Flory-type theory of a knotted ring polymer
Alexander Yu. Grosberg,
1,
* Alexander Feigel,
2
and Yitzhak Rabin
2
1
Department of Physics and Center for Materials Science and Engineering, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139
2
Physics Department, Bar-Ilan University, Ramat-Gan 52900, Israel
Received 11 March 1996
A mean field theory of the effect of knots on the statistical mechanics of ring polymers is presented. We
introduce a topological invariant which is related to the primitive path in the ‘‘polymer in the lattice of
obstacles’’ model and use it to estimate the entropic contribution to the free energy of a nonphantom ring
polymer. The theory predicts that the volume of the maximally knotted ring polymer is independent of solvent
quality and that the presence of knots suppresses both the swelling of the ring in a good solvent and its collapse
in a poor solvent. The probability distribution of the degree of knotting is estimated and it is shown that the
most probable degree of knotting upon random closure of the chain grows dramatically with chain compres-
sion. The theory also predicts some unexpected phenomena such as ‘‘knot segregation’’ in a swollen polymer
ring, when the bulk of the ring expels all the entanglements and swells freely, with all the knots concentrated
in a relatively small and compact part of the polymer. S1063-651X9610012-X
PACS numbers: 61.41.+e
Modern polymer physics is based on the analogy between
a polymer chain and a random walk. While this analogy has
been extremely fruitful in predicting the long-wavelength
static and dynamic properties of polymer chains radius of
gyration, relaxation time, etc., it misses one of the most
important attributes of polymer chains, namely, their ability
to form permanent entanglements, i.e., knots. Although there
is a very successful phenomenological theory the reptation
model 1 of the effect of temporary entanglements on the
dynamics of polymers which give rise to the viscoelasticity
of polymer liquids, there is no comparable theory of perma-
nent entanglements which play an important role in biologi-
cal systems e.g., DNA rings in bacteria. Our understanding
of the latter is limited by the fact that, despite the progress in
the mathematical theory of classification of knots and topo-
logical invariants 2, there are only few rigorous results on
the entropic properties of knots 3,4, and further attempts in
this direction encounter severe mathematical difficulties 5.
Computer simulations 6–8, however insightful, have some
basic limitations.
What appears to be missing in the area of polymer topol-
ogy is a simple ‘‘physical’’ model which could give some
testable physical predictions. The main purpose of the
present work is to trade mathematical sophistication for the
simplicity and effectiveness of such a physical approach and
to formulate a model that goes along the lines of the simple
Flory mean field theory of linear polymers and takes into
account topological constraints, albeit in a very primitive and
incomplete manner.
It is also worth mentioning what we do not attempt to
accomplish here: as we pursue a scaling-type approach, we
cannot capture the subtle differences between simple knots,
such as, for instance, trivial knot, trefoil, figure eight, etc.;
we only hope to be able to describe general tendencies for
very complex knots. As far as biological applications involv-
ing DNA molecules are concerned, our model does not apply
to current experiments on rather small DNA rings 9,10
such as plasmids, but may be relevant to very large circular
DNA e.g., that of bacteria. We also stress that we are
speaking of topological constraints on the dsDNA as an en-
tire thread, and do not have in mind linking of the two
strands in the duplex, that is related to biologically important
issue of superhelicity 11.
We address two questions: 1 What is the equilibrium
size of a polymer ring, depending on both solvent character-
istics or monomer interactions and knot topology? A simi-
lar question for linear polymers, where topology is not an
issue, is discussed in every textbook on polymer physics, and
it is known that the Flory theory yields a very good approxi-
mation for both swelling in good solvent and collapse in
poor solvent. Clearly, almost the same behavior, except for
some subtle chain end-related effects, is expected for a phan-
tom ring polymer which can cross itself. Our goal is to
consider a nonphantom ring, with quenched knot topology.
2 What is the probability distribution of various knots
obtained upon random closure of a linear polymer, or by
random motion of a phantom ring with annealed topology?
As we wish to build up a simple theory, we do not attempt
to characterize knots by sophisticated polynomial invariants.
Instead, we introduce the following construction. Consider a
polymer chain in some spatial conformation and denote by
L the contour length of the chain. Let us first construct a tube
that contains the polymer chain and is sufficiently narrow
such that the topology of the tube as a whole is the same as
that of the polymer. We now inflate the tube such that its
length L is preserved, while its cross section is roughly the
same everywhere along the tube we assume that different
tube portions cannot penetrate each other. This inflation will
eventually end when the inflated tube fills the main part of
the volume within its loops. Let us denote by D the diameter
of the tube in this maximally inflated state. We state that the
aspect ratio of the maximally inflated tube
*
Permanent address: Institute of Chemical Physics, Russian Acad-
emy of Sciences, Moscow 117977, Russia.
PHYSICAL REVIEW E DECEMBER 1996 VOLUME 54, NUMBER 6
54 1063-651X/96/546/66185/$10.00 6618 © 1996 The American Physical Society