arXiv:1001.0009v1  [q-bio.BM]  30 Dec 2009Jamming proteins with slipknots and their free energy lands cape
Joanna I. Su/suppress lkowska1, Piotr Su/suppress lkowski2,3,4and Jos´ e N. Onuchic1
1Center for Theoretical Biological Physics,
University of California San Diego,
Gilman Drive 9500, La Jolla 92037,
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,
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
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
pathway II is shown in Fig. 5 (right).
Summarizing, we have analyzed the process of tighten-
ing of the slipknot in protein 1e2i and determined the cor-
responding free energy landscape. Its main feature is the
presence of a metastable configuration with a tightened
slipknot, which is observed for sufficiently large pulling
forces. This phenomenon does not exist for uniformly
elastic polymers. In this Letter we concentrated on pro-
tein 1e2i but similar behavior has also been observed forother proteins with slipknots, e.g. 1p6x. Our results
provide testable predictions that can now be verified by
AFM stretching experiments.
We appreciate useful comments of O. Dudko. The
work of J.S. was supported by the Center for Theo-
retical Biological Physics sponsored by the NSF (Grant
PHY-0822283) with additional support from NSF-MCB-
0543906. P.S. acknowledges the support of Hum-
boldt Fellowship, DOE grant DE-FG03-92ER40701FG-
02, Marie-Curie IOF Fellowship, and Foundation for Pol-
ish Science.
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