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Knots appear in a wide variety of biophysical systems, ranging from biopolymers, such as DNA and proteins, to macroscopic objects, such as umbilical cords and catheters. Although significant advancements have been made in the mathematical theory of knots and some progress has been made in the statistical mechanics of knots in idealized chains, the mechanisms and dynamics of knotting in biophysical systems remain far from fully understood. We report on recent progress in the biophysics of knotting-the formation, characterization, and dynamics of knots in various biophysical contexts.
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ANRV411-BB39-18 ARI 2 April 2010 11:20
Biophysics of Knotting
Dario Meluzzi,1Douglas E. Smith,2
and Gaurav Arya1
1Department of Nanoengineering and 2Department of Physics, University of California at
San Diego, La Jolla, California 92093; email:,
Annu. Rev. Biophys. 2010. 39:349–66
First published online as a Review in Advance on
February 16, 2010
The Annual Review of Biophysics is online at
This article’s doi:
Copyright c
2010 by Annual Reviews.
All rights reserved
Key Words
polymer physics, entanglement, DNA, proteins, topoisomerase
Knots appear in a wide variety of biophysical systems, ranging from
biopolymers, such as DNA and proteins, to macroscopic objects, such as
umbilical cords and catheters. Although significant advancements have
been made in the mathematical theory of knots and some progress has
been made in the statistical mechanics of knots in idealized chains, the
mechanisms and dynamics of knotting in biophysical systems remain far
from fully understood. We report on recent progress in the biophysics
of knotting—the formation, characterization, and dynamics of knots in
various biophysical contexts.
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Knot: a topological
state of a closed 3D
curve, also a knot-like
conformation of an
open chain
INTRODUCTION .................. 350
TYPES OF KNOTS.................. 350
SYSTEMS ........................ 352
KNOTTING ..................... 354
SYSTEMS ........................ 356
Size of Knots and Knotted
Systems ........................ 356
Knot Localization.................. 357
Strength and Stability
of Knotted Systems ............. 357
INVOLVING KNOTS ............ 358
Knot Diffusion .................... 358
Electrophoresis .................... 359
Unknotting ........................ 359
CONCLUSION ..................... 361
Knots are fascinating topological objects that
have captured human imagination for centuries.
They find a plethora of useful applications,
from tying shoelaces to securing surgical su-
tures. But knots can also be a nuisance, crop-
ping up in long hair, electrical cords, and other
inconvenient places. Equally important, knots
are interesting subjects for scientific inquiry and
have attracted increasing attention from physi-
cists and biophysicists: Various physically rel-
evant systems have an undeniable capacity to
become entangled. Notable examples include
biopolymers such as DNA and proteins. An
understanding of these knots beyond the con-
fines of mathematical topology and theoretical
physics is essential to bring about new discov-
eries and practical applications in biology and
Here we describe some recent experimen-
tal and theoretical efforts in the biophysics
of knotting. We begin with a brief introduc-
tion to knot classification. We then explore a
variety of topics related to the biophysics of
knotting. The organization of these topics re-
flects our attempt to address the following gen-
eral questions: Where and how do knots form?
How likely are knots to form? What are some
properties of knots and knotted systems? In
what processes do knots play a role? When and
how do knots disappear? In addressing these
questions, we aim for a qualitative presentation
of recent works, emphasizing the diversity of
methods and results without delving extensively
into technical details. A more comprehensive
treatment of specific topics can be found in
books (1, 19) and in the various cited reviews.
The ability to discern and classify different
kinds of knots is an essential requirement for
understanding biophysical processes involving
knots. The mathematical field of Knot theory
offers powerful tools for detecting and classify-
ing different knots (1). A knot is a topological
state of a closed, nonintersecting curve. Two
closed curves contain knots of the same type
if one of the curves can be deformed in space
to match the other curve without temporarily
opening either curve. In practice, a 3D knot-
ted curve is mathematically analyzed by first
projecting it onto a 2D plane and then exam-
ining the points, known as crossings, where
the curve crosses itself in the 2D projection
(Figure 1a). Note that when we talk about
knots in open curves, such as a linear string or
DNA molecule, we are imagining that the ends
of those curves are connected using a sensible,
well-defined procedure to yield corresponding
closed curves (Figure 1b).
The absence of a knot is called the unknot
or trivial knot. It can always be rearranged to
yield a projection with zero crossings. Knots, in
contrast, give rise to projections with nonzero
numbers of crossings. The minimum number
of crossings, C, is an invariant for any arrange-
ment of a closed string with a given knot. C
is often used to classify knots into different
types. Specifically, each knot type is denoted
as CS, where Sis a sequence number within
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41 Figure-eight
Closed curve
Open curve
q–1 + q–3q–4
q2q + 1 –q–1 + q–2
q–2 + q–4q–5 + q–6q–7
q–1q–2 + 2q–3q–4 + q–5q–6
Figure 1
Knot types and features. (a) Knots formally exist only in 3D curves (left). Knot projections are 2D
representations of knots (right). (b) Knot-like conformations in open curves are often encountered in
biophysics (left). To analyze such knots, their loose ends must be connected, according to some procedure, to
obtain a closed curve (right). (c) Projections of the four simplest nontrivial knot types, with the corresponding
CSdenominations and Jones polynomials (see text for definition of CS) (adapted from http://katlas.math. Page). (d) The size of a knot, SK, in a polymer may be less than the size of the
polymer, SP, containing the knot. (e) In a slip link arrangement, entropic competition between the knotted
loops causes the ring to squeeze one of the knots. The size of the latter can be deduced from the position of
the ring. Adapted from Reference 64. ( f) The size of a tight knot can be estimated from the volume of the
enclosing ideal knot representation: SK(D2L)1/3, where Dand Lare the diameter and length of the outer
tube. Adapted from Reference 39. ( g) Square and granny knots can tie ropes together but unravel easily at
the molecular scale. Slipknots in proteins have been studied to assess the effects of knots on stability.
the family of knot types having the same C
(Figure 1c). Some common knots are also re-
ferred to by name: 31and 41are called trefoil
and figure-eight, respectively. The number of
different knot types having the same Cincreases
rapidly with C: There are only 3 knots with 6
crossings, but 1,388,705 knots with 16 cross-
ings (42). The number Cserves as a measure of
knot complexity.
Simple knots can be distinguished visually
by comparison with published tables, but ex-
tensively knotted systems require mathematical
methods of knot classification. One ingenious
strategy for classifying knots is to transform
a knot projection into a special polynomial
formula, which depends on the knot type but
not on any particular projection. Comparing
this polynomial with those enumerated in Biophysics of Knotting 351
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Jones polynomial:
a mathematical
expression that can be
computed by analyzing
the crossings in
any particular 2D pro-
jection of a knot, and
serves as fingerprint
to uniquely identify
the type of the knot
double-stranded DNA
enzymes that allow
single or double
strands of DNA to pass
through other single
or double strands of
DNA to change the
topology of a closed
dsDNA molecule
knot tables enables the identification of the
knot type from a given projection. Examples
of such polynomials include the Alexander,
Jones, and HOMFLY polynomials (1). V.F.R.
Jones was awarded the famous Fields Medal in
mathematics in 1990 for his groundbreaking
discovery of the Jones polynomial. These
polynomials occasionally fail to distinguish
different knots and become computationally
prohibitive with projections of many crossings,
but they are invaluable tools for analyzing the
vast majority of simpler knots.
Knots can form via two general mechanisms:
threading of loose ends or breaking and re-
joining of segments. Linear double-stranded
DNA (dsDNA) molecules undergoing random
cyclization in solution exemplify the first mech-
anism. Cyclization is possible when the ends
of a linear dsDNA molecule have comple-
mentary single-stranded overhangs. A knotted
molecule results whenever the molecule’s ends
pass through loops within the same molecule
before joining (90, 95).
Knots can arise from cyclization of viral ge-
nomic DNA from tailless P2 and P4 phages
(57, 58) and intact P4 deletion mutants (119)
(Figure 2a,b). At least half of the knots form
while the DNA is still in the capsid (6). Pro-
duction of knotted DNA from P4 phages (45) is
useful for assessing the activity and inhibition of
enzymes such as DNA topoisomerases, which
can change the topology of DNA. Mutant P4
phages generate knots even in nonnative DNA
molecules. The genomic DNA of phage P4 is
11.2 kbp long, but these capsids produce knots
in plasmids as short as 5 kbp (106). The yields of
knotted DNA were >95%, much greater than
yields from random cyclization of DNA in so-
lution (95). Although the specific mechanism
of knot formation in viruses remains unclear,
both confinement and writhe bias seem to play
an important role (5).
The second DNA knotting mechanism,
which relies on the breaking and rejoining of
chain segments, is facilitated by enzymes such as
topoisomerases and recombinases. Fundamen-
tal insights into the mechanisms of these and
other enzymes have resulted from detailed anal-
yses of knots in DNA (13, 59, 98, 115).
DNA topoisomerases are classified as type I
or type II (93). Type I DNA topoisomerases
temporarily break a single DNA strand and
allow it to pass through the complementary
strand (7). Knotted dsDNA results when cir-
cular dsDNA is nicked or gapped and the en-
zyme breaks a strand at a location opposite
the nick (23). In contrast, type II topoisom-
erases temporarily break both strands in one
segment of dsDNA, allowing one segment to
pass through another intact segment before the
strands are chemically rejoined (72, 93). Type
II DNA topoisomerases introduce knots into
supercoiled circular DNA in vitro (114), pro-
viding a way to assess the DNA supercoiling
activity of other enzymes, such as condensins
(79). In vivo, type II DNA topoisomerases re-
move knots from DNA. Such knots arise nat-
urally during replication, as evidenced by the
presence of knots in partially replicated plas-
mids (73, 96).
Recombinases are responsible for site-
specific genetic recombination of DNA. Like
topoisomerases, they operate by breaking and
rejoining single or double strands. Their func-
tion, however, is to insert, excise, or invert a seg-
ment delimited by appropriate recombination
sites (37). When the substrate is supercoiled
DNA, recombinases yield knotted DNA (13).
The latter was used to assay the unknotting ac-
tivity of Escherichia coli topoisomerase IV (26).
Besides DNA, long peptides may also
become knotted. Several proteins exhibit a
knotted conformation in their native state
(Figure 2g), which only becomes evident
when the backbone is closed and smoothed by
numerical methods (103). Presumably, these
proteins become entangled while they fold into
their native structures (62). Thus, the ability of
protein backbones to form knots complicates
the already difficult problem of explaining
how proteins fold (62, 104, 120). Nevertheless,
recent studies on knotted proteins are rapidly
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100 nm
10 µm
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7
18 18
250 nm
250 nm 250 nm
250 nm
Figure 2
Knotted biophysical systems. (a) Negative stain electron micrograph of P2 virions. Adapted with permission from Reference 21.
(b) Conformations of packed P4 genome as determined by coarse-grained molecular dynamics simulations. Reprinted with permission
from Reference 89. (c) Atomic force microscopy images of knotted DNA, isolated from P4 phage capsids and strongly (left column)or
weakly (right column) adsorbed on mica surface. Reprinted with permission from Reference 34. (d) Optical tweezers tying a trefoil knot
in a fluorescently labeled actin filament. Adapted with permission from Reference 3. (e) Left panel: electrophoretic mobility of knotted
DNA plasmids in agarose gel increases with minimum number of crossings, C. Lane 1: unknotted DNA; lanes 2–7: individual knotted
DNA species isolated by prior gel electrophoresis. I and II are the positions of markers for circular and linear DNA, respectively. Right
panel: electron micrographs of knotted DNA molecules isolated from gel bands (left column), interpretation of crossings (middle column),
and deduced knot types (right column). The molecules were coated with Escherichia coli RecA protein to enhance visualization of DNA
crossings. Adapted with permission from Reference 23. ( f) Knotted DNA from bacteriophage P4 capsids separated by agarose gel
electrophoresis at 25V for 40 h (dimension I) and at 100V for 4 h (dimension II). Adapted with permission from Reference 105.
(g) Structure of the chromophore-binding domain of the phytochrome from Deinococcus radiodurans (left) containing a figure-eight knot
(right). Reprinted with permission from Reference 12. (h) An umbilical cord (diameter 2 cm) with a composite knot. Reproduced with
permission from Reference 20. (i) 3D image, obtained by 4D ultrasonography, of a knotted umbilical cord next to the fetal face.
Adapted with permission from Reference 18. Biophysics of Knotting 353
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Molecular dynamics
(MD): a simulation
technique in which the
Newtonian equations
of motion for a system
of many particles are
approximately, but
efficiently, integrated
over time to observe
the evolution of the
system and to
determine its statistical
mechanical properties
Minimum crossing
number: the
minimum number of
points where a knotted
curve crosses over
itself when viewed in
any 2D projection
gathering new clues. For example, a 52knot
is present in the human protein ubiquitin
C-terminal hydrolase UCH-L3, which is
involved in the recycling of ubiquitin. After
denaturation, this protein folds back into
its native knotted conformation without any
help from chaperones, suggesting that knot
formation in UCH-L3 is encoded by the amino
acid sequence (2). Molecular dynamics (MD)
simulations of the homodimeric α/β-knot
methyltransferases YibK and YbeA, both
of which feature a trefoil knot, and of the
proteins AFV3–109 and thymidine kinase,
both of which feature a slipknot (100), have
suggested that knots form through a slipknot
intermediate, rather than by threading one
terminus through a backbone loop.
Although they arise naturally, nanoscale
knots can also be tied directly by humans. In
particular, polystyrene beads attached to the
ends of actin filaments or dsDNA molecules
were maneuvered with optical tweezers to con-
struct trefoil knots (Figure 2d) (3). Using simi-
lar techniques, Bao et al. (8) tied the more com-
plex knots 41,5
2, and 71in dsDNA. Trefoil
and figure-eight knots can be created also in
single-stranded DNA and RNA by exploiting
self-assembly of nucleic acids (94). A refined
approach, based on annealing and ligation of
DNA oligonucleotides with stem and loop re-
gions, yielded knots with three, five, and seven
crossings (15).
As interesting as the knots found in
biomolecules are those encountered in
biomedical contexts. For example, following a
ventriculoperitoneal shunt operation to relieve
excessive buildup of spinocerebral fluid, the
surgically implanted catheter tube has been
found in some cases to become spontaneously
knotted, thus blocking drainage (33). Also
notable is the knotting of umbilical cords
during human pregnancy, a phenomenon re-
ported in about one percent of live births (35)
(Figure 2h). Although these knots are not
always harmful (20, 61), they can sometimes
be fatal (22, 97). Recent advances in under-
standing the dynamics of knotting in agitated
strings (83) as well as technological advances in
ultrasound imaging (18) (Figure 2i) promise to
facilitate the study and diagnosis of umbilical
To understand the mechanisms of knot-
ting, physicists have studied macroscopic model
systems that are easier to implement and con-
trol than their molecular counterparts. For in-
stance, a hanging bead chain shaken up and
down at constant frequency occasionally pro-
duces trefoil and figure-eight knots (10). Re-
cently, our group investigated tumbling a string
in a rotating cubic box, which rapidly produced
knots (83) (Figure 3a). Determination of the
Jones polynomial for the string after only ten
1-Hz revolutions of the box revealed a vari-
ety of complex knots with a minimum cross-
ing number Cas high as 10. The resulting knot
distribution was well explained by a model that
assumed random braid moves of the ends of a
coiled string (Figure 3c).
As knots arise in several biophysical systems,
one may wonder how likely are such knots to
form. This basic question was posed in 1962 by
the famous biophysicist Max Delbr ¨
uck (27) and
since then has been frequently investigated by
polymer physicists. Grosberg (38) recently re-
viewed some key results on the probability of
knotting in polymers. Most notably, the prob-
ability of finding a knot of any type K, includ-
ing the unknot, in an N-step self-avoiding ran-
dom walk is predicted to be PKeN/N0,
where the constant N0is model dependent,
and the prefactor depends on the knot type.
The overall probability of finding a nontriv-
ial knot and the average complexity of knots
are thus predicted to increase with increasing
polymer length, and the probability of find-
ing the unknot is predicted to approach zero as
N→∞. Besides N, other parameters, such as
solvent quality, temperature, and confinement,
affect knotting probability. These nontrivial ef-
fects have been investigated theoretically or
through computer simulations and are summa-
rized in several excellent reviews (46, 75, 102,
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The knotting probability depends strongly
on the space available to the polymer. Early
numerical studies of self-avoiding random
walks found the knotting probability of ring
polymers to increase with increasing confine-
ment by a sphere (70). More recent Monte
Carlo (MC) simulations of phantom polymer
rings, which are free from topological con-
straints, found that knot formation is inhib-
ited when the radius of the confining sphere
becomes too small (68). Also, in the case of um-
bilical cords, confinement of the growing fe-
tus in the amniotic sac was theorized to hinder
knot formation (35). Thus, effects of confine-
ment depend on the specific physical context or
theoretical assumptions.
Spatial confinement also affects knotting of
DNA in phage capsids. MC simulations of P4
phage DNA, modeled as a semiflexible cir-
cular self-avoiding random walk in a confin-
ing sphere, reproduced the experimentally ob-
served prevalence of chiral knots over achiral
knots (69). However, contrary to experimental
results, 52knots outnumbered 51knots, pos-
sibly owing to insufficient confinement or to
inaccurate modeling of DNA dynamics within
the capsid. In another study, the packaging of
DNA in viral capsids, which has been studied
experimentally (84), was modeled using random
spooling polygons without excluded volume or
electrostatic interactions (4). This work repro-
duced qualitatively both the chiral bias and the
distribution of knot types observed with tailless
mutants of P4 bacteriophages.
Effects of spatial confinement on knotting
probability were evident in our experiments
with macroscopic strings in a rotating box (83).
As the string length was increased, the knot-
ting probability did not approach the theoret-
ical limit of 1 expected for self-avoiding ran-
dom walks (Figure 3b). The lower probability
observed was due to finite agitation time and
to the restricted motion experienced by long
strings of nonzero stiffness within a box of fi-
nite size. In preliminary work (D. Meluzzi &
G. Arya, unpublished data), we reproduced and
further quantitatively studied these effects us-
ing MD simulations of macroscopic bead chains
Box revolutions
Tow ard
Knot probability
String length (m)
String length (m)
Knot probability
Figure 3
Macroscopic string knotting. (a) Examples of initial (left) and final (right)
configurations of a string tumbled in a 30-cm cubic box rotated ten times
at 1 revolution per second. Adapted with permission from Reference 83.
(b) Measured knotting probability versus string length, L, in the rotating
box. Reproduced with permission from Reference 83. (c) Simplified model for
the formation of knots in the random tumbling. Top: End segments lie parallel
to coiled segments. Bottom: Threading of an end segment is modeled by a
series of random braid moves. Reproduced with permission from Reference 83.
(d) Molecular dynamics (MD) simulations of a string in a rotating box,
mimicking the above experiment. The string was represented as a bead chain
subject to bending, excluded volume, and gravitational potentials. (e) Estimated
knotting probability versus string length, based on 33 tumbling simulations per
point. Knots were detected by MD simulations in which the string ends were
pulled either toward (light purple line and dots) or away from (dark purple line
and crosses) each other until the knot was tight or disappeared. ( f) Simulated
knotting probability versus box revolution. Values were determined as in panel e.
MC: Monte Carlo
in a rotating box (Figure 3d,e). We have also
calculated the probability of knot formation as
a function of box revolutions, predicting a rapid
formation of knots: 80% of the simulated tri-
als produced a knot after only two revolutions Biophysics of Knotting 355
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ANRV411-BB39-18 ARI 2 April 2010 11:20
AFM: atomic force
(Figure 3f). Such simulations may offer a con-
venient route for dissecting the mechanisms of
knot formation.
Knotted systems can be studied in greater depth
by analyzing a variety of static properties. Here
we give a few examples of these properties and
describe recent progress in studying biophysi-
cally relevant systems.
Size of Knots and Knotted Systems
Several knot size measures have been investi-
gated experimentally, theoretically, and compu-
tationally (74). In polymers, knot size may dif-
fer from the size of the polymer (Figure 1d).
Polymer size is typically characterized by the ra-
dius of gyration, Rg, i.e., the average root mean
square distance between each segment and the
center of mass. For linear polymers, RgNν,
where ν=0.5 for pure random walk chains
and ν0.588 for self-avoiding random walk
chains (56) or chains with excluded volume (24,
29). The same self-avoiding random walk scal-
ing exponent has been observed for knotted
and unknotted circular polymers in the limit
of N→∞, as determined by MC simulations
(38, 75).
The scaling in Rgwas investigated exper-
imentally via fractal dimensional analysis of
atomic force microscope (AFM) images of cir-
cular DNA molecules strongly and weakly ad-
sorbed on a mica surface (34) (Figure 2c).
Strong adsorption gave ν0.60, close to
ν0.588 for 3D polymers, suggesting that it
projects 3D conformations onto the surface. In
contrast, weak adsorption yielded ν0.66, in-
termediate between ν0.588 for 3D polymers
and ν=0.75 for 2D polymers, suggesting a
partial relaxation of 3D conformations into a
quasi-2D state (34). A similar intermediate scal-
ing exponent was predicted by MC simulations
of dilute lattice homopolymers confined in a
quasi-2D geometry (41).
As knots shrink, their size or length can
be investigated separately from the size of the
overall chain (Figure 1d). In ring polygons,
knot size can be determined from the short-
est portion of the polygon that, upon appropri-
ate closure, preserves the topology of the chain
(50, 64, 65). Another computational method in-
volves introducing a slip link that separates two
knotted loops within the same ring polygon.
Entropic effects expand one loop at the expense
of the other, and the average position of the slip
link defines the length of the smaller knot (64)
(Figure 1e).
The size of tight knots in open chains has
also been studied (81). Open chains cannot be
knotted in a strict mathematical sense. For the-
oretical arguments, knot size can be deduced
from the volume of a maximally inflated tube
containing the knot (39) (Figure 1f). Accord-
ingly,it was predicted that the size of sufficiently
tight and complex knots in an open polymer
should depend on a balance between the en-
tropy of the chain outside the knot and the
bending energy of the chain inside the knot.
If the chain tails are sufficiently long, the knot
should neither shrink nor grow on average (39).
In one study, the size of tight knots in stretched
polyethylene was predicted from the distribu-
tion of bond lengths, bond angles, and torsion
angles along the chain, suggesting that trefoil
knots involve a minimum of 16 bonds (121).
For comparison, ab initio calculations predicted
a minimum of 23 bonds (92). Furthermore, the
extent of tight knots has been determined ex-
perimentally. Fluorescence measurements indi-
cated that trefoil knots in actin filaments can
be as small as 0.36 μm (3). Similar measure-
ments on 31,4
2, and 71knots in linear
dsDNA yielded knot lengths of 250–550 nm for
molecules stretched by a tension of 1 pN (8).
Knots can be tightened on proteins as
well. The figure-eight knot present in the
chromophore-binding domain (CBD) of the
phytochrome from Deinococcus radiodurans
(Figure 2g) was tightened with an AFM to a
final length of 17 amino acids (12). Similarly,
simulations of the 52knot in ubiquitin carboxy-
terminal hydrolase L1 (UCH-L1) using a
Go-like model suggested minimum lengths of
either 17 or 19 residues, depending on the final
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location of the tight knot along the backbone
(101). More accurate all-atom MD simulations
with explicit water found tight 31and 41knots in
stretched model peptides to be about 13 and 19
amino acids long, respectively, in good agree-
ment with the experiments (32). Curiously, in
these simulations, a tight 41knot in polyleucine
was found to trap a single water molecule, which
escaped upon further tightening.
Assessing the size of tight protein knots is
important for understanding their biological
roles. Bulky knots could hamper the threading
of polypeptides through the narrow pore of the
proteasome, possibly protecting certain knot-
ted proteins from rapid degradation (109). This
hypothesis was supported by Langevin dynam-
ics simulations of the translocation of a test pep-
tide through a narrow channel (radius 6.5 ˚
The presence of a 52knot in the peptide re-
duced the translocation rate by two orders of
magnitude, suggesting that knots may indeed
hinder protein degradation by the proteasome
Knot Localization
Several studies have addressed the localization
of knots in a polymer (Figure 1d), and vari-
ous aspects of knot localization, including the
role of entropic and electrostatic effects, have
been reviewed (38, 48, 75). Knot localization
within a closed knotted chain results from the
gain in entropy by a long unentangled loop,
which causes the knotted portion of the chain
to shrink (38). This effect could be mimicked by
vibrating a twisted bead chain on a horizontal
plate (40). The same phenomenon was inferred
from the size distributions of simple knots in
random closed chains of zero thickness (50).
Numerically, knots are localized when
their average size grows slower than the
length Nof the chain, or limN→∞ /N=0.
When ∼Nt, with t<1, the knot is weakly
localized (63). The value of tdepends on
solvent quality. MC simulations of trefoil knots
in circular self-avoiding polygons on a cubic
lattice (64) yielded t0.75 in good solvent
and t1 in poor solvent, indicating that knots
Langevin dynamics:
a computationally
efficient MD
refinement that
accounts for the effects
of random collisions of
solvent molecules with
the system
are weakly localized in the swollen phase but
are delocalized in the collapsed phase. Similar
scaling exponents have been obtained for linear
polyethylene in good and poor solvent via MC
simulations (108). These exponents have been
confirmed by analyzing the moments of the
probability distributions of knot lengths for
different types of knots (65).
Knot localization was observed in AFM im-
ages of circular DNA weakly adsorbed on a mica
surface (34). Moreover, MC simulations of ring
polymers adsorbed on an impenetrable attrac-
tive plane have predicted that lowering the tem-
perature leads to strong knot localization, i.e.,
becomes independent of N(63). Knot lo-
calization in DNA is important because it may
facilitate the creation of segment juxtapositions
and thereby may enhance the unknotting activ-
ity of type II DNA topoisomerases (59).
Strength and Stability
of Knotted Systems
Rock climbers are well aware that knots weaken
the tensile strength of ropes. Similar ef-
fects hold for knotted molecules. Using Car-
Parrinello MD simulations, it was shown that a
linear polyethylene molecule with a trefoil knot
breaks at a bond just outside the entrance of the
knot, where the strain energy is highest, but is
still only 78% of the strain energy needed to
break an unknotted chain (92). Hence, the knot
significantly weakened the molecule. Similarly,
when the ends of single actin filaments con-
taining a trefoil knot were pulled with optical
tweezers, the filaments were found to break at
the knot with pulling forces of 1 pN, indi-
cating a decrease in tensile strength by a fac-
tor of 600 (3). On a macroscopic scale, exper-
iments with fishing lines and cooked spaghetti
confirmed that rupture occurs at the knot en-
trance, where the curvature was predicted to
be the highest, causing local stresses that favor
crack propagation (80).
Ordinary strings can be tied strongly with a
square or granny knot (Figure 1g), but if two
polymer chains were tied in this fashion and
then pulled apart, the knot would invariably Biophysics of Knotting 357
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Brownian dynamics:
Langevin dynamics
with zero average
acceleration, typically
used to simulate
overdamped systems
Wormlike chain: a
semiflexible polymer
slip. However, Langevin dynamics simulations
found that, when pulled strongly, smooth poly-
mers untie more quickly than bumpy polymers
(53). Increasing the pulling force makes the en-
ergy landscape of bumpy polymers more cor-
rugated, thus hindering the thermally activated
slippage of the strands.
Although they weaken tensioned strings,
knots may actually increase the stability of
certain systems. Increased stability could ex-
plain the presence of knots in some proteins
(120). To test this effect, the deep slipknot
(Figure 1g) in the homodimeric protein alka-
line phosphatase from E. coli was cross-linked
via a disulfide bridge between monomers, ef-
fectively increasing the knotted character of the
overall dimer (52). A 10C increase in melting
temperature of this cross-linked dimer, relative
to a control dimer cross-linked outside the slip-
knot loops, suggested that knots can increase
the thermal stability of proteins. Yet, unfolding
experiments with the 41-knotted CBD of the
phytochrome from D. radiodurans found that
the knot did not significantly enhance mechani-
cal stability (12). It was suggested, however, that
this knot might serve to limit the possible mo-
tions induced by the chromophore on the CBD
upon light absorption.
Finding and characterizing knots in biophysi-
cal systems naturally lead to an investigation of
dynamic processes involving knots. We focus
on three prominent examples: diffusion, elec-
trophoresis, and unknotting.
Knot Diffusion
As discussed above, knots may become local-
ized. Once localized, a knot can diffuse along
the chain. The resulting motion is governed
by the inability of intrachain segments to pass
through one another. The same constraints ex-
ist for intermolecular entanglements and thus
dominate the dynamics of concentrated poly-
mer solutions and melts. Such systems are well
described by the reptation model (24, 29), for
which P.G. de Gennes was awarded the Nobel
Prize in Physics in 1991. This model assumes
that each polymer molecule slides within an
imaginary tube tracing the molecule’s contour.
In agreement with this model, experiments have
shown that linear DNA molecules larger than
50 kbp, in solutions more concentrated than
0.5 mg ml1, exhibit tube-like motion, expe-
rience tube-like confining forces, and diffuse as
predicted by reptation theory (77, 85, 86).
The notion that reptation may also govern
knot diffusion was supported experimentally by
Bao et al., with 31,4
2, and 71knots in sin-
gle, fluorescently stained DNA molecules (8).
The knots were seen as bright blobs diffusing
along the host DNA. The diffusion constants,
D, of the knots were strongly dependent on
knot type, and the drag coefficients deduced
from Dwere consistent with a self-reptation
model of knot diffusion (8). Brownian dynamics
simulations of a discrete wormlike chain model
of DNA yielded Dvalues of the same mag-
nitude as the values measured experimentally
(110). Moreover, Langevin dynamics simula-
tions of knot diffusion in tensioned polymer
chains found Dvalues consistent with a sliding
knot model in which the friction between the
solvent and the knot dominates knot dynam-
ics at low tensions, whereas internal friction of
the chain dominates the dynamics at high ten-
sions (43). In the absence of tension, knot dif-
fusion was proposed to consist of two reptation
modes, one due to asymmetric self-reptation of
the chain outside the knot, the other due to
breathing of the knot region. The latter mo-
tion allows the knot to diffuse in long chains
In addition to diffusing along polymers,
knots can affect the diffusion of the polymers
themselves. Brownian dynamics simulations of
ring polymers with knots of up to seven cross-
ings found that the ratio of diffusion coeffi-
cients for knotted and linear polymers, DK/DL,
grows linearly with average crossing number
NAC of ideal knot representations (47). Thus,
intramolecular entanglement seems to speed
up polymer diffusion. Nevertheless, diffusion
358 Meluzzi ·Smith ·Arya
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of knotted polymers may be complicated by
intermolecular topological constraints. For ex-
ample, we have found that circular DNA can
diffuse up to two orders of magnitude slower
when surrounded by linear DNA than when
surrounded by circular DNA of the same con-
centration and length (87). Current reptation
models fail to fully describe these findings, but
qualitatively we believe that unknotted circu-
lar molecules are easily pinned by threading of
linear molecules. Such pinning mechanisms are
likely to affect the diffusion of knotted polymers
as well.
The strong negative charge on DNA molecules
at sufficiently high pH is exploited in agarose
gel electrophoresis to separate DNA molecules
according to size and supercoiling state. For
over two decades, the same technique has
proven invaluable for analyzing knots in re-
laxed circular DNA (31, 55). In seminal exper-
iments with E. coli topoisomerase I, electron
microscopy revealed the topology of knotted
DNA molecules from distinct gel bands (23)
(Figure 2e). Remarkably, each band contained
DNA knots with the same minimum number
of crossings, C, which seemed to control the
electrophoretic mobility of knotted DNA.
A follow-up study (99) uncovered a surpris-
ingly linear relationship between the previously
reported electrophoretic migration distances of
DNA knots and the average number of cross-
ings, NAC, in the ideal geometric representa-
tions (49) of those knots. Because NAC is lin-
early related to the sedimentation coefficient,
which provides a measure of molecular com-
pactness, it was concluded that DNA knots with
many crossings are more compact and there-
fore migrate faster through the gel than DNA
knots with fewer crossings (112). At high elec-
tric fields, however, the linear relationship be-
tween migration rate and NAC no longer holds.
This change in behavior has been exploited in
2D gel electrophoresis to improve the separa-
tion of knotted DNA (105) (Figure 2f) and has
been reproduced in MC simulations of closed
self-avoiding random walks (117). Such change
was attributed to increased trapping of knotted
DNA by gel fibers at high electric fields. The
distribution of trapping times obeyed a power
law behavior consistent with the dynamics of
a simple Arrhenius model (116), thus enabling
the estimation of the critical electric field as-
sociated with the inversion of gel mobility of
knotted DNA.
Despite considerable modeling efforts and
extensive use of DNA electrophoresis, a com-
plete theory that accurately predicts DNA mo-
bility as a function of electric field and poly-
mer properties is still lacking. Novel separation
techniques provide additional motivation for
understanding the dynamics of knotted poly-
mers in electric fields (76, 54).
Knot removal can occur via two main mecha-
nisms: unraveling and intersegmental passage.
Unraveling is the reverse of the threading-
of-loose-ends mechanism that allows knots to
form in open chains. A clear example of unravel-
ing involved the agitation of macroscopic gran-
ular chains on a vibrating plate. A tight trefoil
knot unraveled with an average unknotting time
that scaled quadratically with chain length (11).
This scaling behavior is reminiscent of knot dif-
fusion in linear polymers predicted by a mech-
anism of “knot region breathing” (67).
As with diffusion, the unraveling of knots
in polymers is affected by external constraints.
MD simulations of polyethylene melts found
that macromolecular crowding causes trefoil
knots to unravel through a slithering motion
with alternating hairpin growth and shrink-
age, resulting in a scaling exponent of 2.5 for
the average unknotting time (51). Similarly,
a tight trefoil knot in a polymer constrained
within a narrow channel was predicted to un-
ravel through simultaneous changes in size and
position, with a cubic dependency of mean knot
lifetime on the polymer length (71).
A situation in which knots must unravel
rapidly is during the ejection of DNA from vi-
ral capsids upon cell infection. The electrostatic Biophysics of Knotting 359
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repulsions and entropic penalty experienced by
DNA molecules confined within phage cap-
sids result in high internal forces (78) of up
to 100 pN, according to measurements by
optical tweezers (84). Such forces are capable
of removing DNA knots in some viruses upon
exit from the capsid through a narrow opening,
as confirmed by MD simulations of a coarse-
grained polymer chain initially confined within
a sphere (66). In this system, the ejection dy-
namics were controlled primarily by the rep-
tation of the polymer through the knot (66), a
process presumably similar to the knot diffusion
observed experimentally by Bao et al. (8).
The second general mechanism of unknot-
ting is intersegmental passage, which can also
lead to knot formation. This mechanism con-
sists of passing chain segments through tempo-
rary cuts on other segments of the same chain.
This procedure is carried out at the cellular level
by type II DNA topoisomerases, which use ATP
to lower the fraction of knotted DNA below
the levels observed in random cyclization (91).
Knotting and catenation of DNA interfere with
vital cellular processes (59), including repli-
cation (9), transcription (25), and chromatin
Hairpin-like G segment model
Hooked juxtaposition model
Figure 4
Models of unknotting by type II DNA topoisomerases. (a) In the hairpin-like G
segment model (111), the enzyme binds to the G segment and sharply bends it
into a hairpin-like structure; the T segment is then allowed to pass only from
the inside to the outside of the hairpin. Adapted from Reference 111. (b) The
hooked juxtapositions model (59) assumes that hooked juxtapositions form
frequently in knotted DNA and that the enzyme binds to DNA only at these
juxtapositions. Once bound, the enzyme catalyzes the intersegmental passage.
Adapted from Reference 59.
remodeling (88). Hence, type II DNA topo-
isomerases have been an attractive target for an-
ticancer drugs (28) and antibiotics (107). The
molecular mechanism by which type II DNA
topoisomerases break, pass, and rejoin dsDNA
is fairly well understood (36, 72, 93), but the
higher-level mechanism that leads to a global
topological simplification of DNA is a subject
of continuing debate (59, 111).
A few interesting models of type II DNA
topoisomerases action have been proposed (59,
111). Two of these models seem consistent with
the structure of yeast topoisomerase II (30). In
the first model (113) (Figure 4a), the enzyme
binds to a DNA segment, known as the G seg-
ment, and bends it sharply into a hairpin-like
structure. Next, the enzyme waits for another
DNA segment, called the T segment, to fall
into the sharp bend. Then, the enzyme passes
the T segment through a break in the G seg-
ment, from the inside to the outside of the
hairpin. Indeed, MC simulations of this model
using a discrete wormlike chain found the pres-
ence of hairpin G segments to lower the steady-
state fraction of knots by a factor of 14. This
value, however, is less than the maximum of 90
observed in experiments with type II DNA
topoisomerases (91).
The other model of topoisomerase action
(Figure 4b) is based on two assumptions (14).
First, hooked juxtapositions, or locations where
two DNA segments touch and bend around
each other, occur more frequently in globally
linked DNA than in unlinked DNA. Second,
the enzyme binds preferentially to DNA at
hooked juxtapositions. Once bound, the en-
zyme passes one segment through the other.
Hence, type II DNA topoisomerases disentan-
gle DNA by selectively removing hooked jux-
tapositions. This model’s ability to predict a
significant steady-state reduction of knots and
catenanes below topological equilibrium was
supported by MC simulations with lattice poly-
gons (60) and freely jointed equilateral chains
(17). Nonetheless, these models of DNA may
not be sufficiently accurate (111). Additional
simulations with wormlike chains may clarify
the significance of hooked juxtapositions (59).
360 Meluzzi ·Smith ·Arya
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The negative supercoiling state of DNA also
seems to affect the results of topoisomerase ac-
tion. Early MC simulations of a wormlike chain
model of circular DNA suggested that super-
coiling reduces the free energy of highly chi-
ral knots below that of unknotted DNA, effec-
tively favoring knot formation in the presence
of type II DNA topoisomerases (82). A more re-
cent study explicitly accounted for the changes
in linking number introduced by DNA gyrase
after each intersegmental passage to maintain
a constant level of torsional tension in DNA
(16). The resulting knot probability distribu-
tions suggested that negative supercoiling op-
poses segment passages in directions that lead to
knotting. Thus, the supercoiling action of DNA
gyrase may be the principal driver toward low
levels of DNA knotting in vivo (16).
Knots have been discovered in a wide range of
systems, from DNA and proteins to catheters
and umbilical cords, and have thus attracted
much attention from biophysicists. In this re-
view we have explored a variety of topics in
the biophysics of knotting. Despite the tremen-
dous progress made in this field by theoretical
and experimental studies, many open questions
remain, which are summarized below. These
questions could inspire new research efforts.
In particular, computer simulations and single-
molecule experiments hold great promise in
clarifying knotting mechanisms, while emerg-
ing techniques for high-resolution molecular
imaging should facilitate the study of knotting
processes inside the cell.
1. The Jones, Alexander, and HOMFLY polynomials from knot theory are powerful tools
for analyzing and classifying physical knots.
2. An agitated string forms knots within seconds. The probability of knotting and the knot
complexity increase with increasing string length, flexibility, and agitation time. A simple
model assuming random braid moves of a string end reproduces the experimental trends.
3. Knots are common in DNA and the different knot types can be separated by using elec-
trophoresis techniques, which exploit the varying mobility of knotted DNA in entangled
media in response to electric fields.
4. Knots have recently been discovered in proteins. The formation mechanisms and the
biological function of these knots are just beginning to be studied.
5. Knots can be generated artificially in nanoscale systems and used to study fundamentals of
knot dynamics. Localized knots in DNA diffuse via a random-walk process that exhibits
interesting trends with respect to tension applied across the molecule.
6. Confinement and solvent conditions not only play an important role in determining the
types and sizes of knots that appear in biophysical systems, but also affect the diffusion
and localization of knots.
7. Knots appear to weaken strings under tension but can have a stabilizing effect on knotted
systems such as proteins.
8. DNA topoisomerases are enzymes that play an important role in the disentanglement
of DNA, and their mechanism of topological simplification is only now beginning to be
understood. Biophysics of Knotting 361
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1. The function of knotted structures within proteins and the mechanism by which these
knots form remain mysterious. How do knots form in proteins? Are chaperones needed
to fold knotted proteins? How do proteins benefit from having knotted backbones?
2. The effect of macromolecular crowding on the knotting dynamics of different biopoly-
mers within the cell has not been examined so far. This effect could be important for
understanding knotting in vivo.
3. The transitions of knots from one type to another in both open and closed chains are
far from fully understood. Do these transitions follow thermodynamic probabilities and
patterns or is the process chaotic? What are the dynamics of these transitions? How do
they depend on the type of agitation and chain (open versus closed)?
4. The formation of knots in human umbilical cord and surgically implanted shunt tubes
is undesirable, but the underlying causes are unclear. Can such processes be accurately
studied and modeled? Can such knots then be avoided?
5. Improved imaging approaches for the visualization of knots, both molecular and macro-
scopic, and both in vitro and in vivo, are needed to facilitate the experimental investigation
of knot dynamics.
6. Are there any useful applications for molecular knots in biotechnology, nanotechnology,
and nanomedicine?
The authors are not aware of any affiliations, memberships, funding, or financial holdings that
might be perceived as affecting the objectivity of this review.
D. Meluzzi was supported partly by the NIH Heme and Blood Proteins Training Grant No.
5T32DK007233–33 and by the ARCS Foundation. The authors are grateful to Dr. Martin
Kenward for helpful comments.
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... Since then, knots have reappeared in various forms in a wide variety of scientific fields. In biophysics [5], their possible functions and the mechanisms by which they originate in DNA and proteins [6] or in other long polymers [7,8] are actively being studied. Topology in chemical synthesis is becoming more and more important as molecules can be synthetically knotted [9,10]. ...
... which penalizes length errors if present [50]. Note that the role of this artificial term is only to prevent the accumulation of numerical errors over time, while the actual inextensibility condition is enforced by the computation of the physical tension term in equation (5). We have checked that the value of the arbitrary constant K does not affect our results (we typically use K = 10 3 ). ...
Full-text available
We consider the problem of an inextensible but flexible fiber advected by a steady chaotic flow, and ask the simple question whether the fiber can spontaneously knot itself. Using a 1D Cosserat model, a simple local viscous drag model and discrete contact forces, we explore the probability of finding knots at any given time when the fiber is interacting with the ABC class of flows. The bending rigidity is shown to have a marginal effect compared to that of increasing the fiber length. Complex knots are formed up to 11 crossings, but some knots are more probable than others. The finite-time Lyapunov exponent of the flow is shown to have a positive effect on the knot probability. Finally, contact forces appear to be crucial since knotted configurations can remain stable for times much longer than the turnover time of the flow, something that is not observed when the fiber can freely cross itself.
... Since then, knots have reappeared in various forms in a wide variety of scientific fields. In biophysics [5], their possible functions and the mechanisms by which they originate in DNA and proteins [6] or in other long polymers [7,8] are actively being studied. Topology in chemical synthesis is becoming more and more important as molecules can be synthetically knotted [9,10]. ...
... which penalizes length errors if present [50]. Note that the role of this artificial term is only to prevent the accumulation of numerical errors over time, while the actual inextensibility condition is enforced by the computation of the physical tension term in equation (5). We have checked that the value of the arbitrary constant K does not affect our results (we typically use K = 10 3 ). ...
Full-text available
We consider the problem of an inextensible but flexible fiber advected by a steady chaotic flow, and ask the simple question of whether the fiber can spontaneously knot itself. Using a one-dimensional Cosserat model, a simple local viscous drag model and discrete contact forces, we explore the probability of finding knots at any given time when the fiber is interacting with the ABC class of flows. The bending rigidity is shown to have a marginal effect compared to that of increasing the fiber length. Complex knots are formed up to 11 crossings, but some knots are more probable than others. The finite-time Lyapunov exponent of the flow is shown to have a positive effect on the knot probability. Finally, contact forces appear to be crucial since knotted configurations can remain stable for times much longer than the turnover time of the flow, something that is not observed when the fiber can freely cross itself.
... However, the separation of knots with desired topology from such melt would be virtually impossible with the current state of the art in analytical methods. Besides artificial polymers, natural sources of knotted polymers are DNA and proteins, where by far the most prevalent knot type is the simplest of knots-a trefoil, as well as knots with more complex topology were discovered [11][12][13]. The DNA knots were first discovered in 1976 [14]. ...
Full-text available
By means of coarse-grained molecular dynamics simulations, we explore chiral sensitivity of confining spaces modelled as helical channels to chiral superstructures represented by polymer knots. The simulations show that helical channels exhibit stereosensitivity to chiral knots localized on linear chains by effect of external pulling force and also to knots embedded on circular chains. The magnitude of the stereoselective effect is stronger for torus knots, the effect is weaker in the case of twist knots, and amphichiral knots do exhibit no chiral effects. The magnitude of the effect can be tuned by the so-far investigated radius of the helix, the pitch of the helix and the strength of the pulling force. The model is aimed to simulate and address a range of practical situations that may occur in experimental settings such as designing of nanotechnological devices for the detection of topological state of molecules, preparation of new gels with tailor made stereoselective properties, or diffusion of knotted DNA in biological conditions.
Nanopore has been studied for a lot of applications in DNA sequencing, multifarious single-molecule detection, detection of molecular conformational change, and biomolecular interaction. There are many categories of nanopores, such as biological nanopores, solid-state nanopores and newly developed hybrid nanopores. All of them play an important role in the detection of DNA structures because of their unique properties, such as the nano-scale pore size, dynamic detection in solution, and low sample loading. This review elaborates on the detection of several DNA structures by nanopore, including the dynamic analysis of conformational changes, and the factors that promote conformational changes. In addition, the detection of DNA origami, a type of artificial structured DNA, is also discussed. In these tests, current signals were produced by the interaction of analyte and nanopore. The signals were often studied by the statistical features of the currents’ amplitude, duration time, or both of them. Finally, taking advantage of the nanopore's ability to detect the conformational changes of these folds, the application of DNA structures detection in nanopore is introduced.
Molecular knots are often prepared using metal helicates to cross the strands. We found that coordinatively mismatching oligodentate ligands and metal ions provides a more effective way to synthesize larger knots using Vernier templating. Strands composed of different numbers of tridentate 2,6-pyridinedicarboxamide groups fold around nine-coordinate lanthanide (III) ions to generate strand-entangled complexes with the lowest common multiple of coordination sites for the ligand strands and metal ions. Ring-closing olefin metathesis then completes the knots. A 3:2 (ditopic strand:metal) Vernier assembly produces +31#+31 and -31#-31 granny knots. Vernier complexes of 3:4 (tetratopic strand:metal) stoichiometry selectively form a 378-atom-long trefoil-of-trefoils triskelion knot with 12 alternating strand crossings or, by using opposing stereochemistry at the terminus of the strand, an inverted-core triskelion knot with six alternating and six nonalternating strand crossings.
Experimental data on the interaction between two knots in deoxyribonucleic acid (DNA) confined in nanochannels produced two particular behaviors of knot pairs along the DNA molecules: (i) widely separated knots experience an attractive interaction but only remain in close proximity for several seconds and (ii) knots tend to remain separated until one of the knots unravels at the chain end. The associated free energy profile of the knot–knot separation distance for an ensemble of DNA knots exhibits a global minimum when knots are separated, indicating that the separated knot state is more stable than the intertwined knot state, with dynamics in the separated knot state that are consistent with independent diffusion. The experimental observations of knot–knot interactions under nanochannel confinement are inconsistent with previous simulation-based and experimental results for stretched polymers under tension wherein the knots attract and then stay close to each other. This inconsistency is postulated to result from a weaker fluctuation-induced attractive force between knots under confinement when compared to the knots under tension, the latter of which experience larger fluctuations in transverse directions.
Recent theoretical studies have demonstrated that the behaviour of molecular knots is a sensitive indicator of polymer structure. Here, we use knots to verify the ability of two state-of-the-art algorithms - configuration assembly and hierarchical backmapping - to equilibrate high-molecular-weight polymer melts. Specifically, we consider melts with molecular weights equivalent to several tens of entanglement lengths and various chain flexibilities, generated with both strategies. We compare their unknotting probability, unknotting length, knot spectra, and knot length distributions. The excellent agreement between the two independent methods with respect to knotting properties provides an additional strong validation of their ability to equilibrate dense high-molecular-weight polymeric liquids. By demonstrating this consistency of knotting behaviour, our study opens the way for studying topological properties of polymer melts beyond time and length scales accessible to brute-force molecular dynamics simulations.
Full-text available
We give statistical definitions of the length, 1, of a loose prime knot tied into a long, fluctuating ring macromolecule. Monte Carlo results for the equilibrium, good solvent regime show that (1) similar to N-t, where N is the ring length and t similar or equal to 0.75 is independent of the knot type. In the collapsed regime below the theta temperature, length determinations based on the entropic competition of different knots within the same ring show knot delocalization (t similar or equal to 1).
Full-text available
KNOTS are usually categorized in terms of topological properties that are invariant under changes in a knot's spatial configuration1–4. Here we approach knot identification from a different angle, by considering the properties of particular geometrical forms which we define as 'ideal'. For a knot with a given topology and assembled from a tube of uniform diameter, the ideal form is the geometrical configuration having the highest ratio of volume to surface area. Practically, this is equivalent to determining the shortest piece of tube that can be closed to form the knot. Because the notion of an ideal form is independent of absolute spatial scale, the length-to-diameter ratio of a tube providing an ideal representation is constant, irrespective of the tube's actual dimensions. We report the results of computer simulations which show that these ideal representations of knots have surprisingly simple geometrical properties. In particular, there is a simple linear relationship between the length-to-diameter ratio and the crossing number—the number of intersections in a two-dimensional projection of the knot averaged over all directions. We have also found that the average shape of knotted polymeric chains in thermal equilibrium is closely related to the ideal representation of the corresponding knot type. Our observations provide a link between ideal geometrical objects and the behaviour of seemingly disordered systems, and allow the prediction of properties of knotted polymers such as their electrophoretic mobility5.
This article describes some of the theoretical and simulation results on random entanglement, and give a few scientific applications. I will prove that, on the simple cubic lattice Z 3 , the probability that a randomly chosen n-edge polygon in Z 3 is knotted goes to one exponentially rapidly with length n (Murphy's Law of entanglement); in other words, all but exponentially few polygons of length n in Z 3 are knotted. Measures of entanglement complexity of random knots and random arcs are discussed as well as application of random knotting to viral DNA packing.
First we survey the literature on knots and links in theoretical physics. Next, we report a numerical study in which equilibrium configurations of ring polymers in an infinite space, or confined to the interior of a sphere, are generated. By using a new algorithm, the a priori probability for the occurrence of a knot is determined numerically. The results are compatible with scaling laws of striking simplicity. We also study the mutual entanglement of links, comparing the Gauss invariant with the Alexander polynomial.
The dynamical evolution of a trefoil knot tied on a polyethylene chain embedded in a melt of similar but unknotted chains is discussed. To explore the dynamic evolution, molecular dynamics simulations are used. A tight open trefoil knot unravels via a sequence of expansions and contractions plus migration while maintaining a definite size that grows with chain length. The associated slithering motion is much faster than for an unknotted chain and together with the hairpin geometry of knot expansion, can provide an efficient means to juxtapose distant sites on the same strand in a dense macromolecular environment.
In this paper the influence of a knot on the structure of a polymethylene (PM) strand in the tensile process is investigated by using the steered molecular dynamics (SMD) method. The gradual increasing of end-to-end distance, R, results in a tighter knot and a more stretched contour. That the break in a knotted rope almost invariably occurs at a point just outside the 'entrance' to the knot, which has been shown in a good many experiments, is further theoretically verified in this paper through the calculation of some structural and thermodynamic parameters. Moreover, it is found that the analyses on bond length, torsion angle and strain energy can facilitate to the study of the localization and the size of a knot in the tensile process. The symmetries of torsion angles, bond lengths and bond angles in the knot result in the whole symmetry of the knot in microstructure, thereby adapting itself to the strain applied. Additionally, the statistical property of the force-dependent average knot size illuminates in detail the change in size of a knot with force f, and therefore the minimum size of the knot in the restriction of the potentials considered in this work for a PM chain is deduced. At the same time, the difference in response to uniaxial strain, between a knotted PM strand and an unknotted one is also investigated. The force-extension profile is easily obtained from the simulation. As expected, for a given f, the knotted chain has an R significantly smaller than that of an unknotted polymer. However, the scaled difference becomes less pronounced for larger values of N, and the results for longer chains approach those of the unknotted chains.
Knots, polyhedra, and Borromean rings with specific structural and topological features can be made from DNA. Biotechnologists have been exploiting the programmability of DNA intermolecular associations for a quarter of a century. These operations have now been applied successfully to branched DNA species to produce complex target structures (for example, the cube shown in the picture) and a nanomechanical device. The assembly of two-dimensional crystals with programmed topographic characteristics demonstrates the simplicity of translating design into surface structures.
The effect of confinement on the phase behavior of lattice homopolymers has been studied using grand canonical Monte Carlo simulations in conjunction with multihistogram reweighting. The scaling of critical parameters and chain dimensions with chain length was determined for lattice homopolymers of up to 1024 beads in strictly 2D and quasi-2D (slab) geometries. The inverse critical temperature scales linearly with the Shultz-Flory parameter for quasi-2D geometries, as it does for the bulk system. The critical volume fraction scales as a power law for all systems, with exponents 0.110 (0.024 and 0.129 (0.004 for the strictly 2D and slab geometries, respectively. The influence of confinement on critical behavior persists even in a thick slab due to the diverging correlation length of density fluctuations. The scaling of the radius of gyration with chain length in the quasi-2D system increasingly resembles the scaling in the strictly 2D system as the chain length increases. At the extrapolated infinite chain critical temperature, the radius of gyration of the 2D system scales with chain length with exponent 0.56 (0.01 = (4/7), in agreement with theoretical predictions.
he history of knot tabulation is long established, having begun over 120 years ago. In many ways, the compilations of the first knot tables marked the beginning of the modern study of knots, and it is perhaps not surprising that as knot theory and topology grew, so did the knot tables. Over the last few years, we have extended the tables to include all prime knots with 16 or fewer crossings. This represents more than a 130-fold increase in the number of tabulated knots since the last burst of tabulation that took place in the early 1980s. With more than 1.7 million knots now in the tables, we hope that the census will serve as a rich source of examples and counterexamples and as a general testing ground for our collective intuition. To this end, we have written a UNIX-based computer program called KnotScape which allows easy access to the tables. The account of our methodology is prefaced by a brief history of knot tabulation, concentrating mostly on events taking place within the last 30 years. The survey article [Thil] contains further details on the work of the nineteenth-centu ry tabulators, but, above all, the reader is encouraged to consult the original sources, in particular the excellent series of papers by Tait [Tail. Kirkman's papers make fascinating reading, as they abound with original ideas and ornate language--his definition of the term "knot' is a single sentence of 101 words. Conway's land
Lattice knot statistics, or the study of knotted polygons in the cubic lattice, gained momentum in 1988 when the Frisch-Wasserman-Delbruck conjecture was proven by Sumners and Whittington (J Phys A Math Gen 21:L857–861, 1988), and independently in 1989 by Pippenger (Disc Appl Math 25:273–278, 1989). In this paper, aspects of lattice knot statistics are reviewed. The basic ideas underlying the study of knotted lattice polygons are presented, and the many open problem are posed explicitly. In addition, the properties of knotted polygons in a confining slab geometry are explained, as well as the Monte Carlo simulation of knotted polygons in \mathbb Z3{{\mathbb Z}^3} and in a slab geometry. Finally, the mean behaviour of lattice knots in a slab are discussed as a function of the knot type.