alx-ai/arxiv_papers · Datasets at Hugging Face

text stringlengths 0 44.4k
k2andk3, correspond to the ends of this knot. In the
native state the protein 1e2i contains a slipknot with
k1= 10,k2= 128,k3= 298. These three points divide
the slipknot into two loops, which are called the knotting
loop and thethreaded loop . The former one is the loop of
the trefoil knot and the latter one is threaded through the
knotting loop. Unfolding of the slipknot upon stretch-
ing depends on the relative shrinking velocity of these
two loops (see Fig. 3). When the threaded loop shrinks
faster than the knotting loop, the slipknot unties. In the
opposite case the slipknot gets (temporarily) tightened
or jammed, resulting in a metastable state associated
to a local minimum in the protein’s FEL. Upon further
stretching, this conﬁguration eventually also unties. The
evolution of both loops of the slipknot is encoded in thetime dependence of the points k1,k2,k3, see Fig. 3.
pathway I pathway II
catch−bonds slip−bondspathway II
catch−bonds slip−bondspathway I
FIG. 3: The behavior of the slipknot during stretching (top)
is determined by the relative behavior of its two loops, en-
coded in the time dependence of k1,k2andk3(bottom). If the
threaded loop shrinks faster than the knotting loop, k1merges
withk2(bottom left) and the slipknot untightens (pathway I,
top left). If the knotting loop shrinks faster, k2approaches k3
(bottom right, ≃14000τ) and the slipknot gets temporarily
tightened (pathway II, top right). This is a metastable stat e
which can eventually untie further stretching , with k1ﬁnally
merging with k2(bottom right, ≃19000τ). Kinetic stud-
ies were performed slightly above folding temperature usin g
overdamped Langevin dynamics with typical folding times of
10000τ.
Before discussing the stretching of 1e2i, we explain why
a slipknot formed by a uniformly elastic polymer should
smoothly unfold under stretching. To simplify the discus-
sion we approximate the threaded and knotting loops by
circles of size RtandRk. These two loops shrink during
stretching and, when the threaded one eventually van-
ishes, the slipknot gets untied. If both loops have similar
sizes, the slipknot is very unstable and unties immedi-
ately. When the threaded loop is much larger than the
knotting one, Rt>> R k, untightening can be explained
as follows. The elastic energy associated to local bend-
ing is proportional to the square of the curvature. If the
loop is approximated by a circle of radius R, then its local
curvature is constant and equals R−1. The total elastic
energy is/contintegraltext
dsR−2∼R−1[21]. From the assumption
Rt>> R kwe conclude that upon stretching it is ener-
getically favorable to decrease Rtrather than Rk. This
happens until both radii become equal and then, just as
above, the slipknot gets very unstable and untightens. In
this discussion we have not yet taken into account that
when a slipknot is stretched some parts of a chain slide
along each other. This eﬀect could be incorporated by in-
cluding the friction generated by the sliding [22]. But in
the slipknot the sliding region associated with the knot-
ting loop is much longer than the region associated to3
the threaded loop. Thus this eﬀect results in a faster
tightening of the threaded rather than the knotting loop,
facilitating even more the untightening of the slipknot.
The above argument should apply to slipknots in
biomolecules because they are characterized by a per-
sistence length that in principle is simply related to their
elasticity [23]. For DNA this eﬀect is described by worm-
like-chain models (WLC) [24] and it has been conﬁrmed
experimentally. Although WLC models are too simple
to describe the protein general behavior, they are use-
ful in some limited applications. Thus at ﬁrst sight one
might expect that slipknots in proteins should smoothly
untie upon stretching. Proteins, however, are much more
complicated than DNA or uniformly elastic polymers.
The presence of stabilizing native tertiary contacts leads
to a jumping character during stretching [12]. In addi-
tion their bending energy is not uniform along the chain
due to the heterogeneity of the amino-acid sequence. As
a consequence it turns out that the intuition obtained
through the above analysis of polymers or WLC models
is misleading.
2 3 4 5F/LBracket1Ε/Slash1/Angstrom/RBracket10.51Prob/LParen1pathway I/RParen1
FIG. 4: Dependence on the applied stretching force of the
probability of choosing pathway I rather than II (see Fig. 3) .
This varying probability leads to the complicated dependen ce
of the total unfolding time on the stretching force observed in
Fig. 1.
Our analysis of the evolution of the endpoints k1,k2,k3
(Fig. 3, bottom) reveals that for various stretching forces
unfolding proceeds along two distinct pathways (Fig. 3,
top). In pathway I the slipknot smoothly unties, which
is observed for relatively weak forces. At intermediate
forces pathway II starts to dominate and the knotting
loop can shrink tightly before the threaded one vanishes.
In this regime the protein gets temporarily jammed (Fig.
3, right), leading to much longer unfolding times (catch
pathway). The probability of choosing pathway I at dif-
ferent forces is shown in Fig. 4. This pathway competi-
tion explains the nontrivial total unfolding time depen-
dence observed in Fig. 1.
The two diﬀerent pathways I and II arise from com-
pletely diﬀerent unfolding mechanisms. Pathway I starts
and continues mostly from the C-terminal side, along

k2andk3, correspond to the ends of this knot. In the

native state the protein 1e2i contains a slipknot with

k1= 10,k2= 128,k3= 298. These three points divide

the slipknot into two loops, which are called the knotting

loop and thethreaded loop . The former one is the loop of

the trefoil knot and the latter one is threaded through the

knotting loop. Unfolding of the slipknot upon stretch-

ing depends on the relative shrinking velocity of these

two loops (see Fig. 3). When the threaded loop shrinks

faster than the knotting loop, the slipknot unties. In the

opposite case the slipknot gets (temporarily) tightened

or jammed, resulting in a metastable state associated

to a local minimum in the protein’s FEL. Upon further

stretching, this conﬁguration eventually also unties. The

evolution of both loops of the slipknot is encoded in thetime dependence of the points k1,k2,k3, see Fig. 3.

pathway I pathway II

catch−bonds slip−bondspathway II

catch−bonds slip−bondspathway I

FIG. 3: The behavior of the slipknot during stretching (top)

is determined by the relative behavior of its two loops, en-

coded in the time dependence of k1,k2andk3(bottom). If the

threaded loop shrinks faster than the knotting loop, k1merges

withk2(bottom left) and the slipknot untightens (pathway I,

top left). If the knotting loop shrinks faster, k2approaches k3

(bottom right, ≃14000τ) and the slipknot gets temporarily

tightened (pathway II, top right). This is a metastable stat e

which can eventually untie further stretching , with k1ﬁnally

merging with k2(bottom right, ≃19000τ). Kinetic stud-

ies were performed slightly above folding temperature usin g

overdamped Langevin dynamics with typical folding times of

10000τ.

Before discussing the stretching of 1e2i, we explain why

a slipknot formed by a uniformly elastic polymer should

smoothly unfold under stretching. To simplify the discus-

sion we approximate the threaded and knotting loops by

circles of size RtandRk. These two loops shrink during

stretching and, when the threaded one eventually van-

ishes, the slipknot gets untied. If both loops have similar

sizes, the slipknot is very unstable and unties immedi-

ately. When the threaded loop is much larger than the

knotting one, Rt>> R k, untightening can be explained

as follows. The elastic energy associated to local bend-

ing is proportional to the square of the curvature. If the

loop is approximated by a circle of radius R, then its local

curvature is constant and equals R−1. The total elastic

energy is/contintegraltext

dsR−2∼R−1[21]. From the assumption

Rt>> R kwe conclude that upon stretching it is ener-

getically favorable to decrease Rtrather than Rk. This

happens until both radii become equal and then, just as

above, the slipknot gets very unstable and untightens. In

this discussion we have not yet taken into account that

when a slipknot is stretched some parts of a chain slide

along each other. This eﬀect could be incorporated by in-

cluding the friction generated by the sliding [22]. But in

the slipknot the sliding region associated with the knot-

ting loop is much longer than the region associated to3

the threaded loop. Thus this eﬀect results in a faster

tightening of the threaded rather than the knotting loop,

facilitating even more the untightening of the slipknot.

The above argument should apply to slipknots in

biomolecules because they are characterized by a per-

sistence length that in principle is simply related to their

elasticity [23]. For DNA this eﬀect is described by worm-

like-chain models (WLC) [24] and it has been conﬁrmed

experimentally. Although WLC models are too simple

to describe the protein general behavior, they are use-

ful in some limited applications. Thus at ﬁrst sight one

might expect that slipknots in proteins should smoothly

untie upon stretching. Proteins, however, are much more

complicated than DNA or uniformly elastic polymers.

The presence of stabilizing native tertiary contacts leads

to a jumping character during stretching [12]. In addi-

tion their bending energy is not uniform along the chain

due to the heterogeneity of the amino-acid sequence. As

a consequence it turns out that the intuition obtained

through the above analysis of polymers or WLC models

is misleading.

2 3 4 5F/LBracket1Ε/Slash1/Angstrom/RBracket10.51Prob/LParen1pathway I/RParen1

FIG. 4: Dependence on the applied stretching force of the

probability of choosing pathway I rather than II (see Fig. 3) .

This varying probability leads to the complicated dependen ce

of the total unfolding time on the stretching force observed in

Fig. 1.

Our analysis of the evolution of the endpoints k1,k2,k3

(Fig. 3, bottom) reveals that for various stretching forces

unfolding proceeds along two distinct pathways (Fig. 3,

top). In pathway I the slipknot smoothly unties, which

is observed for relatively weak forces. At intermediate

forces pathway II starts to dominate and the knotting

loop can shrink tightly before the threaded one vanishes.

In this regime the protein gets temporarily jammed (Fig.

3, right), leading to much longer unfolding times (catch

pathway). The probability of choosing pathway I at dif-

ferent forces is shown in Fig. 4. This pathway competi-

tion explains the nontrivial total unfolding time depen-

dence observed in Fig. 1.

The two diﬀerent pathways I and II arise from com-

pletely diﬀerent unfolding mechanisms. Pathway I starts

and continues mostly from the C-terminal side, along