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16α, 15β, 14α, 13β, 12(helices bundle), 11 α(here the
number denotes a consecutive secondary structure ascounted from N-terminal, and αorβspeciﬁes whether
this is a helix or a β-sheet; for more details about the
structure of 1e2i see the PDB). This is followed by unfold-
ing of helices 11 α, 10αthat allows breaking of the con-
tacts inside the β-sheet created by the N-terminal, with
unfolding proceeding also from the N-terminal. Pathway
II also starts from the C-terminal but rapidly (as soon
as helix 15 is unfolded) switches to the N-terminal. In
this case, diﬀerently from pathway I, the β-sheet from
the N-terminal unfolds even before 13 β. These scenarios
indicate that the pathway I should be dominant at weak
forces since they are not suﬃcient to break the β-sheet
during ﬁrst steps of unfolding. The jammed pathway is
typical only if stretching forces are suﬃciently strong for
unfolding to proceed from the two terminals of the pro-
tein.
A similar phenomenon was ﬁrstly proposed in ref. [25]
and referred to as catch-bonds. Experimental evidence
suggesting this mechanism was ﬁrst observed for adhe-
sion complexes [26, 27]. Using AFM, at large forces the
ligand-receptor pair becomes entangled and therefore ex-
pands the unfolding time. A theoretical description of
this mechanism was given in ref. [28, 29, 30].
The kinetic data can also be used to determine the as-
sociated free energy landscape (FEL) [7]. In an initial
simpliﬁcation we associate the barriers along the stretch-
ing coordinate as the the kinetic bottlenecks during the
mechanical unfolding event. Generalizing Bell’s model,
a recent description of two-state mechanical unfolding in
the presence of a single transition barrier has been devel-
oped in [19], with the rate equation
τ(F) =τ0/parenleftBig
1−νFx†
∆G/parenrightBig1−1/ν
e−∆G
kBT/parenleftbig
1−(1−νFx†/∆G)1/ν/parenrightbig
,
(1)
whereνencodes the shape of the barrier. Here x†denotes
the distance between the barrier and the unfolded basin
(in a ﬁrst approximation it can be regarded as Findepen-
dent) and lies on the reaction coordinate along the AFM
pulling direction. It can be experimentally determined
by measuring how the stretching force modulates the un-
folding times τ. The height of the barrier is denoted by
∆G. Fig. 1 (unfolding times are given by solid red line)
shows that this single barrier theory is not suﬃcient for
the full range of forces. As described before, in the higher
force regime, additional basins have to be included in the
energy landscape. Models with several metastable basins
have been called multi-state FEL models [31]. Evidence
supporting the need of multi-states FEL was conﬁrmed
by AFM experiments in diﬀerent systems [32, 33].
To construct a multi-state FEL that incorporates two
unfolding pathways I and II we use a linear combina-
tion of eq. (1)-like expressions with diﬀerent shapes and
barrier heights. Each one of them essentially accounts
for the distinct barrier along a relevant unfolding route.
Fitting the stretching data to eq. (1) with a cusp-like4
2.5 33.5 4F/LBracket1Ε/Slash1/Angstrom/RBracket16.67.6logΤ
N U I
FIG. 5: Pathway II with two barriers. Left: dependence of the
unfolding time on the applied force with the data and the ﬁt
to the formula (1) for the ﬁrst maximum (lower, in green) and
for the second maximum (upper, in blue). Right: schematic
free energy landscape for this pathway, with jammed slipkno t
in a minimum between two barriers.
ν= 1/2 approximation (another possibility ν= 2/3 for
the cubic potential in general leads to similar results [19])
determines accurately the location and the height of the
potential barriers. Pathway II involves two barriers: ﬁrst
until the moment of creation of the intermediate which
is followed the untieing event. They are characterized by
(x1,∆G1) and (x2,∆G2) arising respectively from the
lower and upper ﬁts in Fig. 5 (left). The superposition
of these two ﬁts gives the overall mean unfolding time for
pathway II (dotted-dashed curve in green in Fig. 1). For
the ordinary slipknot unfolding (pathway I), the results
xIandGIarise from the dashed blue curve in Fig. 1.
This analysis leads to the results
x1= 2.3kBT˚A
ǫ, x2= 0.7kBT˚A
ǫ, xI= 1.4kBT˚A
ǫ,
∆G1= 8.0kBT, ∆G2= 4.2kBT, ∆GI= 4.7kBT.
We conclude that the free energy landscape consists of
two “valleys”. The force-dependent probability of choos-
ing one of the valleys during stretching depends on the
details of the protein structure. It is determined from our
simulations as shown in Fig. 4. Using these probability
values and the parameters above for xand ∆G, we can
accurately represent the simulation data using a linear
combination of equations of the form (1). This agreement
supports our analytical analysis and generalizes eq. (1)
for the full of range forces. In addition it demonstrates
that structure-based models suﬃciently capture the ma-
jor geometrical properties of a slipknotted protein. A
schematic representation of the free energy landscape for

16α, 15β, 14α, 13β, 12(helices bundle), 11 α(here the

number denotes a consecutive secondary structure ascounted from N-terminal, and αorβspeciﬁes whether

this is a helix or a β-sheet; for more details about the

structure of 1e2i see the PDB). This is followed by unfold-

ing of helices 11 α, 10αthat allows breaking of the con-

tacts inside the β-sheet created by the N-terminal, with

unfolding proceeding also from the N-terminal. Pathway

II also starts from the C-terminal but rapidly (as soon

as helix 15 is unfolded) switches to the N-terminal. In

this case, diﬀerently from pathway I, the β-sheet from

the N-terminal unfolds even before 13 β. These scenarios

indicate that the pathway I should be dominant at weak

forces since they are not suﬃcient to break the β-sheet

during ﬁrst steps of unfolding. The jammed pathway is

typical only if stretching forces are suﬃciently strong for

unfolding to proceed from the two terminals of the pro-

tein.

A similar phenomenon was ﬁrstly proposed in ref. [25]

and referred to as catch-bonds. Experimental evidence

suggesting this mechanism was ﬁrst observed for adhe-

sion complexes [26, 27]. Using AFM, at large forces the

ligand-receptor pair becomes entangled and therefore ex-

pands the unfolding time. A theoretical description of

this mechanism was given in ref. [28, 29, 30].

The kinetic data can also be used to determine the as-

sociated free energy landscape (FEL) [7]. In an initial

simpliﬁcation we associate the barriers along the stretch-

ing coordinate as the the kinetic bottlenecks during the

mechanical unfolding event. Generalizing Bell’s model,

a recent description of two-state mechanical unfolding in

the presence of a single transition barrier has been devel-

oped in [19], with the rate equation

τ(F) =τ0/parenleftBig

1−νFx†

∆G/parenrightBig1−1/ν

e−∆G

kBT/parenleftbig

1−(1−νFx†/∆G)1/ν/parenrightbig

,

(1)

whereνencodes the shape of the barrier. Here x†denotes

the distance between the barrier and the unfolded basin

(in a ﬁrst approximation it can be regarded as Findepen-

dent) and lies on the reaction coordinate along the AFM

pulling direction. It can be experimentally determined

by measuring how the stretching force modulates the un-

folding times τ. The height of the barrier is denoted by

∆G. Fig. 1 (unfolding times are given by solid red line)

shows that this single barrier theory is not suﬃcient for

the full range of forces. As described before, in the higher

force regime, additional basins have to be included in the

energy landscape. Models with several metastable basins

have been called multi-state FEL models [31]. Evidence

supporting the need of multi-states FEL was conﬁrmed

by AFM experiments in diﬀerent systems [32, 33].

To construct a multi-state FEL that incorporates two

unfolding pathways I and II we use a linear combina-

tion of eq. (1)-like expressions with diﬀerent shapes and

barrier heights. Each one of them essentially accounts

for the distinct barrier along a relevant unfolding route.

Fitting the stretching data to eq. (1) with a cusp-like4

2.5 33.5 4F/LBracket1Ε/Slash1/Angstrom/RBracket16.67.6logΤ

N U I

FIG. 5: Pathway II with two barriers. Left: dependence of the

unfolding time on the applied force with the data and the ﬁt

to the formula (1) for the ﬁrst maximum (lower, in green) and

for the second maximum (upper, in blue). Right: schematic

free energy landscape for this pathway, with jammed slipkno t

in a minimum between two barriers.

ν= 1/2 approximation (another possibility ν= 2/3 for

the cubic potential in general leads to similar results [19])

determines accurately the location and the height of the

potential barriers. Pathway II involves two barriers: ﬁrst

until the moment of creation of the intermediate which

is followed the untieing event. They are characterized by

(x1,∆G1) and (x2,∆G2) arising respectively from the

lower and upper ﬁts in Fig. 5 (left). The superposition

of these two ﬁts gives the overall mean unfolding time for

pathway II (dotted-dashed curve in green in Fig. 1). For

the ordinary slipknot unfolding (pathway I), the results

xIandGIarise from the dashed blue curve in Fig. 1.

This analysis leads to the results

x1= 2.3kBT˚A

ǫ, x2= 0.7kBT˚A

ǫ, xI= 1.4kBT˚A

ǫ,

∆G1= 8.0kBT, ∆G2= 4.2kBT, ∆GI= 4.7kBT.

We conclude that the free energy landscape consists of

two “valleys”. The force-dependent probability of choos-

ing one of the valleys during stretching depends on the

details of the protein structure. It is determined from our

simulations as shown in Fig. 4. Using these probability

values and the parameters above for xand ∆G, we can

accurately represent the simulation data using a linear

combination of equations of the form (1). This agreement

supports our analytical analysis and generalizes eq. (1)

for the full of range forces. In addition it demonstrates

that structure-based models suﬃciently capture the ma-

jor geometrical properties of a slipknotted protein. A

schematic representation of the free energy landscape for