molecular species responsible for
nuclear processes and gene expression are connected
through a complex and extensively coupled network giving
rise to functional coupling between macromolecular
structures and compartments [108,109]. Though many
aspects of these studies are been driven by the experiments,
polymer theory and simulations could contribute
to better understanding of such structure-function
relationships.
Appendix
Effect of persistence length modifications on gene locus
conformation
To study the generic effects of changes in the persistence
length of the chromatin fiber on the conformation
of the gene locus, we have treated the chromatin fiber
as a worm-like chain (WLC) with fixed contour length
Lc. The WLC is the standard model used to describe
semi-flexible polymer chains, and it has been successfully
used to explain the conformational properties of
double-stranded DNA and other biopolymers [110].
Though this model is too idealized to represent in vivo
chromatin (as it neglects fiber-fiber interactions, existence
of stable loops and heterogeneous flexibility), it
can nonetheless provide qualitative insights into the
magnitude of the conformational changes.
We begin by writing down the mean squared end-toend
distance of a WLC with an “effective” persistence
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length Lp of the chromatin fiber that embodies all
effects of the remodeling [111]:
R2
e = 2L2
p
Lc
Lp
− 1 + exp −
Lc
Lp
. (1)
We consider Lp as a variable that changes from an
initial value Li
p to a final value L
f
p due to chromatin
remodeling. Because we anticipate a weak effect of
remodeling on the overall contour length at the locus
level, we hold Lc constant. In the ideal chain limit of the
WLC, Lc ≫ Lp, the ratio of the final to initial mean
square end-to-end distances is given by
R2
e,f / R2
e,i ≈ L
f
p/Li
p. (2)
Plugging in values of initial and final Lp observed by
Heermann and coworkers [78], the ratio becomes 140/
280 = 1/2. We have further confirmed this conformational
collapse by using simple lattice MC simulations of
phantom linear chains of different lengths and regions
of flexibility. The simulations indicate that introduction
of regions of flexibility in an overall stiff chain, differing
by a nominal bending energy of 2kBT per segment,
strongly decreases the mean squared radius of gyration
of the chains (Table 1).
The implications to this modification in size can be
explored through the Flory approximation. According to
the Flory approximation, the number of binary interactions
in a polymer chain nb ~ c2
, where c is the concentration
of segments [112]. Since c ~ 1/V (where V is the
volume of the chain) and V ∼ R2
e
3/2 assuming spherical
symmetry, the ratio of the final to the initial number of
binary interactions is given by:
nb,f /nb,i ≈ ( R2
e,i / R2
e,f )3
≈ (Li
p/L
f
p)3
. (3)
In the case of the IgH locus, this translates to an 8fold
increase in nb.
Persistence length as a function of folding index
We treat the chromatin fiber as a FJC at the genomewide
level with a contour length given by:
Lc = l × Nr, (4)
where Nr is the number of repeat units, each of length
l = 2Lp. The characteristic size of a FJC is given by
[113]:
Cs ≈ l × (Nr)ν
, (5)
where ν is a scaling exponent that connects the size of
the chain to its number of repeat units. Note that the
fractal dimension, df , is the exponent that connects the
mass of the chain M to its size via M ∼ (Cs)df and that
M is also proportional to the number of repeat units M
~ Nr ~ (Cs)1/ν
(from Eq. 5). Hence, ν is the inverse of
the fractal dimension ν = 1/df and determines the organization
of the FJC at all length scales.
We can define the “folding” index F of FJC as the
ratio of fully extended contour length Lc to the folded
size Cs, which can be simplified further using Eqs. (4)
and (5):
≡ Lc/Cs ≈ N1−ν
r = (Lc/2Lp)1−ν
. (6)
Since the contour length Lc = d/c, where d is the
genomic distance in bp and c is the linear mass density
in bp/nm, F increases with decreasing linear mass
density.
The contour length within the FJC framework can be
partitioned arbitrarily, i.e., if the contour length of a
structure is 100 μm then it can be partitioned as, say,
100 repeat units each of 1 μm length or 1000 repeat
units of length 0.1 μm. The flexibility of the contour
length partitioning allows for choice of either number of
repeat units Nr or the persistence length Lp as a variable
in Eq. (6). However, the contour length Lc according to
Eq. (6) is determined by the product of the folding
index F and size Cs. This constraint has to be satisfied
when Lp is chosen as the variable and the contour
length in the framework changes with the change in the
folding index. Thus F can be used like a lens of variable
resolution to view genome organization at different
length scales.
The persistence length Lp can now be computed as a
function of F at different levels of human genome organization.
We begin with the organization at the level of
dsDNA, whose persistence and contour lengths are
known: Lc ≈ 2m for dsDNA with 6 billion bp at a linear
mass density c = 2.94 bp/nm [114]. Given that Lp = 50
nm [114] and ν = 1/3, the number of repeat units Nr ≈
2 × 107
from Eq. (4), the folding index F ≈ 8 × 104
from Eq. (6), and Cs ≈ 25 μm from Eq. (5). Interestingly,
the FJC size Cs is consistent with typical dimensions of
Table 1 Ratio of mean-square radius of gyration of
chains
Nr Fa
R(0)b
R(1) R(2) R(3)
16 5.15 2.38 1.46 1.30 1.19
32 10.11 2.73 1.77 1.622 1.49
64 20.16 2.91 1.91 1.83 1.76
128 40.20 3.02 2.02 1.95 1.91
256 80.46 3.07 2.06 2.00 1.98
512 160.78 3.10 2.11 2.04 2.03
1024 321.84 3.11 2.10 2.05 2.05
Ratio of mean-square radius of gyration of chains of different lengths (Nr) with
different regions of flexibility to that of a fully flexible chain (F).
a
Absolute values of mean square radius of gyration.
b
In R(x), x denotes the number of flexible regions sandwiched between stiff
regions that are of the same length as the stiff regions.
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human cell nuclei. Since, this size is invariant across all
organization levels, we denote it as Cs,0. With this constraint,
Eq. (6) reduces to
= (Cs,0/2Lp)(1−ν)/ν
|ν=1/3 = (Cs,0/2Lp)2
. (7)
Equation (7) thus provides the relationship between
the persistence length Lp and folding index F for all
organization levels, which is plotted in Figure 7.
Acknowledgements
We thank Prof. Cornelis Murre of the Division of Biological Sciences at UCSD
for many delightful and insightful discussions on organization of
chromosomes.
Authors’ contributions
BVSI and GA organized the article, contributed to the writing of all sections
of the article, and developed the theoretical analysis presented in the
Appendices. MK designed some of the figures and contributed to the
writing of the section on secondary structure. All the authors have read and
approved the final manuscript.
Received: 7 January 2011 Accepted: 15 April 2011
Published: 15 April 2011
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doi:10.1186/2046-1682-4-8
Cite this article as: Iyer et al.: Hierarchies in eukaryotic genome
organization: Insights from polymer theory and simulations. BMC
Biophysics 2011 4:8.
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