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The energy components of stacked
chromatin layers explain the morphology,
dimensions and mechanical properties of
metaphase chromosomes
Joan-Ramon Daban
Research
Cite this article: Daban J-R. 2014 The energy
components of stacked chromatin layers
explain the morphology, dimensions and
mechanical properties of metaphase
chromosomes. J. R. Soc. Interface 11:
20131043.
http://dx.doi.org/10.1098/rsif.2013.1043
Received: 11 November 2013
Accepted: 11 December 2013
Subject Areas:
biophysics, nanotechnology, biomechanics
Keywords:
metaphase chromosome structure, chromatin
higher order structure, supramolecular
structures, bionanoscience, biomechanics
Author for correspondence:
Joan-Ramon Daban
e-mail: [email protected]
Electronic supplementary material is available
at http://dx.doi.org/10.1098/rsif.2013.1043 or
via http://rsif.royalsocietypublishing.org.
Departament de Bioquı́mica i Biologia Molecular, Facultat de Biociències, Universitat Autònoma de Barcelona,
Bellaterra 08193, Spain
The measurement of the dimensions of metaphase chromosomes in different
animal and plant karyotypes prepared in different laboratories indicates that
chromatids have a great variety of sizes which are dependent on the amount
of DNA that they contain. However, all chromatids are elongated cylinders
that have relatively similar shape proportions (length to diameter ratio
approx. 13). To explain this geometry, it is considered that chromosomes are
self-organizing structures formed by stacked layers of planar chromatin and
that the energy of nucleosome–nucleosome interactions between chromatin
layers inside the chromatid is approximately 3.6 10220 J per nucleosome,
which is the value reported by other authors for internucleosome interactions
in chromatin fibres. Nucleosomes in the periphery of the chromatid are in
contact with the medium; they cannot fully interact with bulk chromatin
within layers and this generates a surface potential that destabilizes the structure. Chromatids are smooth cylinders because this morphology has a lower
surface energy than structures having irregular surfaces. The elongated
shape of chromatids can be explained if the destabilizing surface potential
is higher in the telomeres (approx. 0.16 mJ m22) than in the lateral surface
(approx. 0.012 mJ m22). The results obtained by other authors in experimental
studies of chromosome mechanics have been used to test the proposed supramolecular structure. It is demonstrated quantitatively that internucleosome
interactions between chromatin layers can justify the work required for elastic
chromosome stretching (approx. 0.1 pJ for large chromosomes). The high
amount of work (up to approx. 10 pJ) required for large chromosome extensions is probably absorbed by chromatin layers through a mechanism
involving nucleosome unwrapping.
1. Introduction
In eukaryotic cells, the extremely long genomic DNA molecules, which are only
2 nm in diameter but up to more than 1 m in length in some species, are associated
with histone proteins to form a huge number of nucleosomes. The nucleosome
core is a short cylinder (11 nm diameter and 5.7 nm height) formed by 147 bp
of DNA wrapped in 1.7 superhelical turns around an octamer of core histones
[1]. Nucleosome cores connected with linker DNA and complemented with histone H1 constitute long chromatin filaments that are folded within the nucleus
[2–6]. During mitosis, in order to avoid entanglements of different filaments
and to preserve DNA integrity, each chromatid contains only one chromatin filament that is densely packaged [7] before being transferred to the daughter cells.
The spatial organization of chromatin within metaphase chromosomes is a fundamental question of biology and several structural models have been suggested on
the basis of results obtained using different experimental approaches. These
models can be grouped into three categories that are schematized in figure 1a.
The loop–scaffold models (scheme a) were based on early electron microscopy
images showing large DNA loops emanating from histone-depleted metaphase
& 2014 The Author(s) Published by the Royal Society. All rights reserved.
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(b)
a
b
2
c
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(a)
chromatin organization
a
b
fibre loops
irregularly
folded fibres
c
stacked
thin plates
Figure 1. Chromosome dimensions and structural models for metaphase chromatin folding. (a) Schematic illustration of different models for chromatin structure in
metaphase chromosomes: (a) chromatin fibres (grey lines) form loops that are bound to a central protein scaffold (brown) [8–10], (b) chromatin fibres are irregularly
folded [11–13], and (c) chromatin has a laminar structure with many staked layers oriented perpendicular to the chromatid axis [2,14–18]; parallel lines represent the
side view of the stacked layers. (b) Examples showing the differences in metaphase chromosome sizes that exist among animal species: crested newt (T. cristatus) (a),
human (b), and D. melanogaster (c); the three chromosome sets are represented at the same magnification; figure reproduced from [19] with permission from the
Royal Society.
chromosomes [8–10]. Micromechanical studies [11], and
immunofluorescence [12] and cryo-electron microscopy [13]
results suggested models in which chromatin is irregularly
folded within condensed chromatids (scheme b). The thinplate model (scheme c) was based on electron microscopy
and AFM studies of the structure and mechanical properties
of the multilayer plates that emanate from partially denatured
chromosomes under metaphase ionic conditions [14,15].
Further studies using polarizing microscopy, cryo-electron
microscopy and electron tomography [16] and AFM-based
nanotribology [17] showed that in each layer the chromatin
filament forms a flexible two-dimensional network (5–6 nm
thick) in which nucleosomes are irregularly oriented, allowing
the interdigitation of the successive layers that are stacked in
the metaphase chromatids.
Single-molecule techniques (optical and magnetic tweezers) and computer modelling have provided a wealth of
information about the mechanochemical properties of chromatin [5,20]. It was found that stretching forces of 4–6 pN cause
the unfolding of individual chromatin fibres [21,22] and that
higher forces (20–40 pN) produce nucleosome unwrapping and eventually dissociation of core histones [21,23,24].
The force–extension curves corresponding to the unfolding
and refolding of the fibres indicated that this process is reversible and allowed the calculation of the energy corresponding
to the disruption of the attractive interactions between nucleosomes in folded fibres. The energy required for breaking
a single nucleosome–nucleosome interaction is between 3.4
and 14 kBT [21,22] (3.6 10220 J considering that kBT ¼
4.1 10221 J; kB is the Boltzmann constant and T corresponds
to the room temperature). These experimental results are compatible with the internucleosome interaction energy obtained
in modelling studies of chromatin fibres having different
degrees of nucleosome stacking [25–27]. Furthermore, the
stretching experiments indicated that the first stage of nucleosome unwrapping is also reversible; the energy required for
the unwrapping of the outer DNA turn of a single nucleosome
is 9–20 kBT [24,28]. Assuming that chromatin fibres form a zigzag structure without internucleosome interactions, it was
possible to model fibre unfolding considering that the work
involved in this transformation corresponds to the partial
unwrapping of nucleosomal DNA [29]. The unwrapping of
the inner DNA turn is an irreversible process that requires
more energy [26,28]; the activation barrier energy for this
transformation is 36–38 kBT per nucleosome [24].
Other laboratories have studied the micromechanical
properties of mitotic chromosomes using microneedles and
micropipette manipulation [30–34]. Individual chromosomes
showed a linear elastic response for stretching forces up to
approximately 5 nN, corresponding to a fivefold extension.
Higher forces were needed to obtain longer extensions,
which were accompanied by an irreversible plastic deformation of chromosomes. In the stretching experiments,
chromosomes showed a viscoelastic behaviour corresponding
to an internal viscosity that is approximately 105 times the viscosity of water [33]. The Young elastic modulus obtained from
force–extension curves in the elastic region for chromosomes
of different species is 0.2–1 kPa. Equivalent values were
obtained for the Young modulus calculated from the bending
rigidity measured experimentally for different metaphase
chromosomes [35]. These results indicated that chromosomes
are equivalent to isotropic elastic rods and that their internal
structure, which is responsible for the observed stretching
and bending elastic properties, is homogeneous across the
entire chromatid cross section. Taking into account the low
elastic stiffness observed for chromosomes, they have to be considered as very soft structures. Nevertheless, chromosomes
have outstanding mechanical properties [31,32]. Chromosomes
recover their initial shape and dimensions after repeated
extension–relaxation cycles in the reversible elastic region of
the stretching experiments, and they can be extended up to
80 times their native length without suffering breakage.
Metaphase chromosomes of different species have different
sizes (see examples in figure 1b), but apparently they have structural similarities. These similarities have never been explained
quantitatively. After many years of advances in many fields of
structural biology, the physical elements that determine the
global shape and basic properties of condensed chromosomes
remain unknown. This represents a major gap in our knowledge
of cell structure [36], which is particularly frustrating because
the metaphase chromosome is one of the most fascinating
structures of living matter. Moreover, probably the tremendous success obtained by X-ray crystallography in the
resolution of relatively large protein–nucleic acids complexes
[1,37] has reduced the current interest in huge structures, for
example metaphase chromosomes, that cannot be crystallized.
J. R. Soc. Interface 11: 20131043
30 mm
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2.1. Metaphase chromosomes of different species show
a great variety of sizes but have relatively similar
shape proportions
As can be seen in the examples presented in figure 1b and in the
electronic supplementary material, figure S1, chromosomes of
different animal and plant species exhibit dramatic differences in size. The chromosome dimensions of selected species
covering a broad range of genome sizes are presented in
table 1. It can be observed that the diameter and length of the
chromatids increases with the amount of DNA that they contain. In agreement with this observation, individual species,
such as Drosophila melanogaster [43] and chicken [44], having
chromosomes that contain very different amounts of DNA,
have karyotypes with a great heterogeneity of chromosome
sizes ([19,45,46]; figure 1b,c). The measurements shown in
table 1 cover from the small chromatids (approx. 35 Mb) of
rice ([47,48]; see the electronic supplementary material, figure
S1a) up to the very large chromatids (approx. 7450 Mb) of
Paris japonica (see the electronic supplementary material,
figure S1b,b), a species having the largest genome so far
reported [49,50]. There are species having chromosomes smaller than those of rice; for instance, chromosome 4 of Drosophila
(4 Mb [43]) and the microcromosomes of chicken (approx.
5 Mb [44]). These small chromosomes (sometimes called dot
chromosomes because they appear as circular objects in the
D
(mm)
L (mm)
F
35
65
0.3
0.4
1.5 – 3.4
3.9
8
10
Chionographis
japonica
125
0.5
1.7 – 3.3
5
humanb
229
0.6
5.4 – 7.2
11
barley
pine
729
1917
0.9
1.0
10 – 14
14 – 21
13
18
T. cristatus
N. viridescens
1958
3382
1.1
1.0
11 – 36
10 – 20
21
15
P. japonica
7450
1.3
14 – 39
20
chromosome
source
NMb
(Mb)
rice
D. melanogaster a
a
Drosophila melanogaster chromosomes have very different sizes and only
the two largest chromosomes were used for the determination of NMb,
D and L.
b
Human chromosomes have significant size differences; in this case NMb,
D and L correspond to the five largest chromosomes.
karyotypes) have not been included in table 1 because their
diameters (less than 0.3 mm) are artefactually enlarged in the
micrographs owing to resolution problems of optical microscopy
(see Methods in the electronic supplementary material).
Despite the large differences in size, metaphase chromatids
of different animal and plant species are elongated cylinders
that are approximately similar from a geometrical point of
view. For the species presented in table 1, the ratio F between
chromatid length (L) and diameter (D) is higher for large chromatids (more than 1000 Mb; average F 19) than for smaller
chromatids (less than 1000 Mb; average F 9). Chromatids
of many more species should be measured to confirm whether
these differences are generally valid, but for the approximate
calculations presented in this work (see below), it is reasonable to use the average value of F (13) obtained from all
chromatids considered in table 1. Taking into account that
metaphase chromosomes contain 0.17 pg of DNA mm23 [51],
which is equivalent to a density r ¼ 166 Mb mm23, a chromatid containing a specific number of Mb of DNA (NMb) has a
volume V ¼ NMb/r. For a cylindrical chromatid V ¼ p(D/2)2L,
but considering that L ¼ FD 13D, the chromatid volume
can be expressed exclusively as a function of D
V¼p
2
D
pD3 13
:
fD 2
4
ð2:1Þ
Therefore, NMb/r pD 313/4 and this leads to a formula
allowing the calculation of the approximate dimensions
(D and L in mm) of any chromatid containing NMb of DNA
D
4NMb
13pr
1=3
0:084ðNMb Þ1=3 ;
L 13D:
ð2:2Þ
3
J. R. Soc. Interface 11: 20131043
2. Results
Table 1. DNA content and dimensions of metaphase chromosomes of
different species. The approximate number of Mb of DNA per chromatid (NMb)
was calculated dividing the genome size by the number of chromosomes (n)
of each species. Genome sizes, n, and chromatid diameters (D) and lengths (L)
were obtained as described in Methods (see the electronic supplementary
material). F denotes the ratio between the mid-range value of L and D for
each species; the average value of F is 13.
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Fortunately, however, cutting-edge nanotechnology research
has demonstrated that self-assembly of different structures of
biological origin produces extremely complex micrometrescale materials that can be modelled using basic concepts of
supramolecular chemistry [38–41]. In this context, recent
research has revealed that multilayer plates (identical to those
found in metaphase chromosomes) can be self-assembled
from chromatin fragments obtained by micrococcal nuclease
digestion of metaphase chromosomes [18]. This finding has
reinforced the thin-plate model for metaphase chromosome
structure (figure 1a,c) and suggests that the whole chromosome
could be a self-organizing structure. The immense capabilities of
soft matter for the building of complex structures, and the possibility to understand them without considering all interactions
on the atomic scale [42], has stimulated the research presented
in this study. If the metaphase chromosome is considered as a
typical supramolecular structure, which has a maximum stability when the total energy of the whole structure is minimum,
it is possible to explain the geometry and mechanical properties
of the condensed chromatids. The energy values obtained in the
fibre extension experiments and in modelling studies considered
above have been used in this work, and the size of chromosomes
of diverse species has been measured from microscopy images
obtained in different laboratories. It is demonstrated that
the symmetry breaking produced by the different internucleosome interaction energies in different regions of the condensed
chromatids allows a consistent physical explanation of the
observed dimensions of metaphase chromatids. The results
obtained in experimental studies of chromosome stretching
performed in other laboratories (see above) have been used to
test the proposed supramolecular structure, and it is shown
that the dynamic properties of this structure can explain the
mechanical behaviour of condensed chromosomes.
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(a)
eT
1.8
experimental
4
1
calculated
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1.5
D (µm)
1.2
e0
0.9
eL
a
0.6
0.3
i–1
i
i+1
L
0
(b)
b
experimental
J. R. Soc. Interface 11: 20131043
30
calculated
25
L (µm)
20
n
15
D
10
5
0
10
100
1000
10 000
c
i–1
i
i+1
i+2
log NMb
Figure 2. Dimensions of chromosomes calculated using equation (2.2). The
calculated diameters (a) and lengths (b) of chromatids containing different
amounts of DNA are indicated with diamonds and the corresponding
experimental values with squares.
The dimensions of chromatids containing different amounts
of DNA calculated using this formula as compared with the
corresponding observed values are represented in figure 2.
There is some deviation of the calculated from the observed
lengths for some large chromosomes (Triturus cristatus and
P. japonica) but, as can be seen in panels a and b, respectively, most
of the values of D and L calculated using this chromosomedimensions formula fits reasonably well with the experimental
results. This indicates that a length to diameter ratio of about
13 represents adequately the approximate shape proportions of
metaphase chromosomes.
2.2. Surface energy differences of stacked chromatin
layers can explain the morphology and dimensions
of metaphase chromosomes
If the short centromere constriction is not taken into account to
simplify modelling, condensed chromatids can be considered
as smooth cylinders of constant diameter that are limited by
two circular plane surfaces, the telomeres. Cylinders and
planes have a constant mean curvature that minimizes surface
area under a volume constraint [52]. This suggests that chromatids subject to the constraint V ¼ NMb/r (see the preceding
section) have an internal structure that favours the minimization
of the surface area and produces the observed cylindrical
shape of chromosomes. The total area (including the two
telomere surfaces) of a chromatid of radius R and length L is
S ¼ 2pR2 þ 2pRL. Considering that V ¼ pR 2L, S can be
expressed as a function of R and V,
S ¼ 2pR2 þ
2V
;
R
ð2:3Þ
and the minimum surface area for a constant volume can
be obtained by equating to zero the partial derivative of S with
Figure 3. Idealized representation of the different energy regions in chromosomes
formed by stacked chromatin layers and description of different variables considered
in the text. The schematized chromatid has a length L and a radius R (¼D/2) and
contains a molecule of DNA of NMb megabases coiled in Nn nucleosomes. The
number of nucleosomes in the surface of the telomeres and in the lateral surface
are NT and NL, respectively. (a) The energy per nucleosome in bulk metaphase chromatin (10) is lower than that of a nucleosome in the chromatid surface.
Nucleosomes in any internal layer (i) are stabilized by interactions within the
layer and by interactions with two adjacent layers (i 2 1 and i þ 1). In the
two telomere surfaces (layers 1 and n, indicated in black), nucleosomes have an
extra energy (1T) because each one of these layers can interact only with one adjacent layer (2 and n 2 1, respectively); this generates a surface potential mT (the
surface area of telomeres is ST). Nucleosomes in the chromatid lateral surface (red)
have also an extra energy (1L) because they are more exposed to the medium and
are less stabilized by the attractive interactions within the layer; the corresponding
surface potential is mL (the lateral surface area is SL). (b) Irregular stacking of chromatin layers destabilizes the structure because there are more nucleosomes exposed
to the medium, increasing the lateral surface area. (c) Schematic representation of
nucleosome organization in successive chromatin layers according to [16] (reproduced with permission from the American Chemical Society); nucleosomes are
represented by two turns of DNA (black) wrapped around the core histone octamer
(grey); layers are indicated by discontinuous red and green lines; the consecutive
nucleosomes in each layer are connected with linker DNA (not shown in the drawing). The energy per nucleosome in bulk chromatin 10 has two components: the
energy 1wl due to all interactions within each layer and the energy 1nn due to the
interactions with the interdigitated nucleosomes of the two adjacent layers.
respect to R
@S
@R
¼ 4pR V
2V
¼ 0:
R2
ð2:4Þ
This gives 2pR ¼ V/R2 and, substituting V by pR 2L, it results
that 2R ¼ L. Therefore, in principle, the minimum area corresponds to cylinders having D ¼ L. However, as indicated
above, native chromatids are much more elongated. In this section, it is shown that the study of the factors that reduce D and
the relative surface of the telomeres leads to a simple physical
explanation of chromosome morphology and dimensions.
In chromatids formed by stacked layers of planar chromatin,
the energy 10 per nucleosome (figure 3) has two components:
the energy 1wl corresponding to all interactions within a layer,
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E0 ¼ Nn 10 ¼ Nn 1wl þ Nn 1nn ¼ Ewl þ Enn :
ð2:5Þ
E ¼ Ewl þ Enn þ ET þ EL
¼ Nn (1wl þ 1nn ) þ NT 1T þ NL 1L ;
ð2:6Þ
where 1T and 1L are the energies required for the transfer of one
nucleosome from the bulk metaphase chromatin to the surface
of a telomere and to the lateral surface of the chromatid,
respectively. ET ¼ NT1T is a positive energy because nucleosomes in the telomere surface can only interact with the
nucleosomes of one layer; EL ¼ NL1L is also positive because
nucleosomes in the lateral surface of the chromatid are exposed
to the medium (they are not completely surrounded by nucleosomes) and are less stabilized by interactions with bulk
chromatin. These considerations are consistent with the
observed smooth surface and constant diameter of the
ETþL ¼ ET þ EL ¼ ST mT þ SL mL ¼ 2pR2 mT þ 2pRLmL ; ð2:7Þ
where mT and mL are surface potentials and correspond,
respectively, to the amount of energy required to increase ST
and SL by unit area. Considering that V ¼ pR 2L, the total surface energy can be expressed as a function of R and V and the
two surface potentials
ETþL ¼ 2pR2 mT þ
2V
m :
R L
ð2:8Þ
To have a maximum stability, native chromatids must have a
minimum surface energy. For a constant volume, the minimum
surface energy corresponds to
@ ETþL
2V
¼ 4pRmT 2 mL ¼ 0:
ð2:9Þ
R
@R V
This leads to mT/mL ¼ V/2pR 3 and, substituting V by pR 2L, it
results that mT/mL ¼ L/2R ¼ L/D. Thus, taking into account
that in native chromatids L/D ¼ V 13 (see the preceding section), it can be concluded that mT/mL 13. Furthermore,
according to equations (2.6) and (2.7), NT1T ¼ STmT and
NL1L ¼ SLmL, and considering that the areas ST and SL must
be directly proportional to the number of nucleosomes NT
and NL, respectively, it can be concluded that 1T/1L ¼ mT/
mL 13. As nucleosomes in bulk chromatin can interact with
nucleosomes of two adjacent layers, but nucleosomes in the telomere surface can interact only with nucleosomes of one
adjacent layer, 1T 1nn/2. Therefore, taking the value of 1nn
considered above, 1T 1.7–7 kBT and 1L 0.13–0.54 kBT per
nucleosome. Finally, from the DNA density of chromatids,
the number of nucleosomes per unit of surface area can be estimated (approx. 9 1015 nucleosomes m22) and this allows the
calculation of the approximate values of the small surface
potentials in telomers and in lateral surface: mT 0.063–
0.26 mJ m22 and mL 0.0048–0.02 mJ m22.
According to the surface energy minimization considered
in the preceding paragraph, mT/mL ¼ L/D ¼ F. Therefore, if
more accurate values of F become available in the future
(see preceding section), the mT/mL ratio should be adjusted
according to new values. However, to justify the elongated
structure of native chromatids, mT must be significantly
higher than mL. From a structural point of view, the fact
that mT . mL indicates that nucleosomes in the telomere surfaces are much more in contact with the medium than those
in the chromatid lateral surface. This is consistent with the
thin-plate model schematized in figure 3, because whereas
nucleosomes in telomeres can interact only with one adjacent
layer, nucleosomes in the lateral surface are less stabilized by
the attractive interactions within the layer but can interact
with the nucleosomes of two adjacent layers.
The results presented in figure 4 demonstrate that the surface potentials ratio mT/mL ¼ 13 can explain the dimensions
of chromatids containing different amounts of DNA and
having a DNA density equal to that found experimentally for
5
J. R. Soc. Interface 11: 20131043
The observed small thickness of layers and other microscopy
results suggested that chromatin plates are interdigitated
favouring face-to-face interactions between nucleosomes of
adjacent layers ([15,16]; figure 3c). An equivalent interdigitation between the successive helical turns of 30 nm chromatin
fibres was proposed previously [2,53,54], and electron
microscopy [55,56], X-ray crystallography [1,4,57,58] and
modelling studies [5,25–27,59] indicated that nucleosomes
have a high tendency to interact through their faces. Furthermore, the intense scattering peak corresponding to a spacing
of approximately 6 nm found in a recent synchrotron X-ray
scattering study of human mitotic chromosomes [60] also
suggests a lateral association between nucleosomes in condensed chromatids. The DNA density in metaphase
chromosomes is high [7,51] and edge-to-edge contacts
between nucleosome cores must also be considered. However, the attractive energy of edge-to-edge interactions is
only approximately 10 times lower than that corresponding
to face-to-face interactions [5,25]. Thus, it is very likely that
the energy 1nn that stabilizes the stacking of the successive
layers in chromosomes is essentially due to face-to-face interactions between nucleosomes. The energy per nucleosome in
internucleosome interactions has been obtained experimentally [21,22] and in modelling studies [25 –27] in several
laboratories (see Introduction); taking into account all the
values obtained in these studies, it is reasonable to consider
that 1nn 3.4 –14 kBT. On the other hand, the easy sliding
between layers in chromatin plates observed in electron
microscopy experiments [15,16], suggested that the forces
holding the chromatin filament within a layer are higher
that the interactions between adjacent layers. This is consistent with the relatively large the Young modulus [15] and
the high mechanical resistance [17] found for chromatin
plates in AFM experiments in aqueous media, and indicates
that the energy 1wl stabilizing the structure of layers is
higher than 1nn.
Nucleosomes in the surface of the chromatid are in contact with the medium. These nucleosomes cannot fully
interact with bulk chromatin and have an extra energy
(figure 3). If NT and NL are, respectively, the number of
nucleosomes in the surface of the telomeres and in the lateral
surface of the chromatid, the total energy is
chromatids. As schematized in figure 3, irregular stacking of
chromatin layers destabilizes the chromatid, because it
increases the lateral surface area and more nucleosomes
become exposed to the medium. The destabilizing energy EL
corresponding to a regular stacking (scheme a) is lower than
EL of an irregular stacking (scheme b).
ET and EL can be related, respectively, to the areas of the
two telomere surfaces (ST ¼ 2pR 2) and of the lateral surface
(SL ¼ 2pRL) of the chromatid. The total surface energy is
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and the energy 1nn corresponding to the interactions with
nucleosomes of two adjacent layers. For a chromatid containing
Nn nucleosomes, the energy stabilizing the whole structure is
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ET
(a) 40
ET + L
NMb = 30
E (arb. units)
30
25
20
15
10
5
0.2
5.8
0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 D (µm)
1.4 0.64 0.36 0.23 0.16 0.12 0.09 0.07 L (µm)
(b) 90
80
NMb = 300
E (arb. units)
70
60
50
40
30
20
10
0
0.2
58
0.4
14
0.6
6.4
0.8
3.6
1.0
2.3
1.2
1.6
1.4
1.2
1.6
0.9
1.8 D (µm)
0.7 L (µm)
(c) 160
140
E (arb. units)
120
100
NMb = 3000
80
60
40
20
0
0.2
0.4
0.6
64
0.8
36
1.0
23
1.2
16
1.4
12
1.6
9.0
1.8 D (µm)
7.1 L (µm)
Figure 4. Surface energy of metaphase chromosomes as a function of their
diameter and length. The plots correspond to chromatids containing different
amounts of DNA: (a) 30, (b) 300 and (c) 3000 Mb. The three chromatids have
the same DNA density r ¼ 166 Mb mm23 [51], and the corresponding
volume V ¼ NMb/r is maintained constant in each plot (i.e. R and L satisfy
the equation pR 2L ¼ NMb/r). The relative energies of nucleosomes in the
surface of the telomeres (ET ¼ 2pR 2mT) and in the lateral surface of the
chromatid (EL ¼ 2pRLmL) were calculated considering that mT/mL ¼ L/
D ¼ 13 (see text); according to the approximate value of the surface potentials mT and mL given in the text, 1 arbitrary unit 1.2 10217 J. The
total surface energy (ETþL ¼ ET þ EL) show minimum values (indicated
by blue arrows) corresponding to diameters of approximately 0.3 (a), approximately 0.6 (b) and approximately 1.2 (c) mm. In each plot, there is a range
of chromatid dimensions (indicated by blue rectangles) that have energies
closely similar to the minimum value.
typical metaphase chromosomes of different species [51]. In
the three plots, ET increases with D, but EL decreases with D,
and this produces minimum values of the total surface
energy ETþL calculated according to equation (2.7). The diameters and lengths corresponding to the ETþL minima are similar
to those observed experimentally; the chromatids of 30, 300
2.3. Consistency test for the proposed supramolecular
structure: the interactions between adjacent
stacked chromatin layers and within layers can
explain quantitatively the experimentally observed
mechanical properties of metaphase chromosomes
As indicated in the Introduction, metaphase chromosomes can
be extended reversibly up to five times their native length. The
Young moduli obtained in chromosome extension experiments
in the reversible elastic region [30–32,34,35] has allowed the
calculation of the stretching work Wst (see Methods in the electronic supplementary material, equation (S1.1)) done by the
pulling forces applied to chromosomes of different sizes
(from approx. 178 (Xenopus laevis) to approx. 3382 (N. viridescens) Mb per chromatid). The results obtained are presented
in table 2. The Wst obtained from experimental results corresponding to larger (irreversible) extensions of N. viridescens
chromosomes (see Methods in the electronic supplementary
material) is also presented in this table. In this section, it is
shown that the comparison of Wst to the energies that stabilize
the structure of chromosomes formed by many stacked layers
of chromatin leads to a simple physical explanation of the
mechanical properties of condensed chromosomes.
The stabilizing energy of chromatin layers within chromosomes containing Nn nucleosomes has two main components
(see equation (2.6)): the energy Ewl ¼ Nn1wl corresponding to
interactions within layers, and the energy Enn ¼ Nn1nn corresponding to the interactions with nucleosomes of adjacent
layers; since NT and NL are much lower than Nn, the extra
energy of nucleosomes in the telomere surfaces (ET ¼ NT1T)
and in the lateral surface of chromatids (EL ¼ NL1L) are not
considered in this section because they cause a relatively
small decrease in the total stabilizing energy of bulk chromatin.
Furthermore, as indicated in the preceding section, and in
accordance with what is expected for a laminar material, Ewl
is larger than Enn. As can be seen in table 2, Wst for a fivefold
elongation is dependent on chromosome size. Note, however,
that for all the chromosomes considered in this table, the values
obtained for Enn are higher than Wst corresponding to the
elastic region. Therefore, these calculations indicate that for
this level of chromosome stretching, it is not necessary to consider the strong stabilizing interactions within the layers; the
breakage of part of the attractive nucleosome–nucleosome
6
J. R. Soc. Interface 11: 20131043
0
and 3000 Mb in these plots correspond approximately to the
chromatids of rice, human and Notophthalmus viridescens,
respectively, in table 1. Note in addition that around the
energy minima (in particular for large chromatids; figure 4c),
there is a small range of diameters and a relatively wide
range of lengths that have a surface energy only slightly
higher than that corresponding to the minimum. There is a
fine tuning of the minimum surface energy of each chromatid,
and relatively small differences in DNA content produce inappreciable changes in D and more significant changes in L. This
can explain the relatively wide variability of L observed in
typical karyotypes for chromosomes having not very large
differences in DNA content. The results of this section demonstrate that the symmetry breaking generated by the different
values of the surface potentials (mT . mL) provides a physical
explanation of the global geometry of metaphase chromatids.
rsif.royalsocietypublishing.org
35
EL
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elastic extensions
large extensions
Y
Wst (5L)
X. laevis
human
0.2 – 0.8
0.42
0.0034 – 0.014
0.012
0.025 – 0.10
0.032 – 0.13
grasshopper
N. viridescens
0.43
0.25– 1
0.020
0.047 – 0.19
0.067 – 0.28
0.47 – 1.9
interactions between layers can justify the work required for
extensions up to five times the native length.
A fivefold extension produces a great deformation of the
chromosome. However, this deformation does not preclude
the recovery of the initial length [32]. From a structural point
of view, it seems difficult to justify such a remarkable elasticity,
but considering that the nucleosome–nucleosome interactions between layers can be regenerated, the retraction of the
chromatid may be produced following a two-dimensional
re-zipping mechanism that restores all the initial attractive
interactions between nucleosomes in adjacent layers. This
reversible transformation is represented with a simplified
solid model in figure 5a,b. During elongation, the partial
unzipping of the interactions between layers could also be
responsible for the high internal viscosity and viscoelastic
behaviour of chromosomes observed in micromechanical
studies [33]. Furthermore, chromosome bending requires a
lateral deformation of the chromosome (figure 5d), but again
according to the multilayer chromosome model, presumably this local deformation gives rise to the dissociation of
nucleosome–nucleosome interactions between layers. As this
mechanism is equivalent to the mechanism of elongation,
this can explain the experimental results obtained with
N. viridescens and X. laevis chromosomes [35] showing that
the Young modulus calculated from bending results is
equivalent to that found in stretching experiments.
The values of Wst calculated for large extensions of
N. viridescens (table 2) cannot be explained by considering Enn
alone. In these irreversible plastic deformations (figure 5c), the
high values obtained for Wst are probably due to breakage of
interactions within layers. The total energy Ewl that stabilizes
the structure of chromatin layers is not known. However,
stretching experiments performed with single chromatin
fibres demonstrated that forces larger than those involved in
breaking internucleosome interactions cause nucleosome
unwrapping and dissociation ([21,23,24,26,28]; see Introduction). Furthermore, nanotribology results obtained with
individual metaphase chromatin plates [17] indicated that
during friction measurements, the kinetic energy of the AFM
tip produce reversible deformations that probably include
nucleosome dissociation, but do not cause DNA breakage
even when the forces applied by the tip were higher than
those required for the scission of the covalent DNA backbone.
Therefore, the work applied to chromosomes for large
Wst (10L)
0.65
Wst (14L)
1.7
Wst (50L)
10
Enn
extensions is probably absorbed by layers through a mechanism
involving different levels of unwrapping and dissociation
of nucleosomes.
The good mechanical strength observed for chromatin
plates is due to the fact that they form a two-dimensional network [17]. Therefore, it is unlikely that the connection between
consecutive layers can take place in specific points as represented schematically in figure 6a, because this would
produce local heterogeneities in the connection points that
could expose the chromatin filament to strong stretching
forces without the protection given by the intralayer interactions of bulk chromatin. Considering early work indicating
that metaphase chromatids have a global helical structure
[9,61,62], it was suggested [16,18] that the chromatin layers
could form a continuous helicoidal structure. This gives a feasible geometric solution (figure 6b) that allows chromatin to
form a homogeneous two-dimensional network within the
chromatids, which protects the integrity of DNA and avoids
the random entanglement of DNA within the chromatid.
Furthermore, from microscopy results and the experimental
observation of mirror-symmetrical positioning of single-copy
genes in sister chromatids, it was suggested [62,63] that sister
chromatids in metaphase have an opposite helical handedness. Thus, the scheme in figure 6b represents a right-handed
helicoid but, according to these observations, both right- and
left-handed helicoids are required for the representation of
chromatin folding in metaphase chromosomes.
3. Discussion
Typical chromatin plates can be self-assembled from chromatin
fragments of metaphase chromosomes, indicating that chromatin structure can hold discontinuous DNA in the correct place
within the plates [18]. However, when the relatively high lateral
force of the AFM tip was applied to micrococcal nucleasedigested plates in nanotribology experiments, the structural
integrity of chromatin layers was maintained only if the
degree of digestion was very low [17]. Metaphase chromosomes
subjected to stretching forces of 0.2–0.8 nN can tolerate a certain
level of digestion with restriction enzymes, but after extensive
digestion with micrococcal nuclease the mechanical strength
of chromosomes disappears completely [11]; these results
demonstrated that chromosomes do not have a mechanically
J. R. Soc. Interface 11: 20131043
chromosome source
7
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Table 2. Work required for chromosome stretching and estimated total energy of the internucleosome interactions between stacked layers. The work Wst (pJ)
done by the stretching forces producing chromosome elongations up to 5, 10, 14 and 50 times their native length (L) was calculated as described in Methods
(see the electronic supplementary material). Young modulus Y (kPa) corresponding to the elastic extension of the chromosomes considered in this table were
obtained from [30 – 32,34,35]. The total energy due to nucleosome– nucleosome interactions Enn (pJ) was calculated from the number of nucleosomes Nn in
each chromosome (Nn number of bp per chromosome/200 bp per nucleosome) and considering previous experimental [21,22] and modelling [25 –27] studies
showing that the energy 1nn of a single internucleosome interaction is between 3.4 and 14 kBT (Enn ¼ Nn1nn; see equation (2.5)). All the values shown in this
table correspond to chromosomes formed by two sister chromatids.
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(b)
(c)
stretched (5L)
8
stretched (10L)
rsif.royalsocietypublishing.org
(a)
part of a native
chromosome (L)
bent (40°)
Figure 5. Simplified solid model for the discussion of some basic aspects of chromosome mechanics. The blue plastic discs represent part of a multilayered
unstretched chromatid (a) and after extension up to 5 (b) and 10 (c) times its initial length. It is represented (using staples placed at random locations) that
part of the interactions between adjacent layers is not broken even in large extensions. This facilitates the complete recovery (five times extension) or partial
recovery (10 times extension) of the native length through a two-dimensional re-zipping mechanism based on the restoration of the initial interactions between
layers. (d ) It is shown that chromatid bending produces the breakage (concentrated in a lateral region) of the same interactions between layers considered in (b).
Reversible and irreversible transformations are indicated by parallel arrows and a single arrow, respectively.
(a)
(b)
h
z
R
R
y
x
j
r
whole
N layers = chromatid = N turns
Figure 6. Structural possibilities for the maintenance of the covalent
continuity of the single chromatin filament contained in a chromatid.
(a) Three-dimensional scheme corresponding to part of a chromatid
formed by stacked planar layers; the connections between layers introduce
local structural heterogeneities that could be located in internal regions or
in the periphery of the chromatid (some peripheral connections are schematically indicated by short vertical lines between layers). (b) A helicoid could
facilitate a continuous folding of the chromatin filament without any structural heterogeneity. Note that the side view of a chromatid with N stacked
circular layers of radius R formed with thin sheets of planar chromatin in close
contact (i.e. having a thickness approximately equal to the distance h
between layers) is essentially equivalent to the side view of a helicoid of
N turns constructed with a single sheet of chromatin having the same thickness than the planar layers. This right-handed helicoid of pitch h can be
defined by the parametric equations: x ¼ r cosw, y ¼ r sinw, z ¼ hw/
2p, with 0 r R and 0 w N2p. Equivalent equations define a
left-handed helicoid, but in this case z ¼ 2hw/2p. According to previous
electron microscopy, AFM and electron tomography measurements of layer
thickness [15,16], h 5– 6 nm.
contiguous scaffold formed by non-histone proteins. Nevertheless, different evidences indicated that condensins and
topoisomerase II, which are located in the chromatid axis,
may play a role in the functional organization of chromosomes
[64]. In particular, it has been found recently that condensin
complexes and chromokinesin KIF4 are required for the reestablishment of the characteristic shape of chromosomes
after two rounds of swelling and refolding produced by treatments with buffers containing, respectively, EDTA and Mg2þ
[65]. These proteins may also contribute to define chromatin
loops with functional roles in the interphase nucleus [6,66],
but the observed uniform structure of the surface and edges
of planar chromatin indicates that these topological domains
are completely buried in the layers of condensed chromatin
within metaphase chromosomes. It will be very interesting to
study the mechanism by which chromosomal proteins interact
with planar chromatin. In these studies, it has to be taken into
account that a significant part of the chromosome volume is
occupied by water [67]. This and the relatively weak association
between chromatin layers [15,16] suggest that chromosome
structure may be dynamic. The rapid diffusion observed for
topoisomerase II and other proteins in mitotic chromosomes
in vivo [68–70] is consistent with the possibility that condensed
chromatids are actually very dynamic structures.
Chromosomes from dinoflagellates do not contain histones
and are birefringent. It was suggested that these chromosomes
are equivalent to cholesteric liquid crystals consistent of many
stacked layers of DNA [71]. Liquid crystals formed in vitro
by parallel columns of stacked nucleosome core particles are
also birefringent [56]. By contrast, metaphase chromosomes
are optically isotropic [16], indicating that they do not contain
parallel columns of nucleosomes. In chromatin layers nucleosomes are irregularly oriented [16], but this high degree of
disorder at short distances does not preclude the formation of
a well-defined multilayered structure that can be considered a
lyotropic lamellar liquid crystal, in which the high water
J. R. Soc. Interface 11: 20131043
(d )
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4. Conclusion
The different energy terms considered above are responsible for
specific structural and dynamic roles that can justify the global
geometry and physical properties of metaphase chromosomes.
It has been shown that the stacked chromatin layers generate
surface energies that destabilize nucleosomes located in the telomeres and in the lateral surface of the chromatids. The
minimization of the lateral surface explains the smooth cylindrical shape of the chromatids, and the symmetry breaking
produced by the different values of the surface potentials in
the telomeres and in the lateral surface explains the elongated
structure of the chromatids. The results obtained in previous
experimental studies of chromosome mechanics have been
used as a test for the proposed supramolecular structure. It is
demonstrated that the intermediate energy of nucleosome–
nucleosome interactions between chromatin layers can account
quantitatively for the work required for elastic stretching
and further reversible recovery of the initial chromosome
length. The intralayer energy, which includes the large term
corresponding to nucleosome unwrapping, can justify the
work required for large chromosome extensions. In summary,
it can be concluded that the results obtained in this work
provide a consistent physical explanation of the observed cylindrical shape, diameter and length, and mechanical
properties of metaphase chromosomes.
Acknowledgements. The author acknowledges Mercè and Queralbs for
providing the best atmosphere for the preparation of this work.
Funding statement. Research supported in part by grant BFU2010-18939
from the Ministerio de Economı́a y Competitividad.
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