Experimental test of Sinais model in DNA unzipping Cathelijne ter Burg1 Paolo Rissone2 Marc Rico-Pasto2 Felix Ritort23and Kay J org Wiese1 1Laboratoire de Physique de lE cole Normale Sup erieure ENS Universit e PSL CNRS Sorbonne Universit e

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Experimental test of Sinai’s model in DNA unzipping

Cathelijne ter Burg1, Paolo Rissone2, Marc Rico-Pasto2, Felix Ritort2,3and Kay J¨

org Wiese1

1Laboratoire de Physique de l’E´

cole Normale Sup´

erieure, ENS, Universit´

e PSL, CNRS, Sorbonne Universit´

Universit´

e Paris-Diderot, Sorbonne Paris Cit´

e, 24 rue Lhomond, 75005 Paris, France.

2Small Biosystems Lab, Condensed Matter Physics Department, Universitat de Barcelona,

Carrer de Mart´

ı i Franqu`

es 1, 08028 Barcelona, Spain.

3Institut de Nanoci`

encia i Nanotecnologia (IN2UB), Universitat de Barcelona, 08028 Barcelona, Spain

The experimental measurement of correlation functions and critical exponents in disordered systems is key

to testing renormalization group (RG) predictions. We mechanically unzip single DNA hairpins with optical

tweezers, an experimental realization of the diffusive motion of a particle in a one-dimensional random force

ﬁeld, known as the Sinai model. We measure the unzipping forces Fwas a function of the trap position w

in equilibrium and calculate the force-force correlator ∆m(w), its amplitude, and correlation length, ﬁnding

agreement with theoretical predictions. We study the universal scaling properties since the effective trap stiffness

m2decreases upon unzipping. Fluctuations of the position of the base pair at the unzipping junction uscales

as u∼m−ζ, with a roughness exponent ζ= 1.34 ±0.06, in agreement with the analytical prediction ζ=

3. Our study provides a single-molecule test of the functional RG approach for disordered elastic systems in

equilibrium.

Introduction. Heterogeneity and disorder pervade physical

and biological matter [1–3]. Since Schr¨

odinger’s conception

of the gene as an a-periodic crystal [4], disorder is recog-

nised as a crucial ingredient for life [5]. The readout of the

genetic information encoded in DNA can be modeled with

polymers in random potentials, such as Sinai’s model [6].

The latter describes the dynamics of a particle diffusing in a

one-dimensional random-force ﬁeld, a suitable model for the

mechanical unzipping of the DNA double helix into single

strands. Sinai’s model is a special case (d= 0) of the univer-

sal ﬁeld theory of disordered elastic systems in ddimensions,

where one can analytically calculate force correlations. The

latter were measured in contact-line depinning (d= 1) [7],

Barkhausen noise (d= 2) [8] and RNA-DNA peeling (d= 0)

[9]. While these experiments are for depinning, i.e. nonequi-

librium, an experimental test of the equilibrium universality

class is lacking. Here we test universality of equilibrium-force

correlations as predicted by Sinai’s model in DNA unzipping.

The model parameters are naturally changed during the exper-

iment allowing us to monitor the functional RG ﬂow.

In the experiment, a DNA hairpin of 6.8k base pairs (BPs)

is held between two beads. One is ﬁxed at the tip of a

micropipette, the other is optically trapped (Fig. 1(a) and

Supp. Mat. Sec. A). By moving the optical trap at a speed

v≈10nm/s, the double-stranded DNA (dsDNA) is mechan-

ically pulled and converted into two single strands (ssDNA).

The measured force-distance curve (FDC) shows a sawtooth

pattern characteristic of stick-slip dynamics (Fig. 1(b), red

curve). The hairpin unzips at a critical mean pinning force

fc≈15pN, ﬂuctuating in the range 12-17pN. Once the hair-

pin is unzipped, the reverse process starts (Fig. 1(b), blue

curve): the optical trap moves backward and the hairpin re-

folds into the dsDNA native conformation. The absence of

hysteresis between rezipping and unzipping FDCs and the fact

that there is a single reaction coordinate, implies that the sys-

tem is in equilibrium.

During unzipping, the base pair at the junction separating

dsDNA from ssDNA is subject to random forces generated by

the neighbouring monomers, and modeled by the motion of a

single particle (d= 0) in a random potential that belongs to

Sinai’s universality class [6]. The number of unzipped BPs

is a well-deﬁned reaction coordinate. Opening (closing) one

BP can be seen as a particle hopping to the right (left). We

changed salt concentration from 10mM to 1000mM NaCl,

Fig. 1(c), modulating the strength of BP interactions.

The Model. The motion of the base pair at the junction can be

modeled by a Langevin equation (see Supp. Mat. Sec. Bfor

the derivation)

∂u

∂t =m2(w−u) + F(u) + ηu(t),(1)

where u(t)is the extension of the molecular construct, wthe

relative trap-pipette position (Fig. 1(a)), and m2the effec-

tive stiffness of the molecular construct. The random force

is F(u) = −V0(u), where V(u)is the free energy stored

in the partially hybridized hairpin. F(u)acts at the hairpin

junction and is determined by hydrogen bonding and stack-

ing interactions between consecutive base pairs. Using the

nearest-neighbour model one can show that these forces are

random, and that their distribution is roughly a Gaussian

(Supp. Mat. Sec. C). In equilibrium, ∂u

∂t ≈0, so the force

F(u)applied to the hairpin in Eq. (1) is counteracted by the

force Fwexerted on the bead by the optical trap. For a ﬁxed

trap position w,Fwand uﬂuctuate due to the thermal noise

and the BP breathing dynamics. The equilibrium force corre-

lations are deﬁned as,

∆m,T (w−w0) = FwFw0

c=FwFw0−FwFw0,(2)

where (. . . )stands for a double thermal and disorder aver-

age. Correlations depend on the value of m2, through the m-

dependence in Eq. (1). They also depend on temperature T,

which leads to a rounding of ∆m,T (w)at small w(see below).

The FDCs in Figs. 1(b) and (c) show a sawtooth pat-

tern characterized by segments of increasing force Fw, fol-

lowed by abrupt drops caused by the cooperative unzipping of

groups of base pairs in the range of 10-100 basepairs [11].

arXiv:2210.00777v2 [cond-mat.dis-nn] 17 Apr 2023

Micropipette

(a) (b)

Fw[pN] (c)

Fw[pN]

FIG. 1. (a) Experimental setup. (b) Unzipping (red) and rezipping (blue) FDC’s demonstrating equilibrium behaviour. The residual hysteresis

at the end of the FDC is due to the DNA end-loop that slows down the initiation of stem formation upon reconvolution. (c) Experimental

FDC’s, Fw, for various salt concentrations. The mean pinning force varies between 12-17pN, and is non-universal.

The slope of each segment, equivalent to the effective stiff-

ness m2, decreases with w, permitting us to measure the scal-

ing of ∆m,T (w)with m2. In fact, m2depends on the com-

bined effects of the optical trap, and the elastic response of the

molecular construct (ssDNA and dsDNA handles). It can be

written as (see Eq. (B27))

m2=1

z1k1

,(3)

with kbthe trap stiffness, and z1, k1the mean extension and

stiffness of one nucleotide at the unzipping force. Model-

ing the elastic response of the hairpin [12] shows that k1≈

130pN/nm and z1≈0.45nm at the unzipping force fc≈15

pN, which gives a slope of about (z1k1)−1≈0.02pN−1.

Eq. (3) implies that the larger the length of the unpaired DNA,

the lower the effective stiffness m2. To verify this, we split

the FDCs into four regions (inset of Fig. 2). While smaller

regions have smaller variations in m2, regions must be taken

sufﬁciently large for a reliable statistics. Eq. (3) agrees with

the experimental data shown in Fig. 2.

Force correlations in Sinai’s model can be framed in terms

of the functional renormalisation group (FRG). The FRG

arises as the ﬁeld theory of disordered systems for interfaces

[13–25], generalising the d= 0 case described by the Sinai

model. The FRG predicts two universality classes, critical de-

pinning (non-equilibrium) and equilibrium (considered here).

In equilibrium, the T→0limit of ∆m,T (w)in Eq. (2), can

be written as

∆m(w) = m4ρ2

m˜

∆(w/ρm), ρm∼m−ζ,(4)

with ˜

∆(w)the shape function, ζthe roughness exponent, and

w=w/ρmthe rescaled dimensionless distance. The FRG

allows for observables to be computed perturbatively in an

expansion around the upper critical dimension, parameterised

by ε= 4 −d. The shape function ˜

∆(w)is the ﬁxed point of

the FRG ﬂow equation

0=(ε−2ζ)˜

∆(w) + ζw˜

∆0(w)−1

2∂2

w˜

∆(w)−˜

∆(0)2+. . .

(5)

The dots represent higher-loop corrections in ε, currently

known up to 3-loop order [19–21,23,24,26]. For the equi-

librium random-ﬁeld, ζ= (4 −d)/3, which gives ζ= 4/3

for d= 0. This result is derived by integrating Eq. (5) from

w=0to w=∞. It is exact to all orders in the loop expan-

sion. Eq. (5) predicts that ˜

∆(w)has a cusp at w=0which

is rounded at ﬁnite T. Generalization of the FRG equation (5)

to ﬁnite Tallows us to estimate the size of the rounded re-

gion. An explicit relation between ∆m(w)and ∆m,T (w)was

derived in [13–15],

∆m,T (w)≈ N∆m(pw2+t2), t =6m2kBT

ε|∆0

m(0)|.(6)

It has been shown that the RG ﬂow (5) preserves the area un-

1000 2000 3000 4000 5000

w[nm]

100

120

140

1000 2000 3000 4000 5000 6000

m−2[pN−1·nm]

FIG. 2. Variation of the effective stiffness m2versus waccording to

Eq. (3). The points correspond to the measured values of 1/m2for

the four FDC regions (each one shown with a different colour in the

inset). The ﬁt to data (dashed line) and the extrapolation to w= 0

gives the stiffness of the optical trap, kb= 0.05 ±0.01pN ·nm−1.

The inset illustrates the four studied regions in a FDC at 1M NaCl.

20 40 60 80

w[nm]

0.1

0.2

0.3

0.4

0.5

0.6

Δm(w)[pN2]

1234567

w[nm]

0.30

0.35

0.40

0.45

0.50

Δm(w)[pN2]

deconvoluted

6raw data

Brownian peak−bead noise



 subtracted peak



+





extrapolation



)

FIG. 3. Measured ∆m,T (w)for the ﬁrst region (red). 1-σerror is

shown as a pink strip. Deconvolution (black solid) and extrapolation

to w= 0 (black dot-dashed). The inset shows ∆m,T (w)at short

range with subtraction of the peak at w= 0, as explained in the

main text.

der ∆m,T (w)for all T[26]. Therefore, we can use the mea-

sured ∆m,T (w)and Eq. (6) to determine the normalization

factor Nand ∆m(w). Details about the procedure are given

in Supp. Mat. Sec. D.

For the Sinai model, the shape function ˜

∆in Eq. (5) is

known analytically [13,27],

∆(w) = −e−w3

4π3

2√wZ∞

−∞

dλ1Z∞

−∞

dλ2e−(λ1−λ2)2

×eiw

2(λ1+λ2)Ai0(iλ1)

Ai(iλ1)2

Ai0(iλ2)

Ai(iλ2)2

×1+2wR∞

0dVewVAi(iλ1+V)Ai(iλ2+V)

Ai(iλ1)Ai(iλ2).(7)

Here Ai is the Airy function, and ζ= 4/3as in FRG.

Data analysis. We analysed 33 FDCs obtained by unzipping

a 6.8kBP DNA hairpin in a broad range of salt conditions

from 10mM to 1000mM NaCl at T= 298K. As illustrated

in Fig. 2, we divided each FDC into four regions measuring

the force correlations (2) for each region. Force correlations

are equal within the experimental resolution for all salt condi-

tions, as shown in Supplementary Fig. 7. Although the effec-

tive stiffness of the molecular construct m2changes with salt,

it changes much less than it does over the different unzipping

regions for a ﬁxed salt condition. To enlarge statistics we av-

eraged ∆m,T (w)over all salts. Results for the ﬁrst region are

shown in Fig. 3(red line with red strip for error bars).

To recover ∆m,T (w)in Eq. (6) we must subtract two

sources of thermal noise, which are visible as a short-range

correlated peak at w≈0: Brownian ﬂuctuations of the bead;

and the breathing dynamics (opening and closing) of the DNA

base pairs at the junction. First, bead-noise subtraction re-

duces the peak’s amplitude ∆m,T (w= 0) from ≈0.6pN2

(red in main plot of Fig. 3) to ≈0.5pN2(magenta line in

the inset). Second, we estimated the effect of the breathing

dynamics from numerical simulations of Sinai’s model [26].

This reduces the peak from ≈0.5pN2to ≈0.35pN2with

a dip of amplitude ≈0.3pN2for w < 1nm (cyan curve in

the inset). This dip is also seen in simulations [26]. From

∆m,T (w)we derive the T= 0 force correlations, ∆m(w), by

plotting the experimental data versus √w2+t2, see Eq. (6),

with tgiven there (T= 298K, ε= 4,m2from Fig. 2).

We initially estimate ∆0

m(0) by extrapolation of the raw data.

This gives ∆m(w)for w > t ≈7nm (black continuous line

in Fig. 3). The extrapolated ∆m(w)for w < t (dot-dashed re-

gion) is obtained by ﬁtting a second-order polynomial (black

dot-dashed line in Fig. 3). The whole procedure is iterated un-

til convergence of ∆m(w)is reached. As a consistency check

we used the T= 0 theory prediction ∆m(w)together with

Eq. (6) to calculate ∆m,T (w)for all regions, see Supp. Fig. 8.

Force correlations in Eq. (6) are described by three parame-

ters: the correlation length ρmin the wdirection, the stiffness

m2of the molecular construct, and the temperature T. With

the measured value of m2(Fig. 2) and kBT= 4.11pN ·nm

we use Eq. (6) to predict t(ε= 4 and ∆0

m(0) obtained from

the small-wextrapolation in Fig. 3). According to Eqs. (4) and

(6), the scale ρmis the only ﬁtting parameter, which we report

on the table in Fig. 5for all four regions. Its value increases

with windicating that FDCs become progressively less rough

as unzipping progresses: For the ﬁrst region, ρm= 26.8nm,

which corresponds to 33 basepairs [12], the typical size of

avalanches that can be resolved in the FDC at the beginning

of the unzipping process.

We now check two predictions of the theory: the result (7)

and the FRG scaling relation (4). In particular, the scaling

function ˜

∆only depends on the dimensionless combination

w/ρm∼wmζ, and its amplitude is universal. The inset of

Fig. 4shows ∆m(w)for the four regions where ρmincreases

while the molecule is unzipped and m2decreases. In Fig. 4

we test the scaling law (4) with ζ= 4/3, as predicted for

Sinai’s model. We can also determine the value of ζindepen-

dently of the collapse in Fig. 4. In Fig. 5we show results for

the scaling of the correlation length ρmand amplitude ∆m(0)

with m. We get ζ= 1.41 ±0.10 and ζ= 1.29 ±0.08 from

the scaling of ρmand ∆m(0), respectively, giving an aver-

age of ζ= 1.34 ±0.06 in agreement with the expected value

ζ= 4/3. Details are given in Supp. Fig. 8.

We can go one step further: In random-ﬁeld systems, the

correlations of the potential V(u)grow linearly at large u-

distances, 1

2[V(u)−V(u0)]2'σ|u−u0|. The constant σis

related to the force correlator ∆mby

σ=Z∞

∆∞(u)du≡Z∞

∆m(w)dw . (8)

This relation holds for the microscopic ∆∞(u)and the mea-

sured ∆m(w), as the area under ∆m(w)is preserved by the

RG ﬂow, as previously discussed. A constant σin Eq. (8) im-

plies ζ= 4/3for all min Eq. (4). Eq. (8) then yields the

analytic prediction

ρm="Rw>0∆∞(w)

m4Rw>0˜

∆(w)#1/3

.(9)

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ExperimentaltestofSinai'smodelinDNAunzippingCathelijneterBurg1,PaoloRissone2,MarcRico-Pasto2,FelixRitort2;3andKayJ¨orgWiese11LaboratoiredePhysiquedel'E´coleNormaleSup´erieure,ENS,Universit´ePSL,CNRS,SorbonneUniversit´e,Universit´eParis-Diderot,SorbonneParisCit´e,24rueLhomond,75005Paris,France.2Small...

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Experimental test of Sinais model in DNA unzipping Cathelijne ter Burg1 Paolo Rissone2 Marc Rico-Pasto2 Felix Ritort23and Kay J org Wiese1 1Laboratoire de Physique de lE cole Normale Sup erieure ENS Universit e PSL CNRS Sorbonne Universit e.pdf

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Experimental test of Sinais model in DNA unzipping Cathelijne ter Burg1 Paolo Rissone2 Marc Rico-Pasto2 Felix Ritort23and Kay J org Wiese1 1Laboratoire de Physique de lE cole Normale Sup erieure ENS Universit e PSL CNRS Sorbonne Universit e

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