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Resonant optimization in the mechanical unzipping of DNA
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2014 EPL 105 68005
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March 2014
EPL, 105 (2014) 68005
doi: 10.1209/0295-5075/105/68005
www.epljournal.org
Resonant optimization in the mechanical unzipping of DNA
Ana Elisa Bergues-Pupo1,2 , Fernando Falo2,3 and Alessandro Fiasconaro4(a)
1
Departamento de Fı́sica, Universidad de Oriente - 90500 Santiago de Cuba, Cuba
Departamento de Fı́sica de la Materia Condensada, Universidad de Zaragoza - 50009 Zaragoza, Spain
3
Instituto de Biocomputación y Fı́sica de Sistemas Complejos (BIFI), Universidad de Zaragoza
50018 Zaragoza, Spain
4
School of Mathematical Sciences, Queen Mary University of London - Mile End Road, London E1 4NS, UK
2
received 16 January 2014; accepted in final form 12 March 2014
published online 27 March 2014
PACS
PACS
PACS
87.15.A- – Theory, modeling, and computer simulation
87.15.-v – Biomolecules: structure and physical properties
05.40.-a – Fluctuation phenomena, random processes, noise, and Brownian motion
Abstract – The mechanical separation of the double-stranded DNA in single-molecule experiments is of fundamental importance in the understanding of the replication or transcription
processes. Time-dependent forces can influence in different ways the dynamics of this separation.
We study here this unzipping of DNA in the framework of the mesoscopic Peyrard-Bishop-Dauxois
model under the influence of a periodic driving. Two different protocols, both of them feasible
experimentally, have been studied under two modes of pulling: controlled force and controlled
position. A strong resonant activation phenomenon has been observed in the magnitudes that
characterize the mechanical unzipping such as the mean opening time, the mean opening force,
and the mean critical opening force, all of them as a function of the frequency of the driving. This
optimal frequency region has been observed for all the cases studied both in a uniform DNA of
adenine-thymine nucleotides and in a real DNA sequence. Importantly, a well precise resonant
frequency can be determined with the use of one of this protocols.
c EPLA, 2014
Copyright Introduction. – Separation of the double-stranded
DNA (dsDNA) into single-stranded DNA (ssDNA) is an
essential step in biological processes like replication and
transcription. For this reason, the mechanical response of
dsDNA under the action of an external force is of great relevance, and in this context the unzipping is one of the most
useful and interesting single-molecule experiments [1–3].
Mechanical unzipping has been conducted either with constant pulling velocity or with constant force. Nevertheless,
the effect of alternate forces on the dynamical response of
DNA and other biomolecules had become more and more
relevant, and has been recently addressed both theoretically and experimentally. First, the linear and nonlinear
resonances induced by radiation with frequency in the order of THz has been argued [4–8], and in ref. [7] a reduction
of the unzipping force was obtained due an inertial resonant mechanism at the frequencies of the normal modes of
the dsDNA chain. Second, the use of mechanical alternate
forces in some single-molecule experiments [9–13] has been
investigated. In [9] a periodic pulling protocol is proposed
(a) E-mail:
[email protected]
for the folding and unfolding cycle of a protein. This periodic protocol presents interesting outcomes, as it allows
a better resolution with respect to the standard linear
pulling in the reconstruction of the free-energy landscape
of the molecule. On the other hand, stochastic effects such
as the stochastic resonance and resonant activation (RA)
were observed during the folding/unfolding experiments
of short DNA hairpins [10], and in numerical simulations
in RNA [11] and Ubiquitin unfolding [12]. It is important to notice that the use of mechanical alternate forces
is biologically relevant because many processes occur in
a periodic way due to the periodic energy consumption
inside the cell [13].
In this letter, we investigate a complementary aspect of
the mechanical unzipping concerning the optimization of
the escape rates with alternate forces. We found a RA
in the mechanical unzipping, i.e. a big reduction of the
opening time of the chain as a function of the frequency for
a broad frequency range [14]. This phenomenon has been
widely studied both theoretically [14–19] and experimentally [20]. The RA has been also observed in a model of
68005-p1
Ana Elisa Bergues-Pupo et al.
the transport of a polymer in a metastable potential which
oscillates periodically in time [21,22] and in the translocation driven by an active pore, which acts with a sinusoidal force in the polymer dynamics [23,24]. We study
here the mechanical unzipping in a broad frequency interval of alternate forces, in the framework of the PeyrardBishop-Dauxois (PBD) model [25] modified to include the
solvation barrier [26,27]. This model has been used for describing the mechanical DNA unzipping [28,29]. We found
that a mechanism similar to the standard RA is obtained
in the model under different ways in which the unzipping
experiment is realized. We consider this result some important news involving both the theoretical background
of the stochastic activations and the possibility to find
optimized parameters (in waiting times and/or required
force intensity) in the DNA unzipping experiments. Interestingly, and surprisingly in the thermal activation processes, the presence of a quite precise resonant frequency
can be observed. This novelty arise as a consequence
of the consecutive ordered opening of the nucleotides
of the chain which synchronize with the period of the
driving.
Model and methods. – The PBD model describes
the 1d dynamics of the strand separation of a dsDNA
in a mesoscopic approach. We use a modified version
of the model where a barrier is included to account for
the interaction of the DNA bps with the solvent [6,7,27].
The potential energy of the system has two contributions:
V (yn ), that describes the interaction of the bases belonging to the same base pair (bp) and W (yn−1 , yn ), that accounts for the stacking interaction between consecutive
2
bps. V (yn ) = D(e−αyn − 1)2 + Ge−(yn −y0 ) /b is the sum of
the Morse potential and a Gaussian barrier. D is the bp
dissociation energy and α sets the width of the potential
well. The parameters G, y0 and b are the barrier height,
position and width, respectively. The stacking interaction
reads W (yn−1 , yn ) = 12 K(1 + ρe−δ(yn +yn−1 ) )(yn − yn−1 )2 ,
with K the coupling constant, ρ sets the anharmonic character of the staking and δ sets the scale length for this
behaviour [25].
The mechanical unzipping is simulated in two modes
that can be realized experimentally: controlled force (CF)
and controlled position (CP). In the CF mode, one end of
the chain is pulled apart from the other at a given force.
In the CP mode, it is the separation of the two strands
that is kept fixed or varied with a given law (generally
constant velocity).
Here, we study both these modes under the two different
protocols depicted in fig. 1, where a force F (t) is differently
applied as explained below. In the first one, called Protocol A, the first bp is pulled either by a constant and an
alternate forces F (t) = [Fc + A cos(2πνt + φ)] (CF mode)
or an alternate displacement in the pulling force F (t) =
Kd [Y (t)−y1 ] (CP mode), with Y (t) = V t+Y cos(2πνt+φ)
the controlled position, and Kd the stiffness of the experimental device. Here ν is the frequency of the alternate
Fig. 1: (Colour on-line) Different protocols used for the
mechanical unzipping. (a) Protocol A: pulling is applied to
the first bp (in the control force or control position modes).
(b) Protocol B: pulling is applied to the first bp (in the control
force or control position modes) and an ac force is applied to
all the bps of the chain.
contributions, A, Y are the amplitudes in the CF and CP
modes, and V is the constant pulling velocity. The harmonic contribution in the CP mode is necessary in order
to check the value of the force which acts in the dynamics.
This protocol is similar to those of [9,10], where unfolding and folding are performed with a periodic driving. In
the second protocol, called Protocol B, the periodic driving A cos(2πνt + φ) is applied uniformly to all bps, as in
refs. [6,7]. In this case the non-oscillating forces Fc and
[Kd (V t − y1 )] are added to the first bp Hamiltonian in
the CF or the CP mode, respectively. An additional term
of potential U (yn , yn+1 ) = 0.5Ks (yn+1 − yn )2 is included
in all cases in order to easily simulate the kinetics of the
ssDNA once the chain is open (yn > y0 ). The dynamics is
developed by solving the overdamped Langevin equations
of motion,
ẏn = −V (yn ) − W (yn , yn+1 ) − W (yn−1 , yn )
+ F (t) + U (yn , yn+1 ) + ξn (t),
(1)
with ξn (t) the Gaussian white noise with zero mean and
ξn (t)ξk (t) = 2kB T δn,k δ(t − t ), which is integrated with
a stochastic Runge-Kutta algorithm of the 4th order with
a dt = 0.01 [30,31].
The values of the potential parameters are the same of
those of [27] for an homogeneous adenine-thymine (AT)
chain: D = 0.05185 eV and α = 4 Å−1 for the Morse potential; G = 3D, y0 = 2/α and b = 1/2α2 for the Gaussian
barrier; and K = 0.03 eVÅ−2 , δ = 0.8 Å−1 and ρ = 3 for
the stacking interaction. For simplicity, we use dimensionless units: length is given in units of α−1 , mass in units of
the nucleotide mass (300 uma) and energy in units of D.
The dimensionless temperature is defined as T̃ = kB T /D.
With these transformations we get the following units for
the force, the frequency and the temperature: Fu = Dα =
332 pN, and Tu = 602 K. The
real time tR results rescaled
according to tR → t1 = α1 m
D = 0.19 ps. Moreover,
the overdamped equation of motion (eq. (1)) requires a
68005-p2
Resonant optimization in the mechanical unzipping of DNA
500
6
Prot. A
Prot. B
a)
b)
5.5
10
200
base
8
400
100
4
300
2
-5
10
-3
10
10
ν
-1
10
1
200
8
2.9
4
2.8
6
250
100
4
2
200
10-5
10-3
ν
10-1
101
2.7
3.5
2.6
0
150
Results. – CF mode. In the CF mode, the following
magnitudes are evaluated: the average opening time top ,
defined as the time needed to open a given number of consecutive bps of the chain starting from the first one (we
use here 10 bps), the residence times tr of each bp, defined
as the time that each base remains inside the Morse potential (yn < y0 ), and the critical force Fcr , defined as the
critical value of the force that is able to open the chain after a certain waiting time ts . The field phase φ is changed
randomly in each realization in the interval [0, 2π]. We
use T = 0.5 (T = 301 K), A = 0.5 and Fc = 1.5 for the
temperature, field amplitude and constant unzipping force
values, respectively.
Figure 2 shows the results of the simulation. The curves
show a clear RA mechanism for both protocols, i.e. the
behaviour of the unzipping times is quite similar to that
of the mean first-passage time of a single Brownian particle crossing a fluctuating potential barrier [14–16]. This
means that an evident signature of three different time
scales is present, identified by the corresponding frequency
regions: at high frequency (HF), for ν 10 (7.7 THz),
top presents a plateau that corresponds to the case where
the effective escape barrier is given by an average potential constant in time. At low frequencies (LF), for
ν ≤ 10−3 (0.77 × 10−3 THz), there is a notable increase
of top times because the period of the driving is much
higher than any specific unzipping time, and the contribution to the average of the events with the potential in
its higher barriers becomes dominant. The most interesting region appears at intermediate frequency values (IF),
where there is a notable decrease in the top , with a minimum value of the same order of magnitude of the period of
the corresponding driving. This synchronization between
time and period of the driving is the fingerprint of the
RA mechanism. Such behaviour is observed in both the
mean opening time and the mean residence time tr . In
fact, the inset of fig. 2 shows the mean residence times for
each base of the chain and it is evident that for each one
of them the synchronization occurs in the same way than
the total opening time top . Differently than in other
works [6,7], the resonant mechanism here does not depend
4.5
0
10
base
350
Prot. A
Prot. B
5
6
〈 Fcr 〉
450
〈 t op 〉
time scaling t1 → t = t1 /γ̃ = 0.192 γ ps2 . Where the
dimensionless damping γ̃ has been used (γ = γ̃/t1 ) (see
footnote 1 ).
The dimensionless parameters of the model are then
D = 1, α = 1, K = 0.0362 and δ = 0.2. We set
Ks = Kd = 1 (= 0, 8296 eVÅ−2).
We use a homogeneous chain of 30 bps. All the bp separations are set to yn = 0 at t = 0. Different driving
frequencies have been spanned, and the outcomes have
been averaged over N realizations.
2.5 -5
10
10-3 10-1
3
101
2.5
100
-5
-4
-3
-2
-1
0
1
10 10 10 10 10 10 10
ν
2
-5
-4
-3
-2
-1
0
1
10 10 10 10 10 10 10
ν
Fig. 2: (Colour on-line) CF mode. (a) Average opening time
at different frequencies under the constant force Fc = 1.5. Insets: average residence times for the first 10 bps of the chain
for Protocol A (upper panel) and Protocol B (lower panel).
(b) Critical unzipping force at different frequencies. Inset:
zoom of the resonant region of Protocol B. In both cases
A = 0.5 and T = 0.5.
on any inertial property of the chain, being the system
overdamped.
The frequency resonant region is also evident in another
important measure: the mean critical opening force Fcr .
We performed a set of calculations in order to find the
dependence of Fcr on the frequency by fixing the amplitude of the oscillating term. Figure 2(b) shows the results
for the two protocols for ts = 180. We can see how the
curves of Fcr exhibits a clear minimum in the resonant
region, reproducing the same behaviour as top .
Both the measures top and Fcr (and also the top equivalent tr measure) provide the evidence of optimized
conditions for the experimental execution of the unzipping
in the two protocols.
Two main differences can be recovered in these plots.
The first concerns the absolute values of the measures. As
expected, at HF, results are independent of the protocol.
At LF, top and Fcr of Protocol B are much higher than
in Protocol A because the oscillation force acts with the
same phase on all bps of the chain. A second difference
concerns the resonant region, which in Protocol B is relatively much deeper than in Protocol A, and shows a well
precise resonant frequency value instead of a large resonant plateau. This last property is related to the special
way in which this protocol is defined and corresponds to a
situation where the period of the force synchronizes with
all the opening times of the different bases of the chain.
To check this effect, a different number of opened bps N0
have been simulated for this protocol, and are shown in
fig. 3. For a small N0 the opening times of the bases involve a wide region of periods. With increasing N0 , the
1 A possible estimation of the damping γ of each bp is the followopening time obviously increases, but also the synchroing. We assume that each bp be a sphere of radius R = 1 nm and
nization of the simultaneous opening of all the bases with
mass m = 300 uma immersed in water with viscosity μ = 1002 μ Pa s.
With these magnitudes we use the Stokes formula: γ = 6πR μ/m = the period of the force at the same phase becomes more
37.9 ps−1 . With this value the time unit is then 1.3 ps.
selective. The result is a narrow minimum able to select
68005-p3
Ana Elisa Bergues-Pupo et al.
20
100
0
NO = 10
t op
300 10-5
-3
10
ν
-1
10
1
10
0
-5
10
〈 tr 〉
10
-3
10
ν
-1
10
1
40
P (top)
0
150
NO = 15
t op
5
100
300 10-5
-3
10
ν
-1
10
1
0
-5
10
〈 tr 〉
15
10
-3
10
ν
-1
10
1
P (top)
10
200 10
5
100 5
20
0.01
10
0.008
10
〈 Δt r 〉
300 15
40
ν = 10
50
-4
-3
-2
-1
0
1
10 10 10 10 10 10
30
0
t op
300 10-5
-3
10
ν
-1
10
1
0
-5
10
10
-3
10
ν
-1
10
1.6
a)
300
Prot. A
Prot. B
1
0.004
Prot. A
Prot. B
1
220
0.8
200
180
0.6
160
-5
-4
-3
-2
-1
0
1
10 10 10 10 10 10 10
-5
10
-4
10
-3
10
-2
10
ν
-1
10
0
10
0.004
0
0.01
c)
-4
ν = 10
-3
ν = 10
-2.5
ν = 10
-2
ν = 10
1
ν = 10
0.006
0
0
1.2
〈 Fop 〉
240
ν
Prot. A - CP
0.006
0
b)
ν = 10-4
-3
ν = 10
ν = 10-2.5
ν = 10-2
ν = 101
0.002
800
t op
Prot. B - CF
0.008
400
800
t op
Prot. B - CP
d)
-4
ν = 10
-3
ν = 10
-2.5
ν = 10
-2
ν = 10
1
ν = 10
0.008
0.006
0.004
0.002
0
400
t op
800
0
0
400
t op
800
10
b)
1.4
〈 t op 〉
260
400
0.002
10
Fig. 3: (Colour on-line) Mean opening time top as a function
of the frequency for a different number of opening monomers
N0 (left). On the right: first column, probability distribution
of top at the frequency of the minimum; second column, the
cumulative resident times for the bases tr , and in the third
column, the differential resident times of the bases as a function
of the frequency (from top to bottom N0 = 5, 10, 15).
280
0
20
ν
320
0
0
-2.25
100
0.004
30
200
5
0.006
0.002
〈 Δt r 〉
300 10
a)
ν = 10-4
-3
ν = 10
ν = 10-2.5
ν = 10-2
ν = 101
0
10
ν = 10-2.25
200
Prot. A - CF
0.008
30
P (top)
No = 5
No = 10
No = 15
10
〈 t op 〉
40
200
P (top)
250
〈 Δt r 〉
300 5
P (top)
Prot B - CF
300
〈 tr 〉
5
-2
ν = 10
P (top)
a)
P (top)
NO = 5
350
1
10
Fig. 4: (Colour on-line) CP mode. (a) Average opening time
at different frequencies. (b) Average opening force. In both
cases V = 0.05, Y = 5, A = 0.5 and T = 0.5.
an optimum frequency value. In this way the repetition
of the escape process with a uniform phase distribution of
the oscillating driving forces, can give rise to a strict selection of the resonant frequency of the system. This feature
is also visible in the right panel of fig. 3 where the tr and the differential resident time for each base Δtr are
shown. Δtr is defined as the difference tr (n) − tr (n − 1),
where tr (n) is the time spent by the n-th bp to open.
In particular, the mean differential time Δtr for both
N0 = 10 and N0 = 15 presents a narrow vertical line at
ν ≈ 10−2.25 that shows that the resident times at that frequency are lower (and similar) for all the bases, while for
N0 = 5, the resident times are uniform in a large frequency
region for quite all the bases opened. This feature is similar to the Protocol-A case. The probability distribution at
the minima (1st column on the right of fig. 3) presents a
Fig. 5: (Colour on-line) Probability distribution of top at differnt frequency values. (a) Protocol A-CF mode. (a) Protocol B-CF mode. (a) Protocol A-CP mode. (a) Protocol B-CP
mode.
second peak clearer and clearer with increasing the number of opening bases, indicating the selection of the opening times at the different periods of the driving.
CP mode. The average opening time top has been
also computed in the CP mode. We determined the average opening force Fop defined as the force on the spring
that pulls the first bp, Fop = Kd (Y (t) − y1 ), averaged
on the time and on the N realizations of the experiment.
Figure 4 shows that the RA mechanism is similar to that
observed in the CF mode: a minimum of both top and
Fop is obtained as a function of the driving frequency.
As in the previous mode, both protocols show the same
HF behaviour. Again, Protocol B presents a well precise
resonant frequency, as in the CF case.
Finally, we analyzed the probability distribution of top
for the two protocols and the two unzipping modes at
different frequencies. The results are shown in fig. 5.
At HF and IF the distributions are nearly Poissonian.
For Protocol B two peaks are present in the distribution,
corresponding to the main resonant exit at the two first
semiperiods corresponding to the upper value of the force.
As expected, for the LF range there is a great dispersion
of top and the distributions are highly asymmetric, with
fast as well as very longly distributed opening times. The
opening time distributions for CP (see panels (c) and (d))
show a more limited region of times, due to the particular way this protocol works, which obliges the opening
to occur in a finite time, as the applied force progressively
increases due to the increase of the probe distance in time.
Concluding remarks. – We have obtained an optimum frequency region of the mean opening times and the
mean critical force for the DNA unzipping induced by force
oscillating in time in the framework of the Peyrard-BishopDauxois model, for all the cases studied in a short uniform
DNA chain of adenine-thymine (AT). This simplification
does not prejudice the generality of the outcomes. Longer
non-homogeneous chains also give the same qualitative
results. Figure 6 shows the mean opening time calculated
68005-p4
Resonant optimization in the mechanical unzipping of DNA
4000
3800
Prot. A - CF
3600
〈 t op 〉
3400
3200
A
A
A
A
A
A
A
A
A
A
〈 tr 〉
T
T
T
A
C
C
C
G
T
G
10-5
2800
2400
10-3
3000
ν
10-1
101
C
C
C
C
C
C
C
C
C
C
2600
2400
-6
-5
-4
-3
-2
-1
0
1
2
10 10 10 10 10 10 10 10 10
-3
10
ν
-1
10
1
10
〈 Δt r 〉
50
40
30
20
10
0
2000
1500
1000
500
-5
10
T
T
T
A
C
C
C
G
T
G
2800
-5
10
3600
3200
〈 Δt r 〉
-3
10
ν
-1
10
1
10
〈 Δt r 〉
0
2000
1500
1000
500
-5
10
-3
10
ν
-1
10
1
10
already performed. Conversely, Protocol B can be useful
for an experiment where the oscillating force is provided by
an electromagnetic or electric field having the wavelength
much longer than the dimension of the DNA chain. In this
case all the bps experience the same oscillating force. Although to our knowledge these frequency optimization has
not been observed experimentally yet, it can be checked
at the actual development of single-molecule experiments.
Indeed, as mentioned in the introduction, a resonant activation mechanism similar to the one here presented has
been already observed in DNA hairpins folding-unfolding
(see sect. VI of ref. [10]).
0
∗∗∗
ν
Fig. 6: (Colour on-line) Protocol A-CF. Left panel: top for
the P5 chain. Inset: the resident time for the bases tr .
Right panel: the mean differential time Δtr for (from top
to bottom) adenine-thymine bps only (A), for the cytosineguanine (C) and the real P5 chain.
This work is supported by the Spanish DGICYT
Projects No. FIS2011-25167, co-financed by FEDER
funds, and by the Comunidad de Aragón through a grant
to the FENOL group. AEB-P also acknowledges the financial support of Universidad de Zaragoza and Banco
Santander.
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