Biology Reference
In-Depth Information
any biological and external structural constraints. We will
briefly discuss the main physical parameters that describe
properties of polymers and illustrate how they apply to
chromosomes.
In the absence of any external constraints there are
three critical physical properties that determine the shape
of a polymer, or chromatin fiber, and the average volume
that it will occupy: its persistence length, the mass
density and the type of polymer state
[14]
. The topic
entitled Giant Molecules, Here, There and Everywhere
by Grosberg and Khokhlov
[15]
is a good introduction to
this field. These three parameters will be discussed below
as they define the ground state of chromosomes on which
other constraints, described in detail
around 100 nm
[18]
. The value of the persistence length
will directly influence the ability of chromatin fibers to
form chromatin loops, as described below, and is therefore
of significant biological interest.
Mass Density
The mass density of a polymer is a measure of the
compaction of the polymer, which is typically expressed as
the amount of mass per unit of contour length. In the case of
DNA or chromatin fibers this is the contour length of
a given number of base pairs. The mass density of a poly-
mer is interesting because it relates its average size to the
internal organization of the polymer itself. In the case of
chromatin the mass density is related to how nucleosomes
are packed together to form the chromatin fiber. For
instance, B-DNA has a mass density of ~3 bp per nm.
A 10 nm beads-on-a-string chromatin fiber has a mass
density of ~15 bp per nm, whereas a more condensed 30 nm
solenoid fiber has a mass density of ~ 90 bp per nm.
Intuitively it seems that increasing the mass density,
e.g., through the formation of increasingly internally
condensed chromatin fibers, will result in a polymer that
occupies a smaller volume, i.e., has a smaller radius of
gyration. However, this is complicated by the fact that
increasing the mass density can also increase the persis-
tence length, which in turn can lead to a larger radius of
gyration. Thus, in order to understand the volume of
a polymer one needs to know both the mass density and its
persistence length. In addition, the specific polymer state
will also affect the conformation and volume of a polymer,
as we will explain below.
in subsequent
sections, will act.
Persistence Length
The persistence length of a polymer is a measure of its
flexibility, or bendability. Owing to the stiffness of a poly-
mer, the bending angles of two positions located very close
along the length of the polymer will be correlated, and at
that length scale the polymer is considered rigid and
straight. The persistence length is defined as the minimal
distance between two positions on the polymer for which
the bending angles are no longer correlated. Positions
separated by a distance along the polymer that is several
times the persistence length will display completely inde-
pendent bending angles, and at that scale the polymer
appears fully flexible. The persistence length of a polymer
has a large impact on the average volume it will occupy.
This volume is often expressed as the radius of gyration,
which is related to the average end-to-end distance of the
polymer. In general, the larger the persistence length the
stiffer the polymer and the larger the radius of gyration of
a given length of polymer becomes. This also directly
implies that in the case of chromosomes modulation of
persistence length, e.g., by chromatin modifications or
DNA-binding proteins, will lead to shrinking or expanding
of the chromatin fiber.
The persistence length of naked DNA is around 150 bp,
which is around 50 nm
[16]
. This means that a segment of
less than 150 bp is rather stiff and cannot easily form a loop
or circle, whereas a DNA segment of several hundreds of bp
can easily fold back on itself. The persistence length of
chromatin fibers is less well established and wildly
different values have been reported over the years, ranging
from as small as a few kb (corresponding to tens of nm of
chromatin in case of a 30 nm fiber, or around 100
Polymer States
A chromatin fiber with a given persistence length and
a given mass density can fold into several distinct types of
spatial conformation that are related to the polymer state.
The type of polymer state affects the overall volume occu-
pied by a given polymer and the spatial distance between
loci as a function of their separation along the length of the
polymer. Intuitively, it can be understood that the longer the
polymer is, the larger the volume it will occupy; and
the farther apart two loci are spaced along the polymer, the
larger their average spatial distance would be. However, the
precise quantitative relationships between these parameters
depend on the polymer state. A recent review by Fudenberg
and Mirny provides a good introduction to these polymer
states
[19]
. Here we describe them briefly.
The best-known polymer state is that of a random coil,
also referred to as a random walk. In this state, the average
spatial distance between two loci (R) scales with the
square root of the distance along the polymer between
them (s): R(s)~s
1/2
. Thus, when the distance along the
150 nm
for 10 nm fibers) to as large as tens of kb (corresponding to
several hundred nm for a 30 nm fiber, or as long as a
e
m for
a 10 nm fiber)
[17]
. Recent experiments and re-evaluations
of earlier work are converging on a chromatin fiber that is
less compact than a 30 nm fiber, with a persistence length of
m
