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2Physikalisches Institute and Bethe Center for Theoretical Physics,
Universit¨ at Bonn, Nussallee 12, 53115 Bonn, Germany
3California Institute of Technology, Pasadena, CA 92215,
4Institute for Nuclear Studies,
Ho˙ za 69, 00-681 Warsaw, Poland
Theoretical studies of stretching proteins with slipknots reveal a surprising growth of their un-
folding times when the stretching force crosses an intermed iate threshold. This behavior arises as
a consequence of the existence of alternative unfolding rou tes that are dominant at diﬀerent force
ranges. Responsible for longer unfolding times at higher fo rces is the existence of an intermediate,
metastable conﬁguration where the slipknot is jammed. Simu lations are performed with a coarsed
grained model with further quantiﬁcation using a reﬁned des cription of the geometry of the slip-
knots. The simulation data is used to determine the free ener gy landscape (FEL) of the protein,
which supports recent analytical predictions.
PACS numbers: 87.15.ap, 87.14.E-, 87.15.La, 82.37.Gk, 87. 10.+e
The large increase in determining new protein struc-
tures has led to the discovery of several proteins with
complicated topology. This new fact has arised the ques-
tion if their energy landscape and the folding mechanism
is similar to typical proteins. One class of such proteins
includes knotted proteins which comprise around 1% of
all structures deposited in the PDB database [1, 2]. A
related class of proteins contains more subtle geometric
conﬁgurations called slipknots [3, 4]. Recent theoretical
studies using structure-based models (where native con-
tacts are dominant) suggest that slipknot-like conforma-
tions act like intermediates during the folding of knotted
proteins [5]. This entire new mechanism is consistent
with energy landscape theory (FEL) and the funnel con-
cept [7, 8]. It was shown that the slipknot formation
reduces the topological barrier. Complementing regular
folding studies, additional information about the land-
scape was obtained by mechanical manipulation of the
knotted protein with atomic force microscopy [9] both
experimentally in [10, 11] and theoretically in [12, 13, 14].
For example, [12] it has been showen that unfolding pro-
ceeds via a series of jumps between various metastable
conformations, a mechanism opposite to the smooth un-
folding in knotted homopolymers.
Motivated by these early results, we now propose a uni-
ﬁed picture for the mechanical unfolding of proteins with
slipknots. In this Letter this question is addressed by
explaining the role of topological barriers along their me-
chanical unfolding pathways. Supported by our previous
results that knotted proteins can still have a minimally
frustrated funnel-like energy landscape, structure-based
theoretical coarse-grained models are used [15] to ana-
lyze the behavior of a slipknot protein under stretching.
Studies are performed for the α/β class protein thymi-dine kinase (PDB code: 1e2i [17]).
2 3 4 5F/LBracket1Ε/Slash1/Angstrom/RBracket17.27.57.88.1logΤ
FIG. 1: Dependence of the unfolding times τon the stretch-
ing force Ffor 1e2i (solid line, in red). In this Letter we
describe this mechanism as a superposition of two unfolding
pathways: I for small forces (dashed (lower) line, in blue),
and II for intermidiate and large forces (dashed-dotted (up -
per) line, green).
Most of our analysis is based on stretching simulations
under constant force [16]. The crucial signature for this
process is the overall unfolding time from the beginning
of the stretching until the protein fully unfolds. Normally
one expects that the transition between the native and
the unfolded basins to be limited by overcoming the free
energy barrier, which gets eﬀectively reduced upon an
application of a stretching force. The rate by which this
barrier is reduced depends on the distance between the
unfolded basin and the top of the barrier measured along
the stretching coordinate x. This idea was ﬁrst devel-
oped in the phenomenological model of Bell [18], which
states that the unfolding time τdecreases exponentially
with applied stretching force Fasτ(F) =τ0e−Fx
kBT. A2
reﬁned analysis performed in ref. [19] revealed that this
dependence is more complicated but still monotonically
decreasing.
The unfolding times for 1e2i measured in our simula-
tions are shown as the red curve in Fig. 1. In contrast to
the above expectations, increasing the force in the range
3-3.5ǫ/˚A surprisingly results in a larger stability of the
protein. ǫis the typical eﬀective energy of tertiary na-
tive contacts that is consistent with the value ǫ/˚A≃71
pN derived in [15]. A solution for this paradox is accom-
plished by realizing that unfolding is dominated by two
distinct, alternative routes that are dominant at diﬀerent
force regimes. A routing switch occurs when threshold is
crossed between weak and intermediate forces. At higher
forces, mechanical unfolding is dominated by a route that
involves a jammed slipknot. This jamming gives rise to
the unexpected dependence of unfolding time on applied
force. Characterizing this mechanism is the central goal
of this Letter.
FIG. 2: A slipknot (left) consists of a threaded loop (k1−k2,
in red) which is partialy threaded through a knotting loop
(k2−k3, in blue). An example of a protein conﬁguration with
a tightened slipknot is shown in the right panel.
To describe the evolution of a slipknot quantitatively
requires a reﬁned description. A slipknot is character-
ized by the three points shown in Fig. 2. The ﬁrst
pointk1is determined by eliminating amino acids con-
secutively from one terminus until the knot conﬁgura-
tion is reached (which can be detected e.g. by applying
the KMT algorithm [20]). The two additional points,

2Physikalisches Institute and Bethe Center for Theoretical Physics,

Universit¨ at Bonn, Nussallee 12, 53115 Bonn, Germany

3California Institute of Technology, Pasadena, CA 92215,

4Institute for Nuclear Studies,

Ho˙ za 69, 00-681 Warsaw, Poland

Theoretical studies of stretching proteins with slipknots reveal a surprising growth of their un-

folding times when the stretching force crosses an intermed iate threshold. This behavior arises as

a consequence of the existence of alternative unfolding rou tes that are dominant at diﬀerent force

ranges. Responsible for longer unfolding times at higher fo rces is the existence of an intermediate,

metastable conﬁguration where the slipknot is jammed. Simu lations are performed with a coarsed

grained model with further quantiﬁcation using a reﬁned des cription of the geometry of the slip-

knots. The simulation data is used to determine the free ener gy landscape (FEL) of the protein,

which supports recent analytical predictions.

PACS numbers: 87.15.ap, 87.14.E-, 87.15.La, 82.37.Gk, 87. 10.+e

The large increase in determining new protein struc-

tures has led to the discovery of several proteins with

complicated topology. This new fact has arised the ques-

tion if their energy landscape and the folding mechanism

is similar to typical proteins. One class of such proteins

includes knotted proteins which comprise around 1% of

all structures deposited in the PDB database [1, 2]. A

related class of proteins contains more subtle geometric

conﬁgurations called slipknots [3, 4]. Recent theoretical

studies using structure-based models (where native con-

tacts are dominant) suggest that slipknot-like conforma-

tions act like intermediates during the folding of knotted

proteins [5]. This entire new mechanism is consistent

with energy landscape theory (FEL) and the funnel con-

cept [7, 8]. It was shown that the slipknot formation

reduces the topological barrier. Complementing regular

folding studies, additional information about the land-

scape was obtained by mechanical manipulation of the

knotted protein with atomic force microscopy [9] both

experimentally in [10, 11] and theoretically in [12, 13, 14].

For example, [12] it has been showen that unfolding pro-

ceeds via a series of jumps between various metastable

conformations, a mechanism opposite to the smooth un-

folding in knotted homopolymers.

Motivated by these early results, we now propose a uni-

ﬁed picture for the mechanical unfolding of proteins with

slipknots. In this Letter this question is addressed by

explaining the role of topological barriers along their me-

chanical unfolding pathways. Supported by our previous

results that knotted proteins can still have a minimally

frustrated funnel-like energy landscape, structure-based

theoretical coarse-grained models are used [15] to ana-

lyze the behavior of a slipknot protein under stretching.

Studies are performed for the α/β class protein thymi-dine kinase (PDB code: 1e2i [17]).

2 3 4 5F/LBracket1Ε/Slash1/Angstrom/RBracket17.27.57.88.1logΤ

FIG. 1: Dependence of the unfolding times τon the stretch-

ing force Ffor 1e2i (solid line, in red). In this Letter we

describe this mechanism as a superposition of two unfolding

pathways: I for small forces (dashed (lower) line, in blue),

and II for intermidiate and large forces (dashed-dotted (up -

per) line, green).

Most of our analysis is based on stretching simulations

under constant force [16]. The crucial signature for this

process is the overall unfolding time from the beginning

of the stretching until the protein fully unfolds. Normally

one expects that the transition between the native and

the unfolded basins to be limited by overcoming the free

energy barrier, which gets eﬀectively reduced upon an

application of a stretching force. The rate by which this

barrier is reduced depends on the distance between the

unfolded basin and the top of the barrier measured along

the stretching coordinate x. This idea was ﬁrst devel-

oped in the phenomenological model of Bell [18], which

states that the unfolding time τdecreases exponentially

with applied stretching force Fasτ(F) =τ0e−Fx

kBT. A2

reﬁned analysis performed in ref. [19] revealed that this

dependence is more complicated but still monotonically

decreasing.

The unfolding times for 1e2i measured in our simula-

tions are shown as the red curve in Fig. 1. In contrast to

the above expectations, increasing the force in the range

3-3.5ǫ/˚A surprisingly results in a larger stability of the

protein. ǫis the typical eﬀective energy of tertiary na-

tive contacts that is consistent with the value ǫ/˚A≃71

pN derived in [15]. A solution for this paradox is accom-

plished by realizing that unfolding is dominated by two

distinct, alternative routes that are dominant at diﬀerent

force regimes. A routing switch occurs when threshold is

crossed between weak and intermediate forces. At higher

forces, mechanical unfolding is dominated by a route that

involves a jammed slipknot. This jamming gives rise to

the unexpected dependence of unfolding time on applied

force. Characterizing this mechanism is the central goal

of this Letter.

FIG. 2: A slipknot (left) consists of a threaded loop (k1−k2,

in red) which is partialy threaded through a knotting loop

(k2−k3, in blue). An example of a protein conﬁguration with

a tightened slipknot is shown in the right panel.

To describe the evolution of a slipknot quantitatively

requires a reﬁned description. A slipknot is character-

ized by the three points shown in Fig. 2. The ﬁrst

pointk1is determined by eliminating amino acids con-

secutively from one terminus until the knot conﬁgura-

tion is reached (which can be detected e.g. by applying

the KMT algorithm [20]). The two additional points,