Biology Reference
In-Depth Information
chromosome between two loci increases fourfold, the
spatial distance between them increases twofold. For
a random walk the probability that two loci will contact
each other, e.g., forming a chromatin loop, decreases very
rapidly with increasing distance along the polymer and
scales as P(s)~s
-3/2
.
When steric and excluded volume effects are dominant,
e.g., when parts of the polymer repel each other, the poly-
mer forms a so-called swollen globule. As one might expect,
in this case the polymer will tend to occupy a larger volume
than a random walk. For a swollen coil the spatial distance
between segments along the polymer scales as R(s)~s
0.6
,
and the contact probability scales as P(s)~s
-1.8
[19]
.
The third polymer state is the fractal globule. Although
this polymer state had been proposed more than two
decades ago
[20,21]
, it has only recently been explored in
more detail because it has been found to describe exper-
imental observations of chromosome folding
[9,22]
.Itwas
originally proposed to be an intermediate or non-equilib-
rium state formed by collapse of a polymer upon itself.
Such a collapse can be solvent induced, due to some
intrinsic attraction between the segments that make up the
polymer, or due to some externally imposed confinement.
The fractal globule state leads to a more compact polymer
than a random walk or a swollen coil. Direct computer
simulation of a fractal globule polymer ensemble found
that the spatial distance between loci scales as R(s)~s
1/3
,
reflecting a more compact conformation than a random
walk polymer of the same length. Similarly, the contact
probability of pairs of loci scales only as P(s)~s
-1
,which
means that contact probability decays less rapidly with
distance along the polymer than a random walk or swollen
coil. The fractal globule state has additional interesting
characteristics. First, in contrast to the other polymer
states, it has no knots or entanglements. Second, the
fractal globule is not in equilibrium, and when left for
a sufficient amount of time will convert to a random walk
or swollen coil.
a number of experimental observations. Then, when the
chromosome folds according to a random walk, as would be
expected in the absence of any confinement, the average
end-to-end distance of the chromosome will only be around
0.02 mm, and the average volume of the chromosome will
be around 1280
m
3
. This is about two to three times larger
than the volume of a typical human cell nucleus (~500
m
m
3
).
When the persistence length increases by a factor of 2, the
chromatin fiber becomes less flexible and will occupy
a much larger volume: around 9200
m
m
3
, or about 20 times
the volume of the cell nucleus. A reduction in mass density,
e.g., by chromatin decondensation, will lead to similar
increases in occupied volume. This illustrates that folding of
chromosomes inside cells must be highly constrained, at the
least by confinement within the relatively small volume of
the nucleus.
m
Polymer Conformation is Probabilistic
One fundamental difference between the structure of
proteins and the folding of flexible chromatin polymers is
that the latter do not form reproducible three-dimensional
structures: each polymer molecule will follow its own
unique three-dimensional path. Thus, the conformation of
a polymer solution, or the folding of chromosomes across
a cell population, should be viewed in statistical terms that
describe a large ensemble of different conformations of
otherwise identical polymer chains. Analysis of the spatial
folding of chromosomes, e.g., by FISH, also reveals
significant cell-to-cell variability, with distances between
loci varying widely in a population of otherwise identical
cells
[23]
. This commonly observed phenomenon, which
can be wrongly interpreted to mean that the nucleus shows
limited organization, is firmly rooted in the polymer prop-
erties of the chromatin fiber. As we will describe below,
additional constraints on chromosomes will greatly reduce
the number of possible conformations of the chromatin
fiber, driving reproducible patterns of nuclear organization.
NUCLEAR CONFINEMENT AND
FORMATION OF CHROMOSOME
TERRITORIES
As outlined above, chromosomes are large, and unless
constrained, occupy volumes that are significantly larger
than the cell nucleus. Thus, the chromosomes must
somehow be confined within the nucleus. Another striking
feature of nuclear organization is that chromosomes do not
readily mix: they occupy distinct territories, although there
appears to be a degree of intermingling where two chro-
mosomes touch each other
[24,25]
. Thus, each chromosome
is in reality confined to a volume that is even smaller than the
nucleus. Chromosome territories may be formed in early
G1 cells when individual mitotic chromosomes decondense,
The Ground State of an Unconstrained
Chromosome
Now that we have discussed the basic parameters that
determine the shape of a polymer, we can estimate how they
will affect the volume of a typical otherwise unconstrained
interphase chromatin fiber. As an example, we will examine
a chromatin fiber with the length of human chromosome 1
(247 Mb). When we assume that this fiber adopts a mass
density that is typical for a 30 nm fiber (1 kb/11 nm), the
contour length of this chromosome will be around 2.7 mm.
The average end-to-end distance, or radius of gyration, of
this chromosome will depend on its persistence length and
the polymer state. We will assume a persistence length of
100 nm, corresponding to 9 kb, which is in accordance to
