In an linear (non-circular) RNA sequence of length , we can express the
position of each monomer (or mer) by an index. If we denote the 5
end by the index
and the 3
end by the index
, then for we
can let index
corresponds to the index of monomer
, where
. We denote the correlation between two monomer by the ordered pair
, where
and
. For a true randomly oriented
polymer, the average correlation between any indices
and
is zero.
However, when there is bonding between two mers, the correlation is much
greater. It is an expression for this correlation that we seek.
One important observable thermodynamic state variable of a polymer is the
radius of gyration (). The radius of gyration is a measure of the
root-mean-square (rms) separation-distance between indices
and
of
the non-circular polymer. If the radius of gyration is known, then we can
find the rms end-to-end separation distance from the relationship
. For monomers that lie somewhere within the sequence, it is
convenient to express these subsequence lengths in terms of the difference
of the indices
, where
. It goes without saying
that rules that apply to
must also apply to
because we
can construct individual polymer fragments of length
and measure them and see that their behavior is a function of their length:
The GPC condition represents a special case of the rms end-to-end separation-distance of a polymer chain where
where is the separation distance between monomers (in
this case nucleic acids), and
is the persistence length. The
latter variable (
is known as the persistence length or the
Kuhn length and expresses the correlation between neighboring
monomers:
, the number of monomers that tend to group together
and behave as though they were one single unit. In the traditional
model,
actually refers to the extreme ends of the polymer
chain where
and
, with
the total number of
monomers in the polymer chain. When the length (
) of the polymer
chain is stretched out from end-to-end in a linear fashion,
.
In the most general expression for the distribution function, the
likelihood of finding mers and
with an end-to-end separation
distance of
and contained within a volume element of
is
where is a weight that accounts for the
self-avoiding character of the polymer and contributes to the excluded
volume,
is the weight on the exponential function, and
is a normalization constant for the
distribution function
where is a Gamma-function. The constant
is a dimensionless scaling parameter that is a
function of the root-mean-square (rms) end-to-end separation-distance
of the polymer chain,
In its most general form,
where is the weight on
. The central limit theorem
predicts that
for a GPC; however, some aspects of RNA
behavior tend to deviate from this ideal situation as we will
show. The persistence length is a measure of the correlation between
segments in a polymer.
This function assumes that the characteristics of the polymer can be
reduced to nearly identical entities to a reasonable
approximation. For the most part, the four bases in RNA can broadly
meet these attributes. When
and
,
becomes a Gaussian function that is the origin of its
name Gaussian polymer chain (GPC).