We used optical tweezers to tie individual DNA molecules in knots. Although these knots become highly localized under tension, they remain surprisingly mobile and undergo thermal diffusion with classical random walk statistics. The diffusion constants of knots with different complexities correlate with theoretical calculations of knot sizes. We show that this correlation can be explained by a simple hydrodynamical model of "self-reptation" of the knot along a polymer

Bao, Xiaoyan R.

Lee, Heun Jin

Quake, Stephen R.

English

Caltech Authors - Main

P H Y S I C A L R E V I E W L E T T E R S week ending31 DECEMBER 2003VOLUME 91, NUMBER 26Behavior of Complex Knots in Single DNA Molecules
Xiaoyan R. Bao, Heun Jin Lee, and Stephen R. Quake
Department of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
(Received 1 August 2003; published 31 December 2003)265506-1We used optical tweezers to tie individual DNA molecules in knots. Although these knots become
highly localized under tension, they remain surprisingly mobile and undergo thermal diffusion with
classical random walk statistics. The diffusion constants of knots with different complexities correlate
with theoretical calculations of knot sizes. We show that this correlation can be explained by a simple
hydrodynamical model of ‘‘self-reptation’’ of the knot along a polymer.
DOI: 10.1103/PhysRevLett.91.265506 PACS numbers: 61.41.+e, 02.10.Kn, 87.15.–vson to models of polymer dynamics, as well as more
specific theoretical results for ideal tight knots.
oxidase, and 0:9 mg=ml catalase) in a 4:1 volume ratio
just before observation.Knots and braids have fascinated observers since an-
cient times; complex topological designs have been pre-
served on tiles and pottery dating back thousands of years
[1]. Kelvin and Tait undertook the first systematic study of
knots in the 19th century [2,3], and in the 20th century a
powerful mathematical framework was constructed to
classify knots according to various topological invariants
[4]. They appear in many different scientific contexts,
ranging from the synthesis of topologically nontrivial
molecules in chemistry [5] to the role of topology in
life itself: topoisomerase enzymes exist to measure and
change the topology of DNA. In physics, it was realized
quite some time ago that there is a deep connection
between knot invariants and theories of statistical me-
chanics [6]. Knots also appear naturally in long polymers
[7]; they are, in fact, the generic state and can profoundly
impact the dynamics of systems in which they appear
[8,9]. The magnitude of these effects depends strongly on
the complexity of the knot [10]. Such polymeric topo-
logical constraints arise naturally in cells during DNA
replication [11], and knotting, in particular, has been
important in elucidating the mechanisms of DNA recom-
bination [12].
Theoretical and numerical studies of knots in long
polymers suggest that they are localized: the amount of
polymer directly engaged within the knot is vanishingly
small compared to the total length of the polymer [13,14].
Distributions of knots have previously been synthesized
in bulk samples [15,16], but only in relatively short cir-
cular DNA molecules; in this limit the knots are neither
localized nor tightly constrained, and, indeed, they were
purified and characterized based upon their effects on the
properties of the polymer as a whole.We sought instead to
confine the knot and study it as an isolated object. By
mechanically knotting linear pieces of DNAwith beads at
the ends that act as handles for optical tweezers, we were
able to keep the DNA under a fixed tension, making the
knot a localized structure whose properties are indepen-
dent of the length of substrate DNA. The confinement of
DNA achievable under these conditions allows compari-0031-9007=03=91(26)=265506(4)$20.00 Mechanical knotting of a molecule is challenging be-
cause the first step in the tying process involves making a
loop in the DNA which then immediately relaxes due to
entropic elasticity. Previous experiments tying the sim-
plest possible knot in our lab [J.-C. Meiners (unpublished
results)] and elsewhere [17] used non-Newtonian fluids to
inhibit this relaxation, but with the unfortunate drawback
of suppressing knot dynamics as well. We have now con-
structed a system capable of tying several different types
of knots into DNA molecules and observing their dynam-
ics within a linearly viscous solution. This is therefore the
first quantitative and systematic investigation of single,
thermally driven polymer knots.
To study how higher-order knotting structure affects
dynamics, we constructed knots following the ordering
scheme of a standard knot table. Knots are designated by
Ck, with C the minimal number of self-crossings dis-
played when the knot is projected into a plane, and k a
cardinal index used to distinguish between topologically
different knots with the same C. By threading the DNA
loop multiple times, or making more complicated loops
before threading them, we were able to consistently con-
struct knots of types 41, 51, 52, and 71, in addition to the
simplest knot 31 [Fig. 1(d); see also supplemental video
[18]]. Approximately 100 different DNA molecules were
successfully tied into knots, and about one-third of the
knots provided useful quantitative data.
To make bead-DNA-bead dumbbells,  genomic DNA
was concatenated with T4 DNA ligase, labeled with bio-
tin at both ends [19], and then mixed with streptavidin-
coated polystyrene beads (1 m diameter, Bangs Labs);
dumbbell lengths typically ranged from 40 to 100 m.
The dumbbells were stained in a solution containing
10 mM Tris-HCl (pH 8), 10 mM NaCl, 1 mM EDTA,
6% (w=v) poly(ethylene glycol) (PEG, molecular weight
35 000, Polysciences), 250 g=ml -casein, 0.1% tween-
20, and 20 nM YOYO-1 [20]. To reduce photodamage,
stained dumbbells were gently mixed with oxygen scav-
enger solution (equal volume mix of 113 mg=ml glucose
solution, 50% (v=v) -mercaptoethanol, 5 mg=ml glucose2003 The American Physical Society 265506-1
(a)
0.4
0.6
0.8
1
1.2
1.4
2 3 4 5 6 7 8
# of crossings
D
iff
us
io
n 
Co
ns
ta
nt
 (µ
m
2 /s
)
2
4
6
8
10
10 15 20 25 30
Knot length / diameter
Fr
ic
tio
n 
co
ef
fic
ie
nt
 (p
N/
(m
m/
s))(b)
31
41
51
52
71
FIG. 2. Knot diffusion constants D vary dramatically with
complexity (a); their respective friction coefficients  
kBT=D correlate well to theoretical knot lengths (b). The slope
of the best fit line is 0:29 0:003 pN=mm=s  2:9 0:3 	
10
10 N=m=s. Only data points with relative extensions be-
tween 0.55 and 0.75 were used. The numbers of knots observed
were 23, 9, 3, 4, and 3 for knot types 31, 41, 51, 52, and 71,
respectively.
FIG. 1. Observing knot diffusion. (a) The stained, knotted
DNA appears as a diffraction-limited contour between two
beads, with an increase in fluorescence at the knot (arrow). The
path of the DNA is found by software and its intensity pro-
file computed; tiling these profiles from sequential frames
(b) reveals a diffusive trace which indicates the knot’s trajec-
tory. Scale bars, 5 m (horizontal) and 5 s (vertical); scale of
(a) is the same as the horizontal scale of (b), The trace is
quantified (offset white trace) and its mean squared traveled
distance as a function of time t computed (c) which obeys a
power law with exponent   1:06 (see text). (d) We have tied
and analyzed open knots of types (from left to right) 31, 41, 51,
52, and 71 [18].
P H Y S I C A L R E V I E W L E T T E R S week ending31 DECEMBER 2003VOLUME 91, NUMBER 26Knotting experiments were performed on a custom in-
verted optical trapping microscope described previously
[21], with the addition of a mercury arc lamp (Oriel) for
fluorescence excitation and an image-intensified charge
coupled device for detection. A collimating lens in the
trapping beam path before the objective was placed on a
translation stage driven by a servo motor. Movement of
the lens gave depth control of the trap while xy translation
was performed by moving the stage supporting the ob-
servation flow cell. Each dumbbell was first stretched in
an applied flow to give a length calibration, then tied into
a knot and observed with the two ends held a fixed
distance apart. The type of knot tied was verified during
analysis by reviewing the video of the tying process.
All of the knots shrank under tension, becoming
highly localized, diffraction-limited spots at tensions
as low as 0.1 pN [Fig. 1(a)]. DNA shapes were traced
using an active contour (‘‘snake’’) algorithm [22], and
intensity profiles were computed from those shapes.When
we tiled intensity profiles from successive frames, bright
tracks became identifiable, indicating that the knots move
over time [Fig. 1(b)]. In control molecules in which we
did not tie a knot, these bright tracks were present about
10% of the time, close to the expected probability of
random thermal knotting in DNA. We quantified these
tracks and computed the mean squared traveled distance
as a function of time. Logarithmic plots [Fig. 1(c)] in-265506-2dicate a power law relation hx2i / t. The value of 
averaged over all knots was 1:06 0:02, in agreement
with classical random walk statistics. We then computed
diffusion constants by plotting the diffusion data on
linear axes and fitting to a line, leaving the offset term
free to compensate for short-time artifacts from both
observation and analysis; slopes obtained from these fits
then become our diffusion constant measurements. We
found no discernible tension dependence in knot proper-
ties at the tensions (0.1–2 pN) used to extend the DNA
molecules. However, the diffusion constant varies dra-
matically with knot complexity (Fig. 2).
Since the knots have conserved topologies, they must
move with a snakelike motion of the polymer through the
knot structure, analogous to the reptating motion of a
polymer in the presence of a fixed obstacle network. We
call this motion ‘‘self-reptation’’ since the same molecule
is both constraining and constrained. One can thus define
a characteristic time rept  L2D equal to the time it takes
the knot to diffuse through the length L of DNA compos-
ing the knot; this time is analogous to the lifetime of the
reptation tube. The total length L of polymer engaged
within a given knot type has been calculated in numerical265506-2
P H Y S I C A L R E V I E W L E T T E R S week ending31 DECEMBER 2003VOLUME 91, NUMBER 26simulations of tight, open knots constructed of ideal
ropes [23]. Comparing our drag coefficients   kBTD
to these lengths yields a roughly linear relationship
[Fig. 2(b)], so D L
1 and hence rept  L3, consistent
with the reptation model [24,25] and with observed be-
havior of DNA in an entangled solution [26].
Besides analyzing the kinetics of knot motion, it is also
possible to extract information about the relative sizes of
the knots from the microscope images. Using the knot
position information from track quantification, we aver-
aged knot intensities and compared them to intensities on
adjacent unknotted parts of the DNA. Because the knot
sizes are below the resolution of the imaging system, we
could measure only the extra amount of DNA present in
the knot as compared with an unknotted strand spanning
the same distance; this will in general be different from
the total knot length L but instead correspond to the
change in length caused by the knot in a tight, ideal
rope. These dimensionless theoretical lengths O are
relative to rope diameter and have been studied computa-
tionally [23]. Comparing them to the absolute knot length
measurements (Fig. 3) gives an average interstrand spac-
ing of 26 6 nm. This is most likely due to electrostatic
self-repulsion of the charged DNA molecule, and the
effect is similar in magnitude to computations and ob-
servations for DNA in bulk experiments [15].
The above analyses assume that the knot conforma-
tions can be approximated as those of ideal tight knots,
i.e., conformations which maximize strand-strand dis-
tance. Another possibility would be that the knot size is
determined by DNA bending rigidity. In that case, topo-
logical crossings in the trefoil (31) knot would occur
along a small part of the loop (as in Fig. 3c of [17]). We
can thus calculate the radius of the rigidity-dominant0.1
0.2
0.3
0.4
0.5
0.6
0.7
10 15 20 25
 1.1
 1.2
 1.3
 0  40  80  120
Position (pix.)
In
te
ns
ity
 (a
.u.
)
Extra knot length / diameter
O
bs
er
ve
d 
ex
tra
 k
no
t l
en
gt
h 
(µm
)
31
41
51
52
71
FIG. 3. Averaged extra knot lengths correlate with those
determined from numerical simulations of ideal open knots
[23]; the slope of the best fit line gives a DNA effective
diameter of 26 6 nm. The ordinate axis represents the differ-
ence between the lengths required for a knot and an unknotted
segment.
265506-3knot using elastic bending theory:
R 

kBTp
2F
s
’ 10 nm 

pN
p
F
p ; (1)
where F is the tension on the rope, kBT the thermal
energy, and p the persistence length. Since most mea-
surements were performed at F 1 pN, elastic return
forces should give an extra length of 2R 61 nm, or
only about one quarter of the 255 nm trefoil knot size.
This argument should be valid for all torus knots, includ-
ing the 51 and 71, so it predicts that the sizes of rigidity-
dominant torus knots are independent of knot type; this
is inconsistent with the data in Fig. 3.
Having quantified both the size and mobility of knots,
we can propose a simple hydrodynamic model to link
these measurements. The constraint network for any
given part of the knot is formed by other parts of the
same knot, so the ‘‘walls’’ of the effective reptation tube
move during diffusion. From computed conformations of
ideal knots [23], it appears that the constraining strands
in the network are generally perpendicular to the con-
strained strand; we should therefore expect little or no net
force in the direction of strand motion from the con-
straining strands within a reptating knot. We can then
model the friction coefficient as arising from the viscous
drag of one long cylinder (the constrained DNA strand)
within another (the constraining tube). From [27], p. 54,
this gives a drag coefficient per unit length =L as a
function of the ratio  of radii of the cylinders and the
viscosity :

L
 2
ln
: (2)
In this case, L is the length of DNA composing the knot.
The inner cylinder radius is half of the hydrodynamic
diameter of 2.5 nm for DNA, and the outer cylinder radius
is the observed mean separation between strands of 26 nm.
This gives an axial force contribution of =L  0:017
0:002 Pa s which is in fair agreement with the value of
0:011 0:003 Pa s derived from the slope in Fig. 2(b),
especially given the simplicity of the present model.
Presumably the flow field associated with the motion of
DNA through the knot will ‘‘bleed’’ beyond the phantom
tube defined by its constraints, reducing the energy dis-
sipation and giving rise to a slightly lower drag than that
predicted by our model.
We have thus shown that knots along an extended DNA
molecule behave in a manner consistent with predic-
tions for ideal, tight, reptating knots. Despite having
dimensions comparable to the persistence length of
DNA, the knots are quite mobile and form one of the
smallest known models of reptation. They are also a
model system for the study of tightly confined DNA
which is present in systems as diverse as eukaryotic
nucleosomes and packaged virus heads. Finally, visible265506-3
P H Y S I C A L R E V I E W L E T T E R S week ending31 DECEMBER 2003VOLUME 91, NUMBER 26DNA knots of known topology are ideal substrates for
studying the enzymology of topoisomerases at the single
molecule level.
We thank D. E. Smith, P. Pieranski, M. Wang, and J. Su
for helpful discussion, and J.-C. Meiners, K. Matthews,
and A. Groisman for help with preliminary experiments.
This work was funded in part by a grant from the Packard
Foundation; X. R. B. thanks the Hertz Foundation for
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Behavior of Complex Knots in Single DNA Molecules

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Behavior of Complex Knots in Single DNA Molecules

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