plcb-02-09-10 1074..1079
Intricate Knots in Proteins:
Function and Evolution
Peter Virnau
, Leonid A. Mirny
, Mehran Kardar
1 Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts, United States of America, 2 Harvard–MIT Division of Health Sciences and
Technology, Cambridge, Massachusetts, United States of America
Our investigation of knotted structures in the Protein Data Bank reveals the most complicated knot discovered to date.
We suggest that the occurrence of this knot in a human ubiquitin hydrolase might be related to the role of the enzyme
in protein degradation. While knots are usually preserved among homologues, we also identify an exception in a
transcarbamylase. This allows us to exemplify the function of knots in proteins and to suggest how they may have
been created.
Citation: Virnau P, Mirny LA, Kardar M (2006) Intricate knots in proteins: Function and evolution. PLoS Comput Biol 2(9): e122. DOI: 10.1371/journal.pcbi.0020122
Introduction
Although knots are abundant and complex in globular
homopolymers [1–3], they are rare and simple in proteins [4–
8]. Sixteen methyltransferases in bacteria and viruses can be
combined into the a/b knot superfamily [9], and several
isozymes of carbonic anhydrase (I, II, IV, V) are known to be
knotted. Apart from these two folds, only a few insular knots
have been reported [5,6,10,11], some of which were derived
from incomplete structures [6,11]. For the most part, knotted
proteins contain simple trefoil knots (31) that can be
represented by three essential crossings in a projection onto
a plane (see Figure 1, left). Only three proteins were identified
with four projected crossings (41, Figure 1, middle).
In this report we provide the first comprehensive review of
knots in proteins, which considers all entries in the Protein
Data Bank (http://www.pdb.org) [12], and not just a subset. This
allows us to examine knots in homologous proteins. Our
analysis reveals several new knots, all in enzymes. In particular,
we discovered the most complicated knot found to date (52) in
humanubiquitin hydrolase (Figure 1, right), and suggest that its
entangled topology protects it against being pulled into the
proteasome. We also noticed that knots are usually preserved
among structural homologues. Sequence similarity appears to
be a strong indicator for thepreservationof topology, although
differences between knotted and unknotted structures are
sometimes subtle. Interestingly, we have also identified a novel
knot in a transcarbamylase that is not present in homologues of
known structure. We show that the presence of this knot alters
the functionality of the protein, and suggest how the knot may
have been created in the first place.
Mathematically, knots are rigorously defined in closed
loops [13]. Fortunately, both the N- and C-termini of open
proteins are typically accessible from the surface and can be
connected unambiguously: we reduce the protein to its Ca-
backbone, and draw two lines outward starting at the termini
in the direction of the connection line between the center of
mass of the backbone and the respective ends [5]. The lines
are joined by a big loop, and the structure is topologically
classified by the determination of its Alexander polynomial
[1,13]. Applying this method to the Protein Data Bank in the
version of January 3, 2006, we found 273 knotted structures in
the 32,853 entries that contain proteins (Table S1). Knots
formed by disulfide [14,15] or hydrogen bonds [7] were not
included in the study.
Results
For further analysis, we considered 36 proteins that
contain knots as defined by rather stringent criteria discussed
in the Materials and Methods section. These proteins can be
classified into six distinct families (Table 1). Four of these
families incorporate a deeply knotted section, which persists
when 25 amino acids are cut off from either terminus.
Interestingly, all knotted proteins thus identified are en-
zymes. Our investigation affirms that all members of the
carbonic anhydrase fold (including the previously undeter-
mined isozymes III, VII, and XIV) are knotted. In addition, we
identify a novel trefoil in two bacterial transcarbamylase-like
proteins (AOTCase in Xanthomonas campestris and SOTCase in
Bacteroides fragilis) [16,17].
UCH-L3—The most complex protein knot. One of our
most intriguing discoveries is a fairly intricate knot with five
projected crossings (52) in ubiquitin hydrolase (UCH-L3 [18];
see Figure 1, right). This knot is the first of its kind and, apart
from carbonic anhydrases, the only identified in a human
protein. Human UCH-L3 also has a yeast homologue [6,19]
with a sequence identity of 32% [20]. Amino acids 63 to 77
are unstructured, and if we connect the unstructured region
by an arc that is present in the human structure, we obtain
the same knot with five crossings. What may be the function
of this knot? In eukaryotes, proteins get labeled for
Editor: Robert B. Russell, European Molecular Biology Laboratory, Germany
Received April 3, 2006; Accepted July 28, 2006; Published September 15, 2006
A previous version of this article appeared as an Early Online Release on July 28,
2006 (DOI: 10.1371/journal.pcbi.0020122.eor).
DOI: 10.1371/journal.pcbi.0020122
Copyright: � 2006 Virnau et al. This is an open-access article distributed under the
terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author
and source are credited.
Abbreviations: AOTCase, N-acetylornithine transcarbamylase; SOTCase, N-succi-
nylornithine transcarbamylase; UCH-L3, ubiquitin hydrolase
* To whom correspondence should be addressed. E-mail: virnau@mit.edu
PLoS Computational Biology | www.ploscompbiol.org September 2006 | Volume 2 | Issue 9 | e1221074
degradation by ubiquitin conjugation. UCH-L3 performs
deconjugation of ubiquitin, thus rescuing proteins from
degradation. The close association of the enzyme with
ubiquitin should make it a prime target for degradation at
the proteasome. We suggest that the knotted structure of
UCH-L3 makes it resistant to degradation. In fact, the first
step of protein degradation was shown to be ATP-dependent
protein unfolding by threading through a narrow pore (;13
Å in diameter) of a proteasome [21,22]. Such threading into
the degradation chamber depends on how easily a protein
unfolds, with more stable proteins being released back into
solution [23] and unstable ones being degraded. If ATP-
dependent unfolding proceeds by pulling the C-terminus into
a narrow pore [21], then a knot can sterically preclude such
translocation, hence preventing protein unfolding and
degradation. While arceabacterial proteasome PAN was
shown to process proteins from its C- to N-terminus [21], it
cannot be ruled out that some eukaryotic proteasomes
process proteins in the N- to C-direction, thus requiring
protection of both termini. Unfolding of a knotted protein by
pulling may require a long time for global unfolding and
untangling of the knot. Unknotted proteins, in contrast, have
been shown to become unstable if a few residues are removed
from their termini [24], suggesting that threading a few (5–10)
residues into a proteasomal pore would be sufficient to
unravel an unknotted structure. At both termini, UCH-L3
contains loops entangled into the knot protecting both ends
against unfolding if pulled. It should also be noted that both
N- and C-termini are stabilized by a number of hydrophobic
interactions with the rest of the protein. The C-terminus is
Figure 1. Examples of the Three Different Types of Knots Found in Proteins
Colors change continuously from red (first residue) to blue (last residue). A reduced representation of the structure, based on the algorithm described in
[1,6,36], is shown in the lower row.
(Left) The trefoil knot (31) in the YBEA methyltransferase from E. coli (pdb code 1ns5; unpublished data) reveals three essential crossings in a projection
onto a plane.
(Middle) The figure-eight knot (41) in the Class II ketol-acid reductoisomerase from Spinacia oleracea (pdb code 1yve [26]) features four crossings. (Only
the knotted section of the protein is shown.)
(Right) The knot 52 in ubiquitin hydrolase UCH-L3 (pdb code 1xd3 [18]) reveals five crossings. Pictures were generated with Visual Molecular Dynamics
(http://www.ks.uiuc.edu/Research/vmd) [43].
DOI: 10.1371/journal.pcbi.0020122.g001
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Synopsis
Several protein structures incorporate a rather unusual structural
feature: a knot in the polypeptide backbone. These knots are
extremely rare, but their occurrence is likely connected to protein
function in as yet unexplored fashion. The authors’ analysis of the
complete Protein Data Bank reveals several new knots that, along
with previously discovered ones, may shed light on such con-
nections. In particular, they identify the most complex knot
discovered to date in a human protein, and suggest that its
entangled topology protects it against unfolding and degradation.
Knots in proteins are typically preserved across species and
sometimes even across kingdoms. However, there is also one
example of a knot in a protein that is not present in a closely related
structure. The emergence of this particular knot is accompanied by a
shift in the enzymatic function of the protein. It is suggested that
the simple insertion of a short DNA fragment into the gene may
suffice to cause this alteration of structure and function.
Intricate Knots in Proteins
particularly stable—residues 223 to 229 are hydrophobic and
form numerous contacts at 5 Å with the rest of the structure.
We would like to stress that this hypothesis needs to be
tested by experiments. Different proteins may also provide
different levels of protection against degradation, depending
on structural details, the depth of the knot, and its complex-
ity. Recently, a knot in the red/far-red light photoreceptor
phytochrome A in Deinococcus radiodurans was identified [11]
(see Materials and Methods). Although sequence similarity
suggests that the knot may also be present in plant
homologues, we cannot be certain. In plants, the red-
absorbing form is rather stable (half-life of 1 wk), but the
far-red–absorbing form is degraded upon photoconversion
by the proteasome with a half-life of 1–2 h in seedlings (and
somewhat longer in adult plants) [25].
Evolutionary aspects. As expected, homologous structures
tend to retain topological features. The trefoil knot in
carbonic anhydrase can be found in isozymes ranging from
bacteria and algae to humans (Table 1). Class II ketol-acid
reductoisomerase comprises a figure-eight knot present in
Escherichia coli [10] and spinach [26] (see Figure 1, middle), and
S-adenosylmethione synthetase contains a deep trefoil knot in
E. coli [5,27] and rat [28]. It appears that particular knots have
indeed been preserved throughout evolution, which suggests a
crucial role for knots in protein enzymatic activity and
binding.
Table 1. List of Knotted PDB Entries (January 2006)
Protein Knot Family Protein Species PDB Code Length Knot Knotted Core
a/b knot YbeA-like E. coli 1ns5 153 31 69–121 (32)
T. maritime 1o6d 147 31 68–117 (30)
S. aureus 1vh0 157 31 73–126 (31)
B. subtilis 1to0 148 31 64–116 (32)
tRNA(m1G37)-methyltransferase TrmD H. influenza 1uaj 241 31 93–138 (92)
E. coli 1p9p 235 31 90–130 (89)
SpoU-like RNA 29-O ribose mtf. T. thermophilus 1v2x 191 31 96–140 (51)
H. influenza 1j85 156 31 77–114 (42)
T. thermophilus 1ipa 258 31 185–229 (29)
E. coli 1gz0 242 31 172–214 (28)
A. aeolicus 1zjr 197 31 95–139 (58)
S. viridochromog. 1x7p 265 31 192–234 (31)
YggJ C-terminal domain-like H. influenza 1nxz 246 31 165–216 (30)
B. subtilis 1vhk 235 31 158–208 (27)
T. thermophilus 1v6z 227 31 103–202 (25)
Hypothetical protein MTH1 (MT0001) A. M. Thermoautotr. 1k3r 262 31 48–234 (28)
Carbonic anhydrases Carbonic anhydrase N. gonorrhoeae 1kop 223 31 36–223 (0)
Carbonic anhydrase I H. sapiens 1hcb 258 31 29–256 (2)
Carbonic anhydrase II H. sapiens 1lug 259 31 30–256 (3)
Bos Taurus 1v9e 259 31 32–256 (3)
Dunaliella salina 1y7w 274 31 37–270 (4)
Carbonic anhydrase III Rattus norv. 1flj 259 31 30–256 (3)
H. sapiens 1z93 263 31 28–254 (9)
Carbonic anhydrase IV H.sapiens 1znc 262 31 32–261 (1)
Mus musculus 2znc 249 31 32–246 (3)
Carbonic anhydrase V Mus musculus 1keq 238 31 7–234 (4)
Carbonic anhydrase VII H. sapiens 1jd0 260 31 28–257 (3)
Carbonic anhydrase XIV Mus Musculus 1rj6 259 31 29–257 (2)
Miscellaneous Ubiquitin hydrolase UCH-L3 H. sapiens 1xd3A 229 52 12–172 (11)
S. cerevisiae (synth.) 1cmxA 214 31 9–208 (6)
S-adenosylmethionine synthetase E. coli 1fug 383 31 33–260 (32)
Rattus norv. 1qm4 368 31 30–253 (29)
Class II ketol-acid reductoisomerase Spinacia oleracea 1yve 513 41 239–451 (62)
E. coli 1yrl 487 41 220–435 (52)
Transcarbamylase-like B. fragilis 1js1 324 31 169–267 (57)
X. campestris 1yh1 334 31 171–272 (62)
‘‘Protein’’ describes the name or the family of the knotted structure. ‘‘Species’’ refers to the scientific name of the organism from which the protein was taken for structure determination.
‘‘PDB code’’ gives one example Protein Data Bank entry for each knotted protein: additional structures of the same protein can be found using the SCOP classification tool [9]. ‘‘Length’’
describes the number of Ca-backbone atoms in the structure. ‘‘Knot’’ refers to the knot type which was discovered in the protein: 31, trefoil; 41, figure-eight knot; 52, 2nd knot with five
crossings according to standard knot tables [13]. The core of a knot is the minimum configuration which stays knotted after a series of deletions from each terminus; in brackets we
indicate how many amino acids can be removed from either side before the structure becomes unknotted (see Materials and Methods).
Structure is fragmented and becomes knotted when missing sections are joined by straight lines. The size of the knotted core refers to the thus-connected structure. The knot is also
present in at least one fragment.
Structure is fragmented and only knotted when missing sections are joined by straight lines. Fragments are unknotted.
1v6z is currently not classified according to SCOP (version 1.69). Sequence similarity suggests that it is part of the a/b knot fold. 1v6zB contains a shallow composite knot (31#41), which
turns into a regular trefoil when two amino acids are cut from the N-terminus. (The random closure [see Materials and Methods] determines a trefoil right away.)
1uch contains the same structure as 1xd3. If the missing section in the center of this structure is joined by a straight line, it becomes knotted (52). In the yeast homologue (1cmx), amino
acids 63 to 77 are unstructured, and if we replace the missing parts by a straight line, we obtain a trefoil knot that has been identified before [6]. If we connect the unstructured region by
an arc present in the human structure, we obtain the same knot with five crossings.
eVisual inspection reveals that the calculated size of the knotted core is too small.
DOI: 10.1371/journal.pcbi.0020122.t001
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Intricate Knots in Proteins
UCH-L3 in human and yeast share only 33% [29] of their
sequences, but contain the same 5-fold knot as far as we can
tell from the incomplete structure in yeast. It is not only likely
that all species in between have the same knot—the link
between sequence and structure may also be used to predict
candidates for knots among isozymes or related proteins for
which the structure is unknown. For example, UCH-L4 in
mouse has 96% sequence identity with human UCH-L3. The
similarity with UCH-L6 in chicken is 86%, and with UCH-L1
about 55%. Indeed, a reexamination of the most recent
Protein Data Bank entries revealed that UCH-L1 contains the
same 52 knot as UCH-L3. (See the Update section—the
structure was not yet part of the January Protein Data Bank
release on which this paper is based.) Unfortunately, the
method is not foolproof because differences between knotted
and unknotted structures are sometime subtle. As we will
demonstrate in the next paragraph, a more reliable estimate
has to consider the conservation of major elements of the
knot, like loops and threads.
AOTCase—How a protein knot can alter enzymatic activity.
Somewhat surprisingly, we also identified a pair of homo-
logues for which topology is not preserved. N-acetylornithine
transcarbamylase (AOTCase [17]) is essential for the arginine
biosynthesis in several major pathogens. In other bacteria,
animals, and humans, a homologous enzyme (OTCase)
processes L-ornithine instead [30]. Both proteins have two
active sites. The first one binds carbamyl phosphate to the
enzyme. The second site binds acetylornithine in AOTCases
and L-ornithine in OTCases, enabling a reaction with
carbamyl phosphate to form acetylcitrulline or citrulline,
respectively [17, 31].
AOTCase in X. campestris has 41% sequence identity with
OTCase from Pyrococcus furiosus [32] and 29% with human
OTCase [31]. As demonstrated in Figure 2, AOTCase contains
a deep trefoil knot which is not present in OTCase (Figure 2,
right) and which modifies the second active site. The knot
consists of a rigid proline-rich loop (residues 178–185),
through which residues 252 to 256 are threaded and affixed.
As elaborated in [17], the reaction product N-acetylcitrulline
strongly interacts with the loop and with Lys252. Access to
subsequent residues is, however, restricted by the knot. L-
norvaline in Figure 2 (right) is very similar to L-ornithine but
lacks the N-e atom of the latter to prevent a reaction with
carbamyl phosphate. As the knot is not present in OTCase,
the ligand has complete access to the dangling residues 263–
268 and strongly interacts with them [31]. This leads to a
rotation of the carboxyl-group by roughly 1108 around the
Ca–Cb bond [17].
This example demonstrates how the presence of a knot can
modify active sites and alter the enzymatic activity of a
protein—in this case, from processing L-ornithine to N-
acetyl-L-ornithine. It is also easy to imagine how this
alteration happened: a short insertion extends the loop and
modifies the folding pathway of the protein.
Discussion
Nature appears to disfavour entanglements, and evolution
has developed mechanisms to avoid knots. Human DNA
wraps around histone proteins, and the rigidity of DNA
allows it to form a spool when it is fed into a viral capsid. One
end also stays in the loading channel and prevents subsequent
equilibration [33]. Knotted proteins are rare, although the
reason is far less well understood. Can the absence of
entanglement be explained in terms of particular statistical
ensembles, or is there an evolutionary bias? And how do these
structures actually fold?
Knots are ubiquitous in globular homopolymers [1–3,8], but
rare in coil-like phases [1,34–36]. It is likely that even a flexible
polymer will at least initially remain unknotted after a
Figure 2. Structures of Transcarbamylase from X. campestris with a Trefoil Knot and from Human without a Knot
(Left) Knotted section (residues 171–278) of N-acetylornithine transcarbamylase from X. campestris with reaction product N-acetylcitrulline (pdb code
1yh1 [17]) and interacting side chains.
(Right) Corresponding (unknotted) section (residues 189–286) in human ornithine transcarbamylase (pdb code 1c9y [31]) with inhibitor L-norvaline and
carbamyl phosphate. Colors change continuously from red (first residue in the section) to blue (last residue in the section). The two proteins have an
overall sequence identity of 29% [41]. Pictures were generated with VMD [43].
DOI: 10.1371/journal.pcbi.0020122.g002
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Intricate Knots in Proteins
collapse from a swollen state. In proteins, the free energy
landscape is considerably more complex, which may allow
most proteins to stay unknotted. The secondary structure and
the stiffness of the protein backbone may shift the length scale
at which knots typically appear, too [8]. If knotted proteins are
in fact more difficult to degrade, it might also be disadvanta-
geous for most proteins to be knotted in the first place.
Unfortunately, few experimental papers address folding
and biophysical aspects of knots in proteins. In recent work
[37], Jackson and Mallam reversibly unfolded and folded a
knotted methyltransferase in vitro, indicating that chaper-
ones are not a necessary prerequisite. In a subsequent study
[38], the authors provide an extensive kinetic analysis of the
folding pathway. In conclusion, we would like to express our
hope that this report will inspire more experiments in this
small but nevertheless fascinating field.
Materials and Methods
To determine whether a structure is knotted, we reduce the
protein to its backbone, and draw two lines outward starting at the
termini in the direction of the connection line between the center of
mass of the backbone and the respective ends. These two lines are
joined by a big loop, and the structure is classified by the
determination of its Alexander polynomial [1,13]. To determine the
size of the knotted core, we delete successively amino acids from the
N-terminus until the protein becomes unknotted [1,6]. The proce-
dure is repeated at the C-terminus starting with the last structure that
contained the original knot. For each deletion, the outward pointing
line through the new termini is parallel to the respective lines
computed for the full structure. The thus determined size should,
however, only be regarded as a guideline. A better estimate can be
achieved by looking at the structure.
In Table 1 we include knotted structures with no missing amino
acids in the center of the protein. (A list of potentially knotted
structures with missing amino acids can be found in Table S3.)
Technically, the numbering of the residues in the mmcif file has to be
subsequent, and no two amino acids are allowed to be more than 6 Å
apart. In addition, the knot has to persist when two amino acids are
cut from either terminus. We have further excluded structures for
which unknotted counterexamples exist (e.g., only one nuclear
magnetic resonance structure among many is knotted or another
structure of the same protein is unknotted). If a structure is
fragmented, the knot has to appear in one fragment and in the
resulting structure obtained from connecting missing sections by
straight lines. Other knotted structures are only considered when at
least one additional member of the same structural family [9]
contains a knot according to the criteria above.
The enforcement of these rules leads to the exclusion of the
bluetongue virus core protein [6] (41) and photoreceptor phyto-
chrome A in D. radiodurans [11] (31), which have been previously
identified as being knotted. Both structures are fragmented and
become knotted only when a few missing fragments are connected by
straight lines. In the viral core protein, the dangling C-terminus
threads through a loose loop and becomes knotted in one out of two
cases. On the other hand, the photoreceptor phytochrome A appears
to contain a true knot. Notably, our analysis suggests that the thus
connected structure of phytochrome A contains a figure-eight knot
instead of a trefoil as reported in [11]. Moreover, we excluded a
structure of the Autographa California nuclear polyhedrosis virus, which
contains a knot according to our criteria. However, the N-terminus is
buried inside the protein and the knot only exists because of our
specific connection to the outside.
To further validate our criteria, we implemented an alternative
method [4,8,39] that relies on the statistical analysis of multiple
random closures. We arbitrarily chose two points on a sphere (which
has to be larger than the protein) and connected each with one
terminus. The two points can be joined unambiguously, and the
resulting loop was analyzed by calculating the Alexander polynomial.
We repeated the procedure 1,000 times, and defined the knot as the
majority type.
Applying this analysis, we discovered 241 knotted structures in the
Protein Data Bank. All 241 structures are also present in the 273
structures (Table S1) that were identified by our method, and the knot
type is the same. Themissing32 structures (Table S2) aremostly shallow
knots and were already rejected according to our extended criteria.
The random closure also correctly discards rare structures with buried
termini. In conclusion, the method used in this paper is considerably
faster but requires a slightly increased inspection effort. Our
observations agree with [8], which provides an extensive comparison
of closures applied to proteins. A complete listing of knotted Protein
Data Bank structures is given in the Supporting Information.
Update. Recently, the structure of human UCH-L1 was solved and
released [40]. The protein shares 55% sequence identity with UCH-L3
[41], and it contains the same 5-fold knot. UCH-L1 is highly abundant
in the brain, comprising up to 2% of the total brain protein [42]. The
structure of UCH-L1 was not yet part of the January Protein Data
Bank edition on which the rest of this study is based. We also noticed
several new structures of knotted transcarbamylase-like proteins.
Supporting Information
Table S1. List of Knotted Protein Data Bank Entries
Found at DOI: 10.1371/journal.pcbi.0020122.st001 (79 KB DOC).
Table S2. List of Knotted Entries from Table S1 That Become
UnknottedWhenEnds AreConnected by theRandomClosureMethod
Found at DOI: 10.1371/journal.pcbi.0020122.st002 (28 KB DOC).
Table S3. List of Structures That Become Knotted When Missing
Sections Are Joined by Straight Lines
Found at DOI: 10.1371/journal.pcbi.0020122.st003 (35 KB DOC).
Accession Numbers
The Protein Data Bank (http://www.pdb.org) accession numbers for
the structures discussed in this paper are human UCH-L3 (1xd3),
UCH-L3 yeast homologue (1cmx), human UCH-L1 (2etl), photo-
receptor phytochrome A in D. radiodurans (1ztu), class II ketol-acid
reductoisomerase in E. coli (1yrl), class II ketol-acid reductoisomerase
in spinach (1yve), S-adenosylmethione synthetase in E. coli (1fug), S-
adenosylmethione synthetase in rat (1qm4), AOTCase from X.
campestris (1yh1), SOTCase from B. fragilis (1js1), OTCase from P.
furiosus (1a1s), OTCase from human (1c9y), bluetongue virus core
protein (2btv), and baculovirus P35 protein in Autographa California
nuclear polyhedrosis virus (1p35).
Acknowledgments
Upon completion of this work we became aware of a related study [8],
which independently identified the knots in UCH-L3 and SOTCase in a
re-examination of protein knots. PV would like to acknowledge
discussions with François Nédélec and with Olav Zimmermann, in
which they proposed the potential link between protein knots and
degradation. LM and PV would also like to thank Rachel Gaudet for a
discussion about the function of ubiquitin hydrolase.
Author contributions. MK conceived the study. PV designed and
wrote the analysis code. PV and LM analyzed the data. PV, LM, and
MK wrote the paper.
Funding. This work was supported by National Science Foundation
grant DMR-04–26677 and by Deutsche Forschungsgemeinschaft grant
VI 237/1. LM is an Alfred P. Sloan Research Fellow.
Competing interests. The authors have declared that no competing
interests exist.
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PLoS Computational Biology | www.ploscompbiol.org September 2006 | Volume 2 | Issue 9 | e1221079
Intricate Knots in Proteins

  A number of recently discovered protein structures incorporate a rather
unexpected structural feature: a knot in the polypeptide backbone. These knots
are extremely rare, but their occurrence is likely connected to protein
function in as yet unexplored fashion. Our analysis of the complete Protein
Data Bank reveals several new knots which, along with previously discovered
ones, can shed light on such connections. In particular, we identify the most
complex knot discovered to date in human ubiquitin hydrolase, and suggest that
its entangled topology protects it against unfolding and degradation by the
proteasome. Knots in proteins are typically preserved across species and
sometimes even across kingdoms. However, we also identify a knot which only
appears in some transcarbamylases while being absent in homologous proteins of
similar structure. The emergence of the knot is accompanied by a shift in the
enzymatic function of the protein. We suggest that the simple insertion of a
short DNA fragment into the gene may suffice to turn an unknotted into a
knotted structure in this protein.

Introduction
Although knots are abundant and complex in globular
homopolymers [1–3], they are rare and simple in proteins [4–
8]. Sixteen methyltransferases in bacteria and viruses can be
combined into the a/b knot superfamily [9], and several
isozymes of carbonic anhydrase (I, II, IV, V) are known to be
knotted. Apart from these two folds, only a few insular knots
have been reported [5,6,10,11], some of which were derived
from incomplete structures [6,11]. For the most part, knotted
proteins contain simple trefoil knots (31) that can be
represented by three essential crossings in a projection onto
a plane (see Figure 1, left). Only three proteins were identified
with four projected crossings (41, Figure 1, middle).
In this report we provide the first comprehensive review of
knots in proteins, which considers all entries in the Protein
Data Bank (http://www.pdb.org) [12], and not just a subset. This
allows us to examine knots in homologous proteins. Our
analysis reveals several new knots, all in enzymes. In particular,
we discovered the most complicated knot found to date (52) in
humanubiquitin hydrolase (Figure 1, right), and suggest that its
entangled topology protects it against being pulled into the
proteasome. We also noticed that knots are usually preserved
among structural homologues. Sequence similarity appears to
be a strong indicator for thepreservationof topology, although
differences between knotted and unknotted structures are
sometimes subtle. Interestingly, we have also identified a novel
knot in a transcarbamylase that is not present in homologues of
known structure. We show that the presence of this knot alters
the functionality of the protein, and suggest how the knot may
have been created in the first place.
Mathematically, knots are rigorously defined in closed
loops [13]. Fortunately, both the N- and C-termini of open
proteins are typically accessible from the surface and can be
connected unambiguously: we reduce the protein to its Ca-
backbone, and draw two lines outward starting at the termini
in the direction of the connection line between the center of
mass of the backbone and the respective ends [5]. The lines
are joined by a big loop, and the structure is topologically
classified by the determination of its Alexander polynomial
[1,13]. Applying this method to the Protein Data Bank in the
version of January 3, 2006, we found 273 knotted structures in
the 32,853 entries that contain proteins (Table S1). Knots
formed by disulfide [14,15] or hydrogen bonds [7] were not
included in the study.
Results
For further analysis, we considered 36 proteins that
contain knots as defined by rather stringent criteria discussed
in the Materials and Methods section. These proteins can be
classified into six distinct families (Table 1). Four of these
families incorporate a deeply knotted section, which persists
when 25 amino acids are cut off from either terminus.
Interestingly, all knotted proteins thus identified are en-
zymes. Our investigation affirms that all members of the
carbonic anhydrase fold (including the previously undeter-
mined isozymes III, VII, and XIV) are knotted. In addition, we
identify a novel trefoil in two bacterial transcarbamylase-like
proteins (AOTCase in Xanthomonas campestris and SOTCase in
Bacteroides fragilis) [16,17].
UCH-L3—The most complex protein knot. One of our
most intriguing discoveries is a fairly intricate knot with five
projected crossings (52) in ubiquitin hydrolase (UCH-L3 [18];
see Figure 1, right). This knot is the first of its kind and, apart
from carbonic anhydrases, the only identified in a human
protein. Human UCH-L3 also has a yeast homologue [6,19]
with a sequence identity of 32% [20]. Amino acids 63 to 77
are unstructured, and if we connect the unstructured region
by an arc that is present in the human structure, we obtain
the same knot with five crossings. What may be the function
of this knot? In eukaryotes, proteins get labeled for
Editor: Robert B. Russell, European Molecular Biology Laboratory, Germany
Received April 3, 2006; Accepted July 28, 2006; Published September 15, 2006
A previous version of this article appeared as an Early Online Release on July 28,
2006 (DOI: 10.1371/journal.pcbi.0020122.eor).
DOI: 10.1371/journal.pcbi.0020122
Copyright: � 2006 Virnau et al. This is an open-access article distributed under the
terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author
and source are credited.
Abbreviations: AOTCase, N-acetylornithine transcarbamylase; SOTCase, N-succi-
nylornithine transcarbamylase; UCH-L3, ubiquitin hydrolase
* To whom correspondence should be addressed. E-mail: virnau@mit.edu
PLoS Computational Biology | www.ploscompbiol.org September 2006 | Volume 2 | Issue 9 | e1221074
degradation by ubiquitin conjugation. UCH-L3 performs
deconjugation of ubiquitin, thus rescuing proteins from
degradation. The close association of the enzyme with
ubiquitin should make it a prime target for degradation at
the proteasome. We suggest that the knotted structure of
UCH-L3 makes it resistant to degradation. In fact, the first
step of protein degradation was shown to be ATP-dependent
protein unfolding by threading through a narrow pore (;13
Å in diameter) of a proteasome [21,22]. Such threading into
the degradation chamber depends on how easily a protein
unfolds, with more stable proteins being released back into
solution [23] and unstable ones being degraded. If ATP-
dependent unfolding proceeds by pulling the C-terminus into
a narrow pore [21], then a knot can sterically preclude such
translocation, hence preventing protein unfolding and
degradation. While arceabacterial proteasome PAN was
shown to process proteins from its C- to N-terminus [21], it
cannot be ruled out that some eukaryotic proteasomes
process proteins in the N- to C-direction, thus requiring
protection of both termini. Unfolding of a knotted protein by
pulling may require a long time for global unfolding and
untangling of the knot. Unknotted proteins, in contrast, have
been shown to become unstable if a few residues are removed
from their termini [24], suggesting that threading a few (5–10)
residues into a proteasomal pore would be sufficient to
unravel an unknotted structure. At both termini, UCH-L3
contains loops entangled into the knot protecting both ends
against unfolding if pulled. It should also be noted that both
N- and C-termini are stabilized by a number of hydrophobic
interactions with the rest of the protein. The C-terminus is
Figure 1. Examples of the Three Different Types of Knots Found in Proteins
Colors change continuously from red (first residue) to blue (last residue). A reduced representation of the structure, based on the algorithm described in
[1,6,36], is shown in the lower row.
(Left) The trefoil knot (31) in the YBEA methyltransferase from E. coli (pdb code 1ns5; unpublished data) reveals three essential crossings in a projection
onto a plane.
(Middle) The figure-eight knot (41) in the Class II ketol-acid reductoisomerase from Spinacia oleracea (pdb code 1yve [26]) features four crossings. (Only
the knotted section of the protein is shown.)
(Right) The knot 52 in ubiquitin hydrolase UCH-L3 (pdb code 1xd3 [18]) reveals five crossings. Pictures were generated with Visual Molecular Dynamics
(http://www.ks.uiuc.edu/Research/vmd) [43].
DOI: 10.1371/journal.pcbi.0020122.g001
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Synopsis
Several protein structures incorporate a rather unusual structural
feature: a knot in the polypeptide backbone. These knots are
extremely rare, but their occurrence is likely connected to protein
function in as yet unexplored fashion. The authors’ analysis of the
complete Protein Data Bank reveals several new knots that, along
with previously discovered ones, may shed light on such con-
nections. In particular, they identify the most complex knot
discovered to date in a human protein, and suggest that its
entangled topology protects it against unfolding and degradation.
Knots in proteins are typically preserved across species and
sometimes even across kingdoms. However, there is also one
example of a knot in a protein that is not present in a closely related
structure. The emergence of this particular knot is accompanied by a
shift in the enzymatic function of the protein. It is suggested that
the simple insertion of a short DNA fragment into the gene may
suffice to cause this alteration of structure and function.
Intricate Knots in Proteins
particularly stable—residues 223 to 229 are hydrophobic and
form numerous contacts at 5 Å with the rest of the structure.
We would like to stress that this hypothesis needs to be
tested by experiments. Different proteins may also provide
different levels of protection against degradation, depending
on structural details, the depth of the knot, and its complex-
ity. Recently, a knot in the red/far-red light photoreceptor
phytochrome A in Deinococcus radiodurans was identified [11]
(see Materials and Methods). Although sequence similarity
suggests that the knot may also be present in plant
homologues, we cannot be certain. In plants, the red-
absorbing form is rather stable (half-life of 1 wk), but the
far-red–absorbing form is degraded upon photoconversion
by the proteasome with a half-life of 1–2 h in seedlings (and
somewhat longer in adult plants) [25].
Evolutionary aspects. As expected, homologous structures
tend to retain topological features. The trefoil knot in
carbonic anhydrase can be found in isozymes ranging from
bacteria and algae to humans (Table 1). Class II ketol-acid
reductoisomerase comprises a figure-eight knot present in
Escherichia coli [10] and spinach [26] (see Figure 1, middle), and
S-adenosylmethione synthetase contains a deep trefoil knot in
E. coli [5,27] and rat [28]. It appears that particular knots have
indeed been preserved throughout evolution, which suggests a
crucial role for knots in protein enzymatic activity and
binding.
Table 1. List of Knotted PDB Entries (January 2006)
Protein Knot Family Protein Species PDB Code Length Knot Knotted Core
a/b knot YbeA-like E. coli 1ns5 153 31 69–121 (32)
T. maritime 1o6d 147 31 68–117 (30)
S. aureus 1vh0 157 31 73–126 (31)
B. subtilis 1to0 148 31 64–116 (32)
tRNA(m1G37)-methyltransferase TrmD H. influenza 1uaj 241 31 93–138 (92)
E. coli 1p9p 235 31 90–130 (89)
SpoU-like RNA 29-O ribose mtf. T. thermophilus 1v2x 191 31 96–140 (51)
H. influenza 1j85 156 31 77–114 (42)
T. thermophilus 1ipa 258 31 185–229 (29)
E. coli 1gz0 242 31 172–214 (28)
A. aeolicus 1zjr 197 31 95–139 (58)
S. viridochromog. 1x7p 265 31 192–234 (31)
YggJ C-terminal domain-like H. influenza 1nxz 246 31 165–216 (30)
B. subtilis 1vhk 235 31 158–208 (27)
T. thermophilus 1v6z 227 31 103–202 (25)
Hypothetical protein MTH1 (MT0001) A. M. Thermoautotr. 1k3r 262 31 48–234 (28)
Carbonic anhydrases Carbonic anhydrase N. gonorrhoeae 1kop 223 31 36–223 (0)
Carbonic anhydrase I H. sapiens 1hcb 258 31 29–256 (2)
Carbonic anhydrase II H. sapiens 1lug 259 31 30–256 (3)
Bos Taurus 1v9e 259 31 32–256 (3)
Dunaliella salina 1y7w 274 31 37–270 (4)
Carbonic anhydrase III Rattus norv. 1flj 259 31 30–256 (3)
H. sapiens 1z93 263 31 28–254 (9)
Carbonic anhydrase IV H.sapiens 1znc 262 31 32–261 (1)
Mus musculus 2znc 249 31 32–246 (3)
Carbonic anhydrase V Mus musculus 1keq 238 31 7–234 (4)
Carbonic anhydrase VII H. sapiens 1jd0 260 31 28–257 (3)
Carbonic anhydrase XIV Mus Musculus 1rj6 259 31 29–257 (2)
Miscellaneous Ubiquitin hydrolase UCH-L3 H. sapiens 1xd3A 229 52 12–172 (11)
S. cerevisiae (synth.) 1cmxA 214 31 9–208 (6)
S-adenosylmethionine synthetase E. coli 1fug 383 31 33–260 (32)
Rattus norv. 1qm4 368 31 30–253 (29)
Class II ketol-acid reductoisomerase Spinacia oleracea 1yve 513 41 239–451 (62)
E. coli 1yrl 487 41 220–435 (52)
Transcarbamylase-like B. fragilis 1js1 324 31 169–267 (57)
X. campestris 1yh1 334 31 171–272 (62)
‘‘Protein’’ describes the name or the family of the knotted structure. ‘‘Species’’ refers to the scientific name of the organism from which the protein was taken for structure determination.
‘‘PDB code’’ gives one example Protein Data Bank entry for each knotted protein: additional structures of the same protein can be found using the SCOP classification tool [9]. ‘‘Length’’
describes the number of Ca-backbone atoms in the structure. ‘‘Knot’’ refers to the knot type which was discovered in the protein: 31, trefoil; 41, figure-eight knot; 52, 2nd knot with five
crossings according to standard knot tables [13]. The core of a knot is the minimum configuration which stays knotted after a series of deletions from each terminus; in brackets we
indicate how many amino acids can be removed from either side before the structure becomes unknotted (see Materials and Methods).
Structure is fragmented and becomes knotted when missing sections are joined by straight lines. The size of the knotted core refers to the thus-connected structure. The knot is also
present in at least one fragment.
Structure is fragmented and only knotted when missing sections are joined by straight lines. Fragments are unknotted.
1v6z is currently not classified according to SCOP (version 1.69). Sequence similarity suggests that it is part of the a/b knot fold. 1v6zB contains a shallow composite knot (31#41), which
turns into a regular trefoil when two amino acids are cut from the N-terminus. (The random closure [see Materials and Methods] determines a trefoil right away.)
1uch contains the same structure as 1xd3. If the missing section in the center of this structure is joined by a straight line, it becomes knotted (52). In the yeast homologue (1cmx), amino
acids 63 to 77 are unstructured, and if we replace the missing parts by a straight line, we obtain a trefoil knot that has been identified before [6]. If we connect the unstructured region by
an arc present in the human structure, we obtain the same knot with five crossings.
eVisual inspection reveals that the calculated size of the knotted core is too small.
DOI: 10.1371/journal.pcbi.0020122.t001
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Intricate Knots in Proteins
UCH-L3 in human and yeast share only 33% [29] of their
sequences, but contain the same 5-fold knot as far as we can
tell from the incomplete structure in yeast. It is not only likely
that all species in between have the same knot—the link
between sequence and structure may also be used to predict
candidates for knots among isozymes or related proteins for
which the structure is unknown. For example, UCH-L4 in
mouse has 96% sequence identity with human UCH-L3. The
similarity with UCH-L6 in chicken is 86%, and with UCH-L1
about 55%. Indeed, a reexamination of the most recent
Protein Data Bank entries revealed that UCH-L1 contains the
same 52 knot as UCH-L3. (See the Update section—the
structure was not yet part of the January Protein Data Bank
release on which this paper is based.) Unfortunately, the
method is not foolproof because differences between knotted
and unknotted structures are sometime subtle. As we will
demonstrate in the next paragraph, a more reliable estimate
has to consider the conservation of major elements of the
knot, like loops and threads.
AOTCase—How a protein knot can alter enzymatic activity.
Somewhat surprisingly, we also identified a pair of homo-
logues for which topology is not preserved. N-acetylornithine
transcarbamylase (AOTCase [17]) is essential for the arginine
biosynthesis in several major pathogens. In other bacteria,
animals, and humans, a homologous enzyme (OTCase)
processes L-ornithine instead [30]. Both proteins have two
active sites. The first one binds carbamyl phosphate to the
enzyme. The second site binds acetylornithine in AOTCases
and L-ornithine in OTCases, enabling a reaction with
carbamyl phosphate to form acetylcitrulline or citrulline,
respectively [17, 31].
AOTCase in X. campestris has 41% sequence identity with
OTCase from Pyrococcus furiosus [32] and 29% with human
OTCase [31]. As demonstrated in Figure 2, AOTCase contains
a deep trefoil knot which is not present in OTCase (Figure 2,
right) and which modifies the second active site. The knot
consists of a rigid proline-rich loop (residues 178–185),
through which residues 252 to 256 are threaded and affixed.
As elaborated in [17], the reaction product N-acetylcitrulline
strongly interacts with the loop and with Lys252. Access to
subsequent residues is, however, restricted by the knot. L-
norvaline in Figure 2 (right) is very similar to L-ornithine but
lacks the N-e atom of the latter to prevent a reaction with
carbamyl phosphate. As the knot is not present in OTCase,
the ligand has complete access to the dangling residues 263–
268 and strongly interacts with them [31]. This leads to a
rotation of the carboxyl-group by roughly 1108 around the
Ca–Cb bond [17].
This example demonstrates how the presence of a knot can
modify active sites and alter the enzymatic activity of a
protein—in this case, from processing L-ornithine to N-
acetyl-L-ornithine. It is also easy to imagine how this
alteration happened: a short insertion extends the loop and
modifies the folding pathway of the protein.
Discussion
Nature appears to disfavour entanglements, and evolution
has developed mechanisms to avoid knots. Human DNA
wraps around histone proteins, and the rigidity of DNA
allows it to form a spool when it is fed into a viral capsid. One
end also stays in the loading channel and prevents subsequent
equilibration [33]. Knotted proteins are rare, although the
reason is far less well understood. Can the absence of
entanglement be explained in terms of particular statistical
ensembles, or is there an evolutionary bias? And how do these
structures actually fold?
Knots are ubiquitous in globular homopolymers [1–3,8], but
rare in coil-like phases [1,34–36]. It is likely that even a flexible
polymer will at least initially remain unknotted after a
Figure 2. Structures of Transcarbamylase from X. campestris with a Trefoil Knot and from Human without a Knot
(Left) Knotted section (residues 171–278) of N-acetylornithine transcarbamylase from X. campestris with reaction product N-acetylcitrulline (pdb code
1yh1 [17]) and interacting side chains.
(Right) Corresponding (unknotted) section (residues 189–286) in human ornithine transcarbamylase (pdb code 1c9y [31]) with inhibitor L-norvaline and
carbamyl phosphate. Colors change continuously from red (first residue in the section) to blue (last residue in the section). The two proteins have an
overall sequence identity of 29% [41]. Pictures were generated with VMD [43].
DOI: 10.1371/journal.pcbi.0020122.g002
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Intricate Knots in Proteins
collapse from a swollen state. In proteins, the free energy
landscape is considerably more complex, which may allow
most proteins to stay unknotted. The secondary structure and
the stiffness of the protein backbone may shift the length scale
at which knots typically appear, too [8]. If knotted proteins are
in fact more difficult to degrade, it might also be disadvanta-
geous for most proteins to be knotted in the first place.
Unfortunately, few experimental papers address folding
and biophysical aspects of knots in proteins. In recent work
[37], Jackson and Mallam reversibly unfolded and folded a
knotted methyltransferase in vitro, indicating that chaper-
ones are not a necessary prerequisite. In a subsequent study
[38], the authors provide an extensive kinetic analysis of the
folding pathway. In conclusion, we would like to express our
hope that this report will inspire more experiments in this
small but nevertheless fascinating field.
Materials and Methods
To determine whether a structure is knotted, we reduce the
protein to its backbone, and draw two lines outward starting at the
termini in the direction of the connection line between the center of
mass of the backbone and the respective ends. These two lines are
joined by a big loop, and the structure is classified by the
determination of its Alexander polynomial [1,13]. To determine the
size of the knotted core, we delete successively amino acids from the
N-terminus until the protein becomes unknotted [1,6]. The proce-
dure is repeated at the C-terminus starting with the last structure that
contained the original knot. For each deletion, the outward pointing
line through the new termini is parallel to the respective lines
computed for the full structure. The thus determined size should,
however, only be regarded as a guideline. A better estimate can be
achieved by looking at the structure.
In Table 1 we include knotted structures with no missing amino
acids in the center of the protein. (A list of potentially knotted
structures with missing amino acids can be found in Table S3.)
Technically, the numbering of the residues in the mmcif file has to be
subsequent, and no two amino acids are allowed to be more than 6 Å
apart. In addition, the knot has to persist when two amino acids are
cut from either terminus. We have further excluded structures for
which unknotted counterexamples exist (e.g., only one nuclear
magnetic resonance structure among many is knotted or another
structure of the same protein is unknotted). If a structure is
fragmented, the knot has to appear in one fragment and in the
resulting structure obtained from connecting missing sections by
straight lines. Other knotted structures are only considered when at
least one additional member of the same structural family [9]
contains a knot according to the criteria above.
The enforcement of these rules leads to the exclusion of the
bluetongue virus core protein [6] (41) and photoreceptor phyto-
chrome A in D. radiodurans [11] (31), which have been previously
identified as being knotted. Both structures are fragmented and
become knotted only when a few missing fragments are connected by
straight lines. In the viral core protein, the dangling C-terminus
threads through a loose loop and becomes knotted in one out of two
cases. On the other hand, the photoreceptor phytochrome A appears
to contain a true knot. Notably, our analysis suggests that the thus
connected structure of phytochrome A contains a figure-eight knot
instead of a trefoil as reported in [11]. Moreover, we excluded a
structure of the Autographa California nuclear polyhedrosis virus, which
contains a knot according to our criteria. However, the N-terminus is
buried inside the protein and the knot only exists because of our
specific connection to the outside.
To further validate our criteria, we implemented an alternative
method [4,8,39] that relies on the statistical analysis of multiple
random closures. We arbitrarily chose two points on a sphere (which
has to be larger than the protein) and connected each with one
terminus. The two points can be joined unambiguously, and the
resulting loop was analyzed by calculating the Alexander polynomial.
We repeated the procedure 1,000 times, and defined the knot as the
majority type.
Applying this analysis, we discovered 241 knotted structures in the
Protein Data Bank. All 241 structures are also present in the 273
structures (Table S1) that were identified by our method, and the knot
type is the same. Themissing32 structures (Table S2) aremostly shallow
knots and were already rejected according to our extended criteria.
The random closure also correctly discards rare structures with buried
termini. In conclusion, the method used in this paper is considerably
faster but requires a slightly increased inspection effort. Our
observations agree with [8], which provides an extensive comparison
of closures applied to proteins. A complete listing of knotted Protein
Data Bank structures is given in the Supporting Information.
Update. Recently, the structure of human UCH-L1 was solved and
released [40]. The protein shares 55% sequence identity with UCH-L3
[41], and it contains the same 5-fold knot. UCH-L1 is highly abundant
in the brain, comprising up to 2% of the total brain protein [42]. The
structure of UCH-L1 was not yet part of the January Protein Data
Bank edition on which the rest of this study is based. We also noticed
several new structures of knotted transcarbamylase-like proteins.
Supporting Information
Table S1. List of Knotted Protein Data Bank Entries
Found at DOI: 10.1371/journal.pcbi.0020122.st001 (79 KB DOC).
Table S2. List of Knotted Entries from Table S1 That Become
UnknottedWhenEnds AreConnected by theRandomClosureMethod
Found at DOI: 10.1371/journal.pcbi.0020122.st002 (28 KB DOC).
Table S3. List of Structures That Become Knotted When Missing
Sections Are Joined by Straight Lines
Found at DOI: 10.1371/journal.pcbi.0020122.st003 (35 KB DOC).
Accession Numbers
The Protein Data Bank (http://www.pdb.org) accession numbers for
the structures discussed in this paper are human UCH-L3 (1xd3),
UCH-L3 yeast homologue (1cmx), human UCH-L1 (2etl), photo-
receptor phytochrome A in D. radiodurans (1ztu), class II ketol-acid
reductoisomerase in E. coli (1yrl), class II ketol-acid reductoisomerase
in spinach (1yve), S-adenosylmethione synthetase in E. coli (1fug), S-
adenosylmethione synthetase in rat (1qm4), AOTCase from X.
campestris (1yh1), SOTCase from B. fragilis (1js1), OTCase from P.
furiosus (1a1s), OTCase from human (1c9y), bluetongue virus core
protein (2btv), and baculovirus P35 protein in Autographa California
nuclear polyhedrosis virus (1p35).
Acknowledgments
Upon completion of this work we became aware of a related study [8],
which independently identified the knots in UCH-L3 and SOTCase in a
re-examination of protein knots. PV would like to acknowledge
discussions with François Nédélec and with Olav Zimmermann, in
which they proposed the potential link between protein knots and
degradation. LM and PV would also like to thank Rachel Gaudet for a
discussion about the function of ubiquitin hydrolase.
Author contributions. MK conceived the study. PV designed and
wrote the analysis code. PV and LM analyzed the data. PV, LM, and
MK wrote the paper.
Funding. This work was supported by National Science Foundation
grant DMR-04–26677 and by Deutsche Forschungsgemeinschaft grant
VI 237/1. LM is an Alfred P. Sloan Research Fellow.
Competing interests. The authors have declared that no competing
interests exist.
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