PHYSICAL REVIEW E 86, 021919 (2012)
Spin-oscilla o model o he unzipping o biomolecules by mechanical o ce
A. P ados,1A. Ca pio,2and L. L. Bonilla3
1F´
ısica Te´
o ica, Uni e sidad de Se illa Apa ado de Co eos 1065, E-41080 Se illa, Spain
2Depa amen o de Ma em´
a ica Aplicada, Uni e sidad Complu ense de Mad id, 28040 Mad id, Spain
3G. Mill´
an Ins i u e o Fluid Dynamics, Nanoscience and Indus ial Ma hema ics, Uni e sidad Ca los III de Mad id, 28911 Legan´
es, Spain
(Recei ed 11 May 2012; e ised manusc ip ecei ed 8 July 2012; published 21 Augus 2012)
A spin-oscilla o sys em models unzipping o biomolecules (such as DNA, RNA, o p o eins) subjec o
an ex e nal o ce. The sys em comp ises a mac oscopic deg ee o eedom, ep esen ed by a one-dimensional
oscilla o , and in e nal deg ees o eedom, ep esen ed by Glaube spins wi h nea es -neighbo in e ac ion and
a coupling cons an p opo ional o he oscilla o posi ion. A a c i ical alue Fco an applied ex e nal o ce
F, he oscilla o es posi ion (o de pa ame e ) changes ab up ly and he sys em unde goes a fi s -o de phase
ansi ion. When he ex e nal o ce is cycled a di e en a es, he ex ension gi en by he oscilla o posi ion
exhibi s a hys e esis cycle a high loading a es, whe eas i mo es e e sibly o e he equilib ium o ce-ex ension
cu e a e y low loading a es. Unde cons an o ce, he loga i hm o he esidence ime a he s able and
me as able oscilla o es posi ion is p opo ional o F−Fcas in an A henius law.
DOI: 10.1103/PhysRe E.86.021919 PACS numbe (s): 87.15.Cc, 05.40.−a, 05.50.+q, 64.60.De
I. INTRODUCTION
Many physical si ua ions can be modeled by a mechanical
sys em coupled o a he mal ba h o o spin sys ems. Examples
abound: The collec i e Jahn-Telle e ec has been analyzed by
spin-phonon sys ems [1–3], mass spec ome y h ough a na-
noelec omechanical oscilla o whose esonan equency de-
c eases as single molecules a e added he e o [4], decohe ence
o a spin ep esen ing a wo-le el sys em due o coupling o a
boson ba h ( he spin-boson sys em) [5], a classical oscilla o
coupled o a spin causing wa e unc ion collapse he eo [6],
a1/2-spin ep esen ing a nonlinea Josephson phase quan um
bi coupled o an oscilla o (supe conduc ing esona o ) and o
a classical signal [7,8], ippling in clamped g aphene shee s
in es iga ed by means o a spin-s ing sys em [9], e c.
Ve y ecen ly we ha e in oduced a simple model in
which a single oscilla o is coupled o a chain o Ising spins
unde going Glaube dynamics in con ac wi h a he mal ba h
[10–12]. In ou model he spins in he chain a e coupled
only o hei nea es neighbo s, bu hei coupling cons an
is p opo ional o he oscilla o posi ion, which makes hei
in e ac ion e ec i ely long ange. In equilib ium, elimina ion
o he oscilla o coo dina es gi es ise o an e ec i e spin
in e ac ion equi alen o a one-dimensional Ising model wi h
mean-field coupling [13]. The e is a second-o de phase
ansi ion a a fini e empe a u e, wi h he oscilla o es
posi ion as i s o de pa ame e . Abo e he c i ical empe a u e,
he oscilla o es posi ion is ze o, he eby coinciding wi h ha
o he uncoupled oscilla o . Below he c i ical empe a u e, wo
symme ic nonze o es posi ions issue o h symme ically
om ze o as in he case o a pi ch o k bi u ca ion. In he limi
o as elaxa ion o he spins compa ed o he na u al pe iod
o he oscilla o , he oscilla o posi ion sa isfies an e ec i e
equa ion ha ing bo h nonlinea o ce and nonlinea ic ion
e ms [10,12]. In e es ingly, his nonlinea ic ion a ises om
he coupling o he mac oscopic elas ic mode wi h he in e nal
deg ees o eedom (modeled in ou sys em by he spins). A
ela ed mechanism has been p oposed o explain he “in e nal
ic ion” obse ed in expe imen s wi h p o eins o polyme s in
solu ion [14].
In ecen yea s, echnological de elopmen has allowed o
manipula e o isualize indi idual molecules and o measu e
mic oscopic o ces wi h high-p ecision ins umen s. These
single-molecule expe imen s (SMEs) p o ide key in o ma-
ion abou he he modynamic and kine ic p ope ies o
biomolecules, o e ing a complemen a y bu di e en pe spec-
i e o unde s and molecula p ocesses. An ex ensi e e iew
o hese echniques can be ound in Re . [15]. Using SMEs,
dis ibu ions desc ibing ce ain molecula p ope ies can be
measu ed, he eby allowing one o cha ac e ize he kine ics
o biomolecula eac ions and he obse a ion o possible
in e media e s a es. A ypical ou come o SMEs a e he
o ce-ex ension cu es o DNA, RNA, and o he biomolecules
like p o eins. In a seminal pape , Lipha d e al. [16] pulled
a RNA hai pin by an inc easing applied o ce un il i un olds
a a c i ical alue o he o ce, Fc≃14.5 pN. A e wa ds,
he molecule is pushed back, by dec easing he o ce, un il
i e olds. A low pulling (pushing) a es, he s e ching and
elaxing o ce-ex ension cu es a e supe imposed, and he
molecule un olds a he c i ical alue o he o ce Fc. Taking a
close look a his ansi ion, hopping be ween he wo possible
ex ension alues is obse ed. This sugges s ha he sys em is
bis able: The e a e wo possible s a es o he molecule wi h
s ochas ic ansi ions be ween hem. This physical pic u e is
confi med by expe imen s ca ied ou a cons an load, in
a na ow egion a ound he c i ical o ce [see Fig. 2(c) o
Re . [16]]. The esidence imes in he zipped and unzipped
s a es ha e an A henius-like dependence. When cycles o
pulling and pushing he molecule a e ca ied ou a high
loading a es, he ex ension o he molecule occu s a a highe
o ce F+>F
c, whe eas he hai pin olds a a lowe o ce
F−<F
c. Thus a hys e esis cycle a ises, and some au ho s
ha e been claimed his o be a signa u e o i e e sible
nonequilib ium beha io [16–19]. Mo e ecen ly many wo ks
ha e ied o unde s and hese unzipping expe imen s om a
physical poin o iew [17–26].
The elas ic esponse o DNA [27–36] and o he
biomolecules like p o eins [37–41] exhibi s simila beha io .
Despi e some di e ences in mino de ails, in all cases he
021919-1
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A. PRADOS, A. CARPIO, AND L. L. BONILLA PHYSICAL REVIEW E 86, 021919 (2012)
“basic” beha io is like ha ound in Re . [16]: The e is a
c i ical alue o he applied o ce Fc, a which he molecule
unzips (o un olds) and i s leng h inc eases discon inuously.
Some peculia i ies a ise when going in o fine de ail hough.
On he one hand, in expe imen s wi h DNA o RNA hai pins
he molecule is a ached o wo beads and he dis ance be ween
hem is con olled. Thus he o al leng h o he molecule is kep
fixed in hese expe imen s, and i is he con ol pa ame e . The
s e ching ansi ion is accompanied by a d op in he measu ed
o ce [16–19]. In e y ecen expe imen s, he applied o ce is
he con ol pa ame e , and he unzipping ansi ion occu s a a
cons an alue o he o ce [25]. Fo p o eins, on he o he hand,
a ypical saw oo h pa e n wi h egula ly spaced o ce peaks
is o en obse ed a he ansi ion. This beha io is usually
in e p e ed as he successi e un olding o each o he p o ein
domains in a single p o ein molecule [37–39]. Pe o ming
pulling and pushing back cycles, bo h a “simple” e e sible
beha io and hys e esis cycles simila o hose o Re . [16]ha e
been obse ed [40,41]. Finally, in he case o DNA, he e a e
di e en ansi ions ha depend on he expe imen de ails [30].
The e is a so-called o e s e ching (o B-DNA o S-DNA)
ansi ion a Fc≃65 pN. The base pai dis ance inc eases
by app oxima ely 70% o his alue o he o ce [26–28,36],
esul ing in a sudden elonga ion o he molecule. In addi ion,
he unzipping ansi ion occu s a a c i ical o ce Fcin he
ange 10–20 pN [24,28,32] (o he same o de o magni ude
o he c i ical o ce o RNA hai pins in Re . [16]).A his
alue o he o ce, many base pai s o double-s anded DNA
(dsDNA) a e “opened,” and he leng h suddenly inc eases. As
has been al eady poin ed ou [29,33,34], his beha io is a
signa u e o a fi s -o de ansi ion, wi h i s associa ed egion
o me as abili y [32].
In his pape we add an ex e nal o ce o ou p e ious
oscilla o -spin model [10] and analyze he esul ing o ce-
ex ension cu es. Quali a i ely hese cu es ha e he same
ea u es as hose o he o ce-ex ension cu es measu ed in
expe imen s wi h DNA, RNA, and o he biopolyme s de-
sc ibed abo e. A subc i ical empe a u es, ou spin-oscilla o
sys em has a fi s -o de phase ansi ion a a c i ical o ce Fc
wi h he oscilla o es s a e as i s o de pa ame e . We find
ha he DNA o ce-ex ension cu es co espond o cycling a
di e en a es he cu es o he fi s -o de phase ansi ion.
As in he expe imen s, we find a egion o me as abili y in
a ce ain ange o o ces, close enough o Fc. Mo eo e , he
esidence ime spen a he basin o a ac ion o bo h he s able
and me as able s a es obey an A henius law: I s loga i hm is
p opo ional o F−Fc. A no e abou e minology: We will
use he e ms olded o zipped o deno e he sho es alue
o he leng h, and un olded o unzipped o deno e he longes
one. These e ms a e equi alen in he model, bu hey should
be p ope ly “ ansla ed” o he conc e e physical si ua ion a
hand. Fo ins ance, olded and un olded a e mos app op ia e
in he con ex o p o ein olding, al hough hey a e also used
in he unzipping o RNA hai pins, while zipped o unzipped is
he mos adequa e o he ansi ion om double-s anded o
single-s anded DNA.
The es o he pape is as ollows. In Sec. II he oscilla o -
spin model is mo i a ed in a biological con ex , and i s
equilib ium p ope ies analyzed. The dynamical beha io o
he model is analyzed in Sec. III. The oscilla o obeys New on’s
second law wi h a mean-field o ce due o he coupling wi h
he spins. The la e flip s ochas ically ollowing Glaube
dynamics a empe a u e T[42]. This causes he oscilla o
posi ion o become a s ochas ic p ocess. In he limi o as
spins, he spin-oscilla o coupling gi es ise o a nonlinea
ic ion e m, which d i es he oscilla o o equilib ium. In
Sec. IV, we p esen Mon e Ca lo simula ions o he sys em
and analyze hem in he ligh o he e ec i e po en ial ac ing
on he oscilla o . Sec ion Vcon ains he main conclusions o
he p esen wo k.
II. THE MODEL
We conside a one-dimensional chain o leng h L0. The e
is a la ge numbe N+1 o “in e nal” deg ees o eedom
si ing a egula ly spaced la ice si es ha a e modeled by
Ising spins. Thus, he dis ance be ween spins is d0=L0/N
(Fig. 1). Assume ha we s e ch he chain so ha i s leng h
becomes L=L0+. Fo he sake o simplici y, we will
assume ha he spins a e egula ly spaced a e he s e ching,
so ha he dis ance be ween wo neighbo ing spins changes o
d=d0+/N. This assump ion amoun s o a “mean-field”
app oxima ion. The po en ial ene gy o he sys em is
V(,σ)=1
2mω22+J()
N+1

i=1
σiσi+1,(1a)
whe e
J()=J0−μ (1b)
is a unc ion o . The po en ial Vcon ains a ha monic mac o-
scopic elas ic e m mω2/2 and a spin ene gy a ising om
a nea es -neighbo in e ac ion. The spin coupling cons an J
depends linea ly on he sepa a ion be ween si es, and equals J0
o he ini ial chain leng h L0. This simple choice is easonable
o L. The in e ac ion be ween nea es neighbo spins
mimics (in a e y simple way) he sho - anged in e ac ion
be ween he in e nal deg ees o eedom o complex biological
molecules like nucleic acids. We assume ha bo h J0and μ
a e posi i e. Le us define
x=−J0
μ,(2)
such ha J(x) anishes o x=0. Fo x<0 ( olded s a e) he
in e ac ion be ween he spins is e omagne ic, whe eas o
x>0 (un olded s a e) i is an i e omagne ic. In e ms o x,V
becomes
V=1
2mω2x2+Fcx−μx
N+1

i=1
σiσi+1,F
c=mω2J0
μ(3)
excep o an i ele an addi i e cons an . The pa ame e Fc
has he dimensions o a o ce. I an ex e nal load Fis applied
d0
L0
FIG. 1. Ske ch o he model desc ibed in he main ex . The spin
ep esen ing he in e nal deg ee o eedom a each la ice si e is
shown.
021919-2
SPIN-OSCILLATOR MODEL FOR THE UNZIPPING OF ... PHYSICAL REVIEW E 86, 021919 (2012)
o he sys em, a new e m −Fx is added o (3) so ha he
po en ial ene gy is now
V=1
2mω2x2−Hx −μx
N+1

i=1
σiσi+1,H=F−Fc.(4)
This po en ial ene gy is he same as in oduced in Re . [10–12],
excep o he ex a e m −Hx. In e es ingly, in he “ze o field”
case, ou sys em has a second-o de phase ansi ion a a c i ical
empe a u e Tc, gi en by [10–12]
Tc=μ2(N+1)
mω2kB
,(5)
whe e kBis Bol zmann’s cons an . The es posi ion xis he
o de pa ame e o he ansi ion: Fo T>T
c,x=0, whe eas
o T<T
c he e a e wo equally p obable equilib ium s a es
wi h es posi ions x=±x0(x0>0).
The po en ial ene gy (4) has he key ing edien s o model
DNA/RNA beha io in un olding o e olding expe imen s.
Fo T<T
cand applied o ce F<F
c,H<0 s abilizes he
solu ion wi h es s a e −x0( olded s a e), whe eas o F>F
c,
H>0 and he s able solu ion has es s a e +x0(un olded
s a e). The main e ec o he “ex e nal field” His ha he
sys em unde goes a fi s -o de phase ansi ion a H=0(F=
Fc)[43]. Thus, Fcis he a c i ical alue o he o ce F:A any
empe a u e T<T
c he oscilla o es posi ion as a unc ion
o he o ce changes ab up ly a F=Fc. Fu he mo e he e is
a egion o me as abili y a ound Fc, as discussed in Sec. II A.
To analyze ou model, i is con enien o ende i s equa ions
o mo ion dimensionless fi s . Le ω−1be he ime uni .
The elas ic and he spin e m in he po en ial a e o he
same o de i he scale o he oscilla o posi ion is [x]=
μ(N+1)/(mω2). The o ce and he spin e m a e o he same
o de p o ided he scale o Fis [F]=μ(N+1), and he e o e
he o de o magni ude o he po en ial is [F][x]=μ2(N+
1)2/(mω2)=(N+1)kBTc. This scaling is easonable because
makes Vex ensi e. Fo he c i ical empe a u e Tc o be
size-independen , we mus assume ha μscales as (N+1)−1/2
in he limi o la ge sys em size [10–12]. Thus we can define
nondimensional a iables acco ding o x∗=x/[x], ∗= /[ ],
V∗=V/[V],..., whe e he uni s [x], [ ], [V],... a e as
defined in Table I. The dimensionless po en ial is
V∗=V
(N+1)kBTc
=x∗2
2−H∗x∗−x∗
N+1
N+1

i=1
σiσi+1,
(6a)
H∗=F∗−F∗
c,(6b)
wi h
F∗
c=Fc
μ(N+1) =mω2J0
μ2(N+1) =J0
kBTc
.(7)
TABLE I. Nondimensional uni s and pa ame e s.
x F Vθ
μ(N+1)
mω2
1
ωμ(N+1) (N+1)kBTcT
Tc
We will d op he as e isks in he ollowing (so as no o clu e
ou o mulas), and om now on e e y exp ession will be
w i en in e ms o he dimensionless a iables and pa ame e s.
A. Equilib ium s a e: E ec i e po en ial
In equilib ium, he join p obabili y dis ibu ion o
he oscilla o posi ion xand he spin configu a ion σ=
{σ1,...,σ
N+1}is he canonical dis ibu ion, which, in nondi-
mensional a iables, is
Peq(x,σ)=1
Zexp[−(N+1)V(x,σ)/θ].(8)
He e Zis a no maliza ion cons an . Le us s udy he equilib-
ium alues o he oscilla o posi ion x. Then we sum o e he
spin a iables o ob ain he ma ginal dis ibu ion p obabili y
Peq(x)=
σ
Peq(x,σ)=1
˜
Zexp[−(N+1)Ve (x)/θ],(9)
whe e ˜
Z=2NZ.InEq.(9),Ve is an e ec i e po en ial o
he x a iable,
Ve (x)=x2
2−Hx −θln cosh x
θ,(10)
whose minima will be he s able equilib ium alues o x.
The e o e,
xeq =H+ anh xeq
θ(11)
gi es he oscilla o es posi ion xeq in equilib ium as a
unc ion o he dimensionless ex e nal field H=F−Fcand
empe a u e θ.
Fo H=0, we eco e he model analyzed in
Re s. [10–12], in which xeq =0 is always a solu ion o
any θ.Fo θ>1, i is he only solu ion; i co esponds o
a maximum o Peq and is he e o e s able. A θ=1 wonew
s able equilib ia co esponding o wo di e en maxima o Peq
bi u ca e om ha ha ing xeq =0. Fo θ<1, he posi ions o
hese maxima a e ±x(0)
eq .Asθ→1−,weha e
x(0)
eq ∼3(1 −θ).(12)
On he o he hand, o θ→0+,weha ex(0)
eq →1, which is he
maximum alue o x(0)
eq >0. Fo H=0, he wo (posi i e o
nega i e) equilib ium es posi ions ±x(0)
eq a e equip obable
because hey co espond o equally deep minima o he
e ec i e po en ial (10), which is an e en unc ion o x o
H=0.
Fo H= 0 and empe a u es below c i ical, θ<1, he field
e m −Hxin Eq. (10) b eaks he symme y be ween x>0 and
x<0, and he e ec i e po en ial is no longe an e en unc ion
o x.I H<0, he field e m a o s he nega i e b anch x<0,
since i gi es a nega i e con ibu ion o he e ec i e po en ial.
The e o e, we expec o find he sys em in he “ olded” s a e
x<0 o low alues o he loading o ce, FFc.On he
con a y, o H>0, he ene gy o he posi i e b anch x>0
will be lowe ed by he field e m, s abilizing i . Thus, he
sys em will be in he “un olded” s a e x>0 o high alues
o he loading o ce, FFc.
021919-3
A. PRADOS, A. CARPIO, AND L. L. BONILLA PHYSICAL REVIEW E 86, 021919 (2012)
Di e en ia ing Eq. (11) wi h espec o H, we find
∂Hxeq =1−1
θsech2xeq
θ−1
.(13)
The igh -hand side o his exp ession is he ecip ocal o he
second de i a i e o he e ec i e po en ial, and he e o e we
can ew i e (13) as
∂2
xV|x=xeq ∂Hxeq =1.(14)
Then he e ec i e po en ial has a local minimum, and he
co esponding equilib ium es posi ion is s able o ∂Hxeq >
0, while he equilib ium es s a e is uns able o ∂Hxeq <0.
Since ∂Hxeq =θ/(θ−1) o xeq =0, he ze o es posi ion
o he oscilla o is s able a θ>1 and uns able a subc i ical
empe a u es θ<1.
The equilib ium posi ion xeq is a mono onically inc easing
unc ion o he applied load F(o he applied field H=F−
Fc) o supe c i ical empe a u es, θ>1. The e is a unique
alue o xeq o each alue o he applied o ce Fi θ>1,
and i is s able. As shown in Fig. 2,His nonmono onic as
a unc ion o xeq o subc i ical empe a u es θ<1: I has a
local maximum a −xband a local minimum a +xb, wi h
cosh2xb
θ=1
θ.(15)
Fo each alue o Hbe ween −Hband +Hb, wi h
Hb=xb− anh xb
θ,(16)
eq eq
FIG. 2. (Colo online) Plo o H s xeq o θ=0.5. The alues
±Hbbe ween which he e a e h ee equilib ium alues o he
oscilla o posi ion a e shown. Fo a gi en alue o H,−Hb<H<
+Hb, he wo locally s able equilib ium poin s xLand xR(big g een
ci cles) and he uns able one xU(small ed ci cle) a e indica ed. The
wo symme ic s able equilib ium poin s ±x(0)
eq co esponding o he
ze o field case a e also shown. The quali a i e shape o he cu e is
he same o all he subc i ical empe a u es θ<1.
he e a e h ee possible alues o xeq:xUbe ween −xband xb
is he e o e uns able, while he o he wo, xL<−xband xR>
xb, a e locally s able. The absolu e minimum o he po en ial
co esponds o he alue o xha ing he same sign as he
applied field H. The o he local minimum, wi h sgn(x)=
sgn(H), is a me as able s a e in he he modynamic sense.
Then we expec o find bis abili y in he sys em o |H|<H
b,
i.e., o a gi en ange o loadings |F−Fc|<H
ba ound he
“c i ical” o ce Fc.
III. DYNAMICS
The Hamil on equa ions o mo ion co esponding o he
nondimensional Hamil onian unc ion
H(x,p,σ)=p2
2+V(x,σ),(17)
wi h po en ial ene gy gi en by Eq. (6a),a e
˙
x=p, ˙
p=−∂xV(x,σ),(18)
so ha
¨
x=−x+H+1
N+1
N+1

i=1
σiσi+1.(19)
Acco ding o he s ochas ic Glaube dynamics, he spins flip
a a a e [42]
Wi(x,σ)=α
21−γ(x)
2σi(σi−1+σi+1),(20a)
γ(x)= anh 2x
θ.(20b)
He e Wi(x,σ) is he ansi ion a e om configu a ion σ o
Riσ, he same as σexcep o he sign o he i h spin. Since he
oscilla o e olu ion equa ion (19) includes a e m ha depends
on he s ochas ically changing spin configu a ion, he oscilla o
posi ion becomes an s ochas ic p ocess.
The a e age alues o he spin co ela ions,
Ci,n =σiσi+n,(21)
sa is y he sys em o equa ions
dCi,n
d =α−2Ci,n+1
2γ(x)(Ci,n−1+Ci,n+1
+Ci−1,n+1+Ci+1,n−1),(22)
o n⩾1, wi h he bounda y condi ion Ci,0=1. I he oscilla-
o posi ion xwe e ime independen , he spins would each he
equilib ium dis ibu ion co esponding o he cons an alue
x. Then he a e age spin co ela ions Ci,n=[ anh(x/θ)]n
would be independen o idue o he spa ial ansla ion
in a iance. Some hing simila occu s in he limi o la ge
sys em size, N1. Bo h xand he spin co ela ions Ci,n
become mac oscopic sel -a e aging a iables; i.e., hey end o
hei espec i e “mac oscopic alues,” xand 
Cn(independen
o i), which coincide wi h hei a e ages and a e he mos
p obable alues o he co esponding s ochas ic a iables [44].
Spli ing bo h xand he co ela ions Ci,n in hei co esponding
mac oscopic and fluc ua ing pa s,
x=x+δx, Ci,n =
Cn+δCi,n,(23)
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whe e δx and δCi,n a e O(N−1/2), Eqs. (19),(22), and (23)
yield he e olu ion equa ions,
¨
x=−x+H+
C1,(24a)
d
Cn
d =α[−2
Cn+γ(x)(
Cn−1+
Cn+1)],n⩾1,(24b)
o be sol ed wi h 
C0=1 and app op ia e ini ial condi ions.
In Eq. (24b) we ha e neglec ed e ms o o de 1/N such as
(δx)2,δxδCi,n,e c.
As he spins ep esen he in e nal deg ees o eedom o
he molecule, we assume ha hey e ol e apidly compa ed
wi h he ime scale o he mac oscopic deg ee o eedom
modeled by he oscilla o posi ion. Thus, he dimensionless
cha ac e is ic a emp a e sa isfies α1 ( ecall ha he
uni o ime has been chosen as ω−1). Thus, we can sol e
app oxima ely he sys em o Eqs. (24b) using a powe se ies
in he small pa ame e α−1[45–47], a p ocedu e akin o he
Hilbe me hod in kine ic heo y. I xwe e ime independen ,
he spin co ela ions 
Cnwould each he equilib ium alues
co esponding o x,

Cn,eq =[
C1,eq]n,
C1,eq = anh x
θ,(25)
in he long ime limi . The leading o de co ec ion o his
esul is [10]

C1=
C1,eq −τd
C1,eq
d ,(26)
whe e τis he spins a e age elaxa ion ime [45,46]
τ=1
2α
1+
C1
2
eq
1−
C1
2
eq2.(27)
Equa ion (26) does no depend on he ini ial condi ion o 
C1
which is o go en a e a ime much sho e han he oscilla o
na u al pe iod. Inse ing Eqs. (25)–(27) in o (24a), we ob ain
d2x
d 2+1
2αθ
1+ anh2x
θ
1− anh2(x
θ)
dx
d +x−H− anh x
θ=0,
(28)
which can be ew i en in e ms o he nondimensional e ec i e
po en ial (10) and he ic ion coe ficien
R(x)=1+ anh2(x
θ)
1− anh2(x
θ),(29)
as
d2x
d 2=−V
e (x)−1
2αθ R(x)dx
d .(30)
This app oxima e e olu ion equa ion gi es he dynamics o
he mac oscopic alue xo he oscilla o posi ion o as
spins: The nonlinea ic ion e m d i es he sys em owa ds
equilib ium, which co esponds o he minima o Ve .Bo h
he “ eno maliza ion” o he po en ial o Ve and he nonlinea
ic ion e m a e a consequence o he coupling be ween he
oscilla o and he in e nal deg ees o eedom. Equa ion (30)
ceases o hold as θ→0+because he spin elaxa ion ime
gi en by (27) di e ges. A de ailed discussion on his poin can
be ound in Re s. [10,11]. In he emainde o he pape , we
will es ic ou sel es o a empe a u e ange o which he
spins change apidly compa ed o he oscilla o mo ion and
Eq. (30) holds.
A. Me as abili y egion
Fo H=0 and subc i ical empe a u es, θ<1, he e ec-
i e po en ial has wo equally deep minima a he symme ic
posi ions ±x(0)
eq o Sec. II, and a maximum a x=0. Fo |H|<
Hb(see Fig. 2 o a quali a i e pic u e), he e ec i e po en ial
has wo minima a xR>0 and xL<0 and a me as abili y
egion appea s. The globally s able posi ion sa isfies xiH>0
(i=R,L), while he o he one is a me as able s a e in he
he modynamic sense. I mus be s essed ha his bis abili y
is p esen o all he subc i ical empe a u es θ<1, i is no
limi ed o a egion nea he c i ical empe a u e.
Le us analyze in mo e de ail he si ua ion o weak fields.
Expansion o Eq. (11) in powe s o H=F−Fc1gi es
xR,L =±x(0)
eq +χH +O(H2),(31)
whe e x(0)
eq is gi en by he solu ion o Eq. (11) o ze o field,
and χis he ze o-field “suscep ibili y”
χ=∂HxeqH=0=1−1
θ+x(0)
eq
2
θ−1
.(32)
The e o e, xR−xL=2x(0)
eq o lowes o de in H. Close o
he c i ical empe a u e, θ→1−, subs i u ion o Eq. (12) in o
(32) yields χ∼θ/[2(1 −θ)]. On he o he hand, he uns able
posi ion xUchanges om ze o o
xU=−1
θ−1H+O(H2).(33)
Acco ding o (9), he ansi ions be ween he wo minima a e
hinde ed by he p esence o la ge ene gy ba ie s,
BR,L =(N+1)[Ve (xU)−Ve (xR,L)],(34)
which a e p opo ional o he sys em size N+1. Fo H=0,
bo h equilib ium s a es ha e he same ene gy ba ie ,
B(0) =−(N+1)Ve x(0)
eq >0.(35)
Fo H= 0, he ba ie co esponding o he s able equilib ium
poin e i ying xiH>0(i=R,L) is la ge han he one
o he me as able equilib ium poin . The ba ie om he
me as able s a e ends o ze o as |H|→Hbbecause xUand he
me as able equilib ium posi ion (xL o H>0, xR o H<0)
coalesce in ha limi . The maximum e ec i e po en ial a xU
sepa a es he basins o a ac ion o he minima a xLand
xR. The sys em spends long pe iods o ime oscilla ing in he
icini y o ei he xLo xRun il i is able o hop o he o he
minimum ia a he mally ac i a ed p ocess. The esidence
imes in each basin o a ac ion inc ease exponen ially wi h
N+1, and we ha e o conside a sys em o mode a e size in
o de o see hopping be ween he wo minima on a easonable
ime scale.
Le us es ima e he ba ie heigh o weak fields. F om
Eqs. (10) and (31),weha e
Ve (xR,L)=Ve x(0)
eq ∓Hx(0)
eq +O(H2),(36)
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A. PRADOS, A. CARPIO, AND L. L. BONILLA PHYSICAL REVIEW E 86, 021919 (2012)
while Ve (xU)=O(H2). In Eq. (36) and o he es o his
sec ion, he uppe sign co esponds o xRand he lowe sign
o xL. The espec i e ba ie s om he equilib ium s a es xR,L
defined in Eq. (34) a e
BR,L ≃B(0) ±(N+1)Hx(0)
eq .(37)
The esidence imes in he espec i e basins o he minima
should ha e he A henius o m [32],
τR,L =τ0exp(BR,L/θ),(38)
whe e τ0is some cha ac e is ic ime. By inse ing Eq. (37) in o
(38), we ge
τR,L =τcexp ±(N+1)x(0)
eq
θH,(39)
whe e τc=τ0exp(B(0)/θ) is he esidence ime o ze o field in
each basin ( he same o hem bo h). Equa ion (39) is he main
esul o his sec ion; we should s ess ha i is alid o weak
fields H1, bu (N+1)Hcan be o he o de o uni y o
e en a la ge numbe . Fo H>0, we ha e τR>τ
L, because xR
is he globally s able s a e, while o H<0i isτR<τ
L, since
xRis he me as able s a e in ha case. In e es ingly, he a io
o he a e age li e imes τR/τLgi es he so-called equilib ium
cons an K o olding and un olding a ha o ce alue [16].
The e o e, o ou model we a i e a
K≡τR
τL
=exp 2(N+1)x(0)
eq
θH,H=F−Fc,(40)
ln Kis a linea unc ion o he applied o ce F. A comple ely
analogous beha io has been obse ed expe imen ally [16].
The exponen in Eq. (40) is simply N+1 imes he di e ence
o alues o he e ec i e po en ial (which plays he ole o he
ee ene gy pe pa icle) be ween bo h s a es, as eadily seen
by making use o Eq. (36).
IV. NUMERICAL RESULTS
We ha e pe o med Mon e Ca lo simula ions o he sys em
dynamics in oduced abo e. In all he cases p esen ed he e,
he dimensionless empe a u e has been chosen o be θ=0.9.
The quali a i e shape o he equilib ium H e sus xcu e is
simila o he one shown in Fig. 2.Fo θ=0.9, he oscilla o
es posi ion a equilib ium and ze o field is ±x(0)
eq ≃±0.525,
as gi en by Eq. (11). In e es ingly, he app oxima ion in
Eq. (12), which is expec ed o be alid e y close o he c i ical
empe a u e, gi es qui e a good es ima e x(0)
eq ≃0.547. The
poin s a which he cu e H e sus xhas ei he a maximum o
a minimum a e ±xb=±0.295; he co esponding alues o
he field a e ∓Hb=∓2.15 ×10−2. The e is me as abili y o
applied fields in he in e al |H|<H
b.
A. Fo ce-ex ension cu es
Le us analyze he o ce ex ension cu es o he model. The
pulling-pushing cycle is as ollows. We s a a he equilib ium
configu a ion co esponding o xmin <0 ( olded s a e), so ha
he ini ial alue o he field is Hmin =xmin − anh(xmin/θ)
gi en by Eq. (11). We pull he sys em by a s epwise inc emen
o he field: A each s ep, he field is inc eased by H, and
hen he sys em is allowed o e ol e du ing a gi en ime  .A
he end o his pe iod, we eco d he oscilla o posi ion x. Then
we inc ease again he field by H and con inue he p ocess
in he same ein. The pulling p ocess ends when we each
a posi i e alue o he field Hmax =−Hmin. Then we s a
o push back, dec easing he field by H a each s ep, un il
we each again he minimum field alue Hmin. As du ing he
pulling p ocess, we eco d he alue o xa fixed Ha e he
e olu ion ime  . This p ocess is comple ely analogous o
ha ca ied ou in unzipping expe imen s wi h biomolecules.
Figu e 3shows some ypical pulling-pushing cycles wi h
di e en loading a es H/ .Weha eusedasys emwi h
N+1=1000 spins, and a spin a emp a e α=4, la ge
enough o he spins o be as as compa ed o he oscilla o
[10]. Fo all he cu es, he minimum alue o he oscilla o
posi ion is xmin =−1.5, and he field inc emen a each s ep
is H =10−3, which is smalle han Hb, and i allows he
sys em o isi he me as abili y egion. The sys em beha io
is quali a i ely simila o o he pa ame e alues, as long
as he empe a u e θ<1. The loading a e is changed by
a ying he amoun o ime  a each s ep. The ed (solid)
lines co espond o he un olding p ocess ( =5, 100, and
104 om op o bo om), and he g een (dashed) lines o
he e olding p ocess ( =5, 100, and 104 om bo om o
op). These nume ical cu es a e quali a i ely simila o hose
obse ed in unzipping expe imen s wi h nucleic acids [16].
The e is always some hys e esis as he un olding and olding
cu es a e no supe imposed. The a ea o he hys e esis cycle
inc eases wi h he loading a e, being la ge o he la ges
loading a e conside ed and almos ze o o he smalles one.
The main di e ence wi h he expe imen al esul s is ha , in
ou model, he ex ension o he molecule is no linked o a
d opping o he loading o ce (in o de o see his e ec in
a eal expe imen , see, o ins ance, Figs. 2(A) and 2(E) o
Re . [16]). In he expe imen s o Re . [16], he o al leng h
be ween he beads localizing he molecule is con olled. This
x
H
10.50-0.5-1
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
FIG. 3. (Colo online) Hys e esis cycles o di e en alues o
he loading a e H/ . The ed (solid) lines co espond o he
un olding p ocess ( =5, 100, and 104 om op o bo om), and
he g een (dashed) lines o he e olding p ocess ( =5, 100, and
104 om bo om o op). The H s xcu e a equilib ium is plo ed
wi h blue s a s.
021919-6
SPIN-OSCILLATOR MODEL FOR THE UNZIPPING OF ... PHYSICAL REVIEW E 86, 021919 (2012)
co esponds o fixing he leng h Lin ou model, no he load
gi en by Has we ha e done. When he o ce is ex e nally
con olled in expe imen s, he e is no d op o he loading
o ce a he ex ension ansi ion, and he hys e esis cycle de-
sc ibed by he molecule ex ension is comple ely analogous o
ou s [25].
Ou nume ical esul s can be explained as ollows. As
depic ed in Fig. 2and obse ed in he las pa ag aph o
Sec. II A, o |H|>H
bwe ha e a unique s able equilib ium
poin o |H|>H
b, he zipped s a e xeq <−xb<0 o H<
−Hb<0 and he unzipped s a e xeq >+xb>0 o H>H
b.
Fo |H|<H
b, he e a e wo (locally) s able es posi ions
xL<0 and xR>0 and an uns able posi ion xUbe ween
hem. The ue he modynamic equilib ium s a e o he sys em
co esponding o he global minimum o he po en ial sa isfies
xiH>0(i=L,R), whe eas he me as able s a e has es
posi ion such ha xiH<0.
Le us conside again he pulling p ocesses in Fig. 3, s a ing
om he equilib ium configu a ion co esponding o a low
alue o he applied o ce, H<0, |H||Hb|. When we
inc ease he o ce a a mode a e a e, he oscilla o ollows
he equilib ium cu e o nega i e alues o H, wi h xeq <0,
because i s elaxa ion ime is small compa ed o  , and he
ic ion e m in Eq. (30) can d i e i o equilib ium. When
he field eaches H=0 (o a e y small alue), he s able
equilib ium posi ion o he oscilla o changes discon inuously
om −x(0)
eq o x(0)
eq . The impo an ques ion is now whe he
 insu ficien ly long o he oscilla o posi ion o o e come
he ene gy ba ie B(0) a ze o field, gi en by Eq. (35).I he
answe is posi i e, he sys em jumps du ing he ime in e al
 o he o he s able b anch whe e x>0, and i s ays on
ha b anch when he o ce is u he inc eased. In he pushing
back expe imen , he sys em e e ses i s pa h and he beha io
is almos e e sible. This beha io is obse ed o he la ges
alue  =104, co esponding o a loading a e H/ =
10−7. Fo la ge pulling a es, such as he o he ones conside ed
in he same figu e, he sys em does no ha e enough ime o
jump o e he ene gy ba ie a H=0. The e o e, xmo es
o e he me as able b anch wi h xiH<0, un il he ba ie
dec eases su ficien ly o he oscilla o posi ion o jump o he
mos s able b anch, wi h x>0. O cou se, he ac ual pa o he
me as able egion isi ed by he sys em depends on he loading
a e. Fo he highes loading a e conside ed, co esponding o
 =5, he sys em isi s he whole me as able b anch up
o he maximum. A simila line o easoning explains he
beha io obse ed in he e olding cu e. I is in e es ing o
no e ha he hys e esis cycle ound o high loading a es is no
a nonequilib ium beha io , as p e iously sugges ed [16–19],
bu i a ises om he sampling o he egions o me as abili y
o subc i ical empe a u es.
A pulling expe imen co esponding o a a e e en slowe
han he smalles loading a e in Fig. 3is plo ed in Fig. 4.
Again, he minimum alue o he oscilla o posi ion has been
chosen o be xmin =−1.5, and he field inc emen a each s ep
H =10−3, bu he ime spen by he sys em a each alue
o he o ce is e y la ge, namely,  =105. Fo he sake o
cla i y, he egion a ound he c i ical o ce F=Fc(H=0) has
been zoomed in. We obse e se e al jumps be ween he olded
(x<0) and un olded (x>0) s a es o |H|=|F−Fc|<
5×10−3. This beha io is a clea signa u e o he bis abili y
x
H
10.50-0.5-1
0.02
0.015
0.01
0.005
0
-0.005
-0.01
-0.015
-0.02
FIG. 4. (Colo online) De ail o he me as abili y egion |H|⩽
Hb o a pulling expe imen wi h a e y slow loading a e, H =10−3
and  =105. Hopping be ween he zipped and unzipped s a e is
clea ly seen o |H|=|F−Fc|<5×10−3.
shown by ou sys em in he egion |H|<H
b. A ze o field,
he ene gy ba ie s be ween he wo s able s a es a e iden ical,
τR=τL=τcin (39), and he e e se ansi ion is equally
likely. Fo  τc, he oscilla o posi ion has enough ime o
su pass he ene gy ba ie s se e al imes, and i may go back
and o h om one s a e o he o he , as shown in Fig. 4.
B. Cons an o ce expe imen s
We ha e also ca ied ou Mon e Ca lo simula ions a
cons an o ce. We ha e chosen θ=0.9, a small alue o
he field, H=3×10−3<H
b, and a smalle size, N+1=
500, in o de o keep he simula ion ime unde con ol.
Equa ion (11) gi es he wo locally s able oscilla o es
posi ions xL=−0.509 (me as able) and xR=0.540 (globally
s able), sepa a ed by he uns able oscilla o posi ion xU=
−0.027. These alues ag ee wi h he weak field exp essions
(31)–(33). Figu e 5shows a ime ace o he sys em. The
oscilla o jumps s ochas ically be ween he wo alues xLand
xRco esponding o he wo locally s able equilib ium poin s.
The sys em spends mo e ime in he s able s a e xR( ecall
H=3×10−3>0), because i has o su pass a la ge ene gy
ba ie so as o escape he e om.
In o de o check Eqs. (39) o (40) o he esidence imes
in each basin o a ac ion, we ha e measu ed he a e age ime
spen in each basin o di e en alues o he applied field in he
me as abili y egion |H|<H
b. We find ha ln τR,L inc eases
linea ly wi h H, wi h a slope ha ag ees wi h he heo e ical
p edic ion o Eq. (39). We ha e plo ed he a io o a e age
li e imes K=τR/τLdefined in Eq. (40) as a unc ion o he
applied field Hin Fig. 6. The ein, ln Kshows a linea beha io ,
simila ly o ha seen in ac ual expe imen s [16]. The slope m=
dln K/dH ob ained nume ically, m=542.5 ag ees well wi h
he heo e ical p edic ion calcula ed om Eq. (40),m=583.8.
This esul s ongly suppo s he physical pic u e de eloped in
Sec. III A o he beha io o he sys em in he me as abili y
egion, bo h om a quali a i e and a quan i a i e poin o
iew.
021919-7
A. PRADOS, A. CARPIO, AND L. L. BONILLA PHYSICAL REVIEW E 86, 021919 (2012)
x
2500020000150001000050000
1
0.5
0
-0.5
-1
FIG. 5. (Colo online) Time ace o he oscilla o posi ion o a
cons an o ce expe imen in he me as abili y egion, namely, H=
3×10−3. Hopping be ween he wo locally s able equilib ium poin s
o he oscilla o is obse ed.
V. CONCLUSIONS
We ha e modeled biomolecules’ olding and un olding
unde an ex e nal load o ce by a mac oscopic linea oscilla o
coupled wi h in e nal deg ees o eedom ep esen ed by
Ising spins ha unde go Glaube dynamics. The simple
mean-field cha ac e o he model p e en s us om doing
quan i a i e compa isons wi h he eal expe imen s, and we
ha e o se le o quali a i e compa isons. We canno simula e
posi ion-con olled expe imen s, only o ce-con olled ones.
To o e come he limi a ions o his simple sys em, we should
conside mo e ealis ic models. One possibili y is o use
spin-s ing models, in which a disc e ized s ing is coupled
o Ising-like a iables. These models do no ha e a mean-field
cha ac e because he couplings be ween he s ing a iables
and he spins a e local, he eby exhibi ing a much iche
and mo e complex phenomenology [9]. In he con ex o
DNA unzipping, i is wo h conside ing ex ensions o he
H
ln K
0.010.0050-0.005-0.01
6
4
2
0
-2
-4
-6
FIG. 6. (Colo online) Loga i hm o he a io K≡τR/τL
(squa es) o he esidence imes o he wo equilib ium s a es xR,L in
he me as abili y egion. The numbe o spins is N+1=500. Also
plo ed is he bes fi o he A henius law (solid line) [Eq. (40)].
well-known Poland-Sche aga and Pey a d-Bishop models
[29,33], which ake in o accoun he base pai ing be ween
he di e en DNA s ands. This ing edien , which is missing
in he spin oscilla o model p esen ed he e, is c ucial o
unde s anding he impo ance o he en opic e ec due o
bubble o ma ion in DNA dena u a ion.
Despi e i s sho comings, he p esen spin-oscilla o model
is an use ul minimal model o unde s and expe imen s in
which biomolecules a e unzipped by mechanical o ce. This
simple model allows us o ob ain exac analy ical esul s om
which a physically appealing and gene al pic u e a ises. The
hys e esis cycles show ha he sys em exhibi s a me as able
equilib ium beha io in he unzipping expe imen s, no a
ue nonequilib ium beha io , as was sugges ed p e iously
[16–19]. In his ega d, he un olding and e olding cycles a e
qui e di e en om he uly nonequilib ium hys e esis cycles
exhibi ed by glass o me s in cooling and hea ing p ocesses
(see Re s. [46,47] and e e ences he ein). In he cooling p o-
cess, he glass o me s depa om he equilib ium cu e and
end in a a om equilib ium s a e a low empe a u es. In he
ehea ing p ocess, hey e u n o equilib ium ollowing a cu e
di e en om he cooling one, which ypically o e shoo s he
equilib ium cu e. In con as wi h his beha io , he hys e esis
cycles in ou sys em a ise because he me as able equilib ium
b anches o he x e sus Hcu e a e swep a high loading
a e (such ha he sys em does no ha e enough ime o find
he ue minimum o he po en ial in he bis able field in e al
|H|<H
b). Due o he shape o he equilib ium x e sus H
cu e in Fig. 2, he sys em unzips a a highe alue o he field
han he one a which i ezips. We expec ha his gene al
pic u e emains alid o mo e ealis ic models and/o ac ual
biomolecules.
As in he expe imen s, he sys em hops be ween he zipped
and he unzipped s a e a e y small loading a es. I has
hen enough ime o su pass he ene gy ba ie sepa a ing
he un olded and he olded s a es. When he o ce is held
cons an wi h |H|<H
b(bis able egion), he sys em hops
back and o h be ween he wo possible equilib ium alues
o he oscilla o posi ion. The a e age li e imes show an
A henius-like dependence on he applied field H=F−Fc,
again in ag eemen wi h he beha io o eal sys ems.
Fo all subc i ical empe a u es T<T
c, hese beha io s
occu due o he fi s -o de ansi ion and i s associa ed egion
o me as abili y. The phase ansi ion is a consequence o he
coupling be ween he oscilla o and he one-dimensional spin
sys em, which in oduces an e ec i ely long- ange in e ac ion
be ween he spins. Simila hidden one-dimensional long- ange
e ec i e co ela ions enable phase ansi ions in biological
sys ems [48].
ACKNOWLEDGMENTS
This esea ch has been suppo ed by he Spanish Minis e io
de Econom´
ıa y Compe i i idad h ough G an s FIS2011-
24460 (A.P., pa ially financed by FEDER unds), FIS2011-
28838-C02-02 (A.C.), FIS2011-28838-C02-01 (L.L.B.), and
FIS2010-22438-E (Spanish Na ional Ne wo k Physics o Ou -
o -Equilib ium Sys ems), and by UCM/BSCH CM 910143
(A.C.).
021919-8
SPIN-OSCILLATOR MODEL FOR THE UNZIPPING OF ... PHYSICAL REVIEW E 86, 021919 (2012)
[1] J. Fede and E. Py e, Phys.Re .B8, 3978 (1973).
[2] P. A. Rik old, Z. Phys. B 26, 195 (1977).
[3] P. A. Rik old, Z. Phys. B 30, 339 (1978).
[4] A. Boisen, Na . Nano echnol. 4, 404 (2009).
[5] A. J. Legge , S. Chak a a y, A. T. Do sey, M. P. A. Fishe ,
A. Ga g, and W. Zwe ge , Re . Mod. Phys. 59, 1 (1987).
[6] L. L. Bonilla and F. Guinea, Phys. Re . A 45, 7718 (1992).
[7] M. Ho heinz, H. Wang, M. Ansmann, R. C. Bialczak, E. Luce o,
M. Neeley, A. D. O’Connell, D. Sank, J. Wenne , J. M. Ma inis,
and A. N. Cleland, Na u e (London) 459, 546 (2009).
[8] A. D. O’Connell, M. Ho heinz, M. Ansmann, R. C. Bialczak,
M. Lenande , E. Luce o, M. Neeley, D. Sank, H. Wang,
M. Weides, J. Wenne , J. M. Ma inis, and A. N. Cleland, Na u e
(London) 464, 697 (2010).
[9] L. L. Bonilla, A. Ca pio, A. P ados, and R. R. Rosales, Phys.
Re . E 85, 031125 (2012).
[10] A. P ados, L. L. Bonilla, and A. Ca pio, J. S a . Mech. (Theo .
Exp.) (2010) P06016.
[11] L. L. Bonilla, A. P ados, and A. Ca pio, J. S a . Mech. (Theo .
Exp.) (2010) P09019.
[12] A. P ados, L. L. Bonilla, and A. Ca pio, AIP Con . P oc. 1332,
232 (2011).
[13] W. L. B agg and E. J. Williams, P oc. R. Soc. London A 145,
699 (1934).
[14] H.-X. Zhou and R. Zwanzig, J. Phys. Chem. A 106, 7562
(2002).
[15] F. Ri o , J. Phys.: Condens. Ma e 18, R531 (2006).
[16] J. Lipha d , B. Onoa, S. B. Smi h, I. Tinoco, J ., and
C. Bus aman e, Science 292, 733 (2001).
[17] J. Lipha d , S. Dumon , S. B. Smi h, I. Tinoco, J ., and
C. Bus aman e, Science 296, 1832 (2002).
[18] F. Ri o , in P og ess o Ma hema ical Physics ol. 38 (Poinca ´
e
Semina 2003), J. Daliba d and V. Ri asseau eds. (Bi kh¨
ause
Ve lag, Basel, 2004), p. 193–227.
[19] D. Collin, F. Ri o , C. Ja zynski, S. B. Smi h, I. Tinoco, J ., and
C. Bus aman e, Na u e (London) 437, 231 (2005).
[20] M. C. Williams and I. Rouzina, Cu . Opin. S uc . Biol. 12, 330
(2002).
[21] M. Manosas and F. Ri o , Biophys. J. 88, 3224 (2005).
[22] J.-D. Wen, M. Manosas, P. T. X. Li, S. B. Smi h, C. Bus aman e,
F. Ri o , and I. Tinoco, J ., Biophys. J. 92, 2996 (2007).
[23] M. Manosas, J.-D. Wen, P. T. X. Li, S. B. Smi h, C. Bus aman e,
I. Tinoco, J ., and F. Ri o , Biophys. J. 92, 3010 (2007).
[24] J. M. Hugue , N. Fo ns, and F. Ri o , Phys.Re .Le .103,
248106 (2009).
[25] J. M. Hugue , Ph.D. hesis, Uni e si y o Ba celona, 2010.
[26] P. G oss, N. Lau ens, L. B. Odde shede, U. Bockelmann, E. J.
G. Pe e man, and G. J. L. Wui e, Na u e Phys. 7, 731 (2011).
[27] S. B. Smi h, Y. Cui, and C. Bus aman e, Science 271, 795 (1996).
[28] M. Rie , H. Clausen-Schaumann, and H. E. Gaub, Na . S uc .
Biol. 6, 346 (1999).
[29] D. K. Lubensky and D. R. Nelson, Phys.Re .Le .85, 1572
(2000).
[30] C. Bus aman e, S. B. Smi h, J. Lipha d , and D. Smi h, Cu .
Opin. S uc . Biol. 10, 279 (2000).
[31] D. Anselme i, J. F i z, B. Smi h, and X. Fe nandez-Busque s,
Single Mol. 1, 53 (2000).
[32] S. Cocco, R. Monasson, and J. F. Ma ko, P oc. Na l. Acad. Sci.
USA 98, 8608 (2001).
[33] D. Ma enduzzo, S. M. Bha acha jee, A. Ma i an, E. O landini,
and F. Seno, Phys.Re .Le .88, 028102 (2001).
[34] D. Ma enduzzo, A. Ma i an, E. O landini, F. Seno, and
A. T o a o, J. S a . Mech. (Theo . Exp.) (2009) L04001.
[35] J. an Mame en, P. G oss, G. Fa ge, P. Hooijman,
M. Modes i, M. Falkenbe g, G. J. L. Wui e, and E. J. G.
Pe e man, P oc. Na l. Acad. Sci. USA 106, 18231 (2009).
[36] A. Fiascona o and F. Falo, a Xi :1205.6126.
[37] T. E. Fishe , A. F. Obe hause , M. Ca ion-Vazquez, P. E.
Ma szalez, and J. M. Fe nandez, T ends Biochem. Sci. 24, 379
(1999).
[38] T. E. Fishe , P. E. Ma szalek, and J. M. Fe nandez, Na . S uc .
Biol. 7, 719 (2000).
[39] A. F. Obe hause and M. Ca ion-Vazquez, J. Biol. Chem. 283,
6617 (2008).
[40] B. Bulla d, T. Ga cia, V. Benes, M. C. leake, W. A. Linke, and
A. F. Obe hause , P oc. Na l. Acad. Sci. USA 103, 4451 (2006).
[41] C. Cecconi, E. A. Shank, C. Bus aman e, and S. Ma qusee,
Science 309, 2057 (2005).
[42] R. J. Glaube , J. Ma h. Phys. 4, 294 (1963).
[43] L. D. Landau and E. M. Li shi z, Cou se o Theo e ical Physics
Vol. 5, S a is ical Physics Pa 1 (Pe gamon, Ox o d, 1980).
[44] N. G. Van Kampen, S ochas ic P ocesses in Physics and
Chemis y (No h-Holland, Ams e dam, 1997).
[45] J. J. B ey and A. P ados, Phys. Re . E 47, 1541 (1993).
[46] J. J. B ey, A. P ados, and M. J. Ruiz-Mon e o, J. Non-C ys .
Solids 172–174, 371 (1994).
[47] A. P ados, J. J. B ey, and B. S´
anchez-Rey, Phys.Re .B55,
6343 (1997); A. P ados and J. J. B ey, Phys.Re .E64, 041505
(2001).
[48] M. Pey a d, Na u e Phys. 2, 13 (2006).
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