Inferring the effective thickness of polyelectrolytes from stretching measurements at various ionic strengths: applications to DNA and RNA

0.45 MB·English

by  Ngo Minh Toan

6
0 Inferring the effective thickness of polyelectrolytes from
0
stretching measurements at various ionic strengths:
2

n applications to DNA and RNA
a
J

6 NgoMinhToan1,2andCristianMicheletti1
1
E-mail:[email protected],[email protected]

] 1InternationalSchoolforAdvancedStudies(S.I.S.S.A.)andINFM,ViaBeirut2-4,34014
t Trieste,Italy
f
o 2InstituteofPhysics,10DaoTan,Hanoi,Vietnam
s
.
t Abstract. By resorting to the thick-chain model we discuss how the stretching response
a
of a polymer is influenced by the self-avoidance entailed by its finite thickness. The
m
characterization oftheforceversusextension curveforathickchainiscarriedoutthrough
- extensive stochastic simulations. The computational results are captured by an analytic
d
expression that is used to fit experimental stretching measurements carried out on DNA
n and single-stranded RNA (poly-U) in various solutions. This strategy allows us to infer
o theapparent diameteroftwobiologically-relevant polyelectrolytes, namelyDNAandpoly-
c U, for different ionic strengths. Due to the very different degree of flexibility of the two
[ molecules, the results provide insight into how the apparent diameter is influenced by the

interplaybetweenthe(solution-dependent)Debyescreeninglengthandthepolymers’“bare”
1
thickness. ForDNA,theelectrostatic contributiontotheeffective radius,∆,isfoundtobe
v
about5timeslargerthantheDebyescreeninglength,consistentlywithprevioustheoretical
8
predictionsforhighly-chargedstiffrods.Forthemoreflexiblepoly-Uchainstheelectrostatic
4 contributionto∆isfoundtobesignificantlysmallerthantheDebyescreeninglength.
3
1
0
6
0
/
t
a
m
-
d
n
o
c
:
v
i
X
r
a
Inferringtheeffectivethicknessofpolyelectrolytesfromstretchingmeasurements 2
Introduction
In recent years, the remarkable advancement of single-molecule manipulation techniques
has made possible to characterise with great accuracy how various biopolymers respond
to mechanical stretching [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. The wealth of
collectedexperimentaldatahaveconstitutedandstillrepresentaninvaluableandchallenging
benchmarkfor models of polymers’ elasticity [15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. The
interpretationofsingle-moleculestretchingexperimentsoftenreliesonone-dimensionalnon-
self-avoiding models of polymers. It is physically appealing that the schematic nature of
such descriptions often conjugates with the capability of reproducing well experimental
measurements. Two notable instances are represented by the freely-jointed chain [25, 15]
and the worm-like chain models which, in their original or extended forms, constitute the
most commonly-usedtheoreticalframeworksfor biopolymers’stretching[26, 17, 27]. Both
modelsareendowedwithaparameter,theKuhnlengthorthepersistencelength,thatprovides
a phenomenological measure of the polymer stiffness and that is obtained by fitting the
experimental data. It is important to notice, however, that it is possible to go beyond this
phenomenologicalapproachandconnectthepersistencelengthtothefundamentalstructural
properties of a polymer. A strong indication of the feasibility of this scheme is provided
by the fact that, for a large number of biopolymers, the observed persistence length shows
anapproximatequarticdependenceonthepolymerdiameter,aspredictedfor“ideal”elastic
rods[15].
From thisperspectiveitappearsnaturalto investigatein detailthe connectionbetween
structuralpropertiesandstretchingresponseofbiopolymers. We haverecentlypursuedthis
objectivebymodellingexplicitlytheintrinsicthicknessofahomo-polymer(treatedasatube
with uniformcross-section)andcharacterisingthe stretchingresponse[21]. Thetheoretical
andnumericalresultswereemployedinanappealingchemico-physicalframeworkwherethe
diameter of a biopolymerwas notprobeddirectly butinferredthroughthe mereknowledge
ofthestretchingresponse.Theadoptedthick-chainmodel[28,29,30],brieflyoutlinedinthe
nextsection, wasusedtofitstretchingmeasurementsobtainedforavarietyofbiopolymers:
DNA[5],thePEVK-domainofthetitinprotein[7,8]andcellulose[31,10]. Foruncharged
polymers,suchastitinandcellulose,theeffectivediameterrecoveredfromfittingtheforce-
extension curves were very consistent with the stereochemical ones, thereby validating the
thick-chainmodelapproach[21]. Evenmoreinterestingisthecaseofpolymerspossessinga
substantiallinearchargedensity,suchasDNAandRNAwhichwillbethefocusofthepresent
study. The properties of polyelectrolytes, in fact, depend very strongly on the electrostatic
screeningprovidedbytheionspresentinsolution.Theinfluenceoftheelectrostaticscreening
on the behaviour of polyelectrolytes has been extensively investigated both experimentally
andtheoretically[32,33, 34, 35,36, 37,38]. Fromthelatter perspective,itiscustomaryto
introduceapparent(oreffective)physico-chemicalparameterstodescribethepropertiesofa
polyelectrolyteinagivensolutionwithreferencetotheunchargedpolymercase.Forexample,
inthecontextofelasticity,oneintroducesaneffective(solution-dependent)bendingrigidity
toaccountfortheadditionalelectrostaticcontributiontothe“bare”persistencelengthofthe
hypothetically-neutralpolymer[37,38]. Also,inthecontextofcolloidaldispersionsofstiff
polyelectrolytes,onecandescribethepolymerasunchargedcylindersandresorttothetheory
for second virial coefficient to derive its solution-dependenteffective diameter[34, 39, 40].
ForDNAinsolutionsoflowionicstrength,boththeeffectivepersistencelengthandeffective
diametercanexceedbyseveralfactorsthebareones. Sofar,theseeffectiveDNAproperties
have been typically probed by distinct methodologies. For example stretching experiments
were employed to establish the dependence of the the persistence length on ionic strength
Inferringtheeffectivethicknessofpolyelectrolytesfromstretchingmeasurements 3
[41, 6] while measurements of second virial coefficients, knotting probabilities or braiding
propertieswereusedfortheeffectivediameter[42,40,43,44,45]).
The thick-chain framework is used to obtain, starting from stretching measurements
data, the effective diameter of a polyelectrolyte and to further relate it to its the effective
persistence length. Besides the implications in the general context of polyelectrolytes the
proposed method can be used to establish the effective structural parameters to be used in
coarse-grainedstudiesoflooping,knottingandpackagingofbiopolymers[46,47].
Thethickchainmodel
To characterisethe stretching responseof a polymerwith finite thicknesswe shall view the
latterasatubewithauniformcircularsectionintheplaneperpendiculartothetubecentreline.
Thechainthickness, ∆, is definedas theradiusofthe circularsection. Severalframeworks
havebeenintroducedtocapturetheuniformthicknessconstraintinawayaptfornumerical
implementation. Theseapproachestypicallyrelyonadiscretisedrepresentationofthethick
chain[48,49, 50, 28,51]. Inthisstudywe shallemploythepiece-wiselinearmodellingof
thechaincenterlineintroducedbyGonzalezandMaddocks[28].
WeshallindicatewithΓthecenterlineofthechainconsistingofasuccessionofpoints
{~r ,~r ,...}equispacedatdistancea. Weshallfurtherdenotewith~b thevirtualbondjoining
0 1 i
the ith and i + 1th points,~b = ~r −~r . In order for the succession of points {~r } to
i i+1 i i
be a viable centerline for a chain of thickness ∆, it is necessary that the radii r of the
ijk
circlesgoingthroughanytriplesofpointsi,jandk,arenotsmallerthan∆. Accordingly,the
Hamiltonianforthethickchain(tube)modelcanbewrittenas
H (Γ)= V (r ) (1)
TC 3 ijk
ijk
X
whereV isthethree-bodypotentialusedtoenforcethethickness∆ofthechain[28,49,52,
3
29, 30, 53]. Asanticipated,the argumentofV isthe radiusofthecircle goingthroughthe
3
tripletofdistinctpointsi, j, kandhastheform
0 ifr>∆,
V (r)= (2)
3 +∞ otherwise.
(cid:26)
Physically,themodelofeqn.1introducesconformationalrestrictionsforthecenterline
that are both local and non-local in character, as depicted in Fig. 1. The local constraints
are those where the triplet i, j and k identifies three consecutive points. The limitation on
theradiusoftheassociatedcircumcirclereflectsthefactthat,toavoidsingularities,thelocal
radiusofcurvaturemustnotbesmallerthan∆. Thisreflectsonthefollowingboundonthe
angleformedbytwoconsecutivebonds:
~b ·~b a2
i i+1 ≥1− (3)
a2 2∆2
Ontheotherhandthereisalsoanon-localeffectduetothefactthatanytwoportionsof
thecenterlineatafinitearclengthseparationcannotinterpenetrate. Ithasbeenshowninref.
[28]thatthissecondeffectcanbeaccountedforbyrequiringthattheminimumradiusamong
circlesgoingthroughanytripletofnon-consecutivepoints,isalsogreaterthan(orequalto)
∆. Theseamlesswayinwhichthelocalandnon-localstericeffectsareaccountedformake
themodelparticularlyappealing. Otherdiscretemodelsrelyingonpairwiseinteractionsfor
theexcludedvolume(suchasthecylindricalmodelofrefs. [44,45])maybeadopted,though
Inferringtheeffectivethicknessofpolyelectrolytesfromstretchingmeasurements 4
adhocprescriptionsfordealingwithe.g.overlappingconsecutiveunitsneedtobeintroduced
[54].
In the present context we will consider the application of a stretching force, f~, to the
endsofachainΓ=r~,...,~r ofthickness∆(thecontourlengththereforebeingL =Na).
0 N c
TheHamiltonianofeqn.1needstobecomplementedwiththestretchingenergy
H= V (r )−f~·(~r −~r ). (4)
3 ijk N 0
ijk
X
As customary we shall characterize the force dependence of the average normalised
projectionoftheend-to-enddistance,~r −~r ,alongthedirectionofappliedforce:
N 0
(~r −~r )·f~
x=h N 0 i (5)
Na|f~|
wherethebracketsdenotethecanonicalensembleaverage.Owingtoitsself-avoidingnature,
the stretching response of the chain cannot be characterised exactly by available analytical
methods. We shall therefore resort to extensive Monte Carlo samplings, based on the
Metropolisscheme,toevaluatetheensembleaveragesofeqn.5.
Besides the numerical study of the tube model subject to stretching it is interesting to
illustrateasimplificationofthemodelofeqn.1,whichisamenabletoanextensiveanalytical
characterization. Todosoweretainonlythelocalthicknessconstraintandthusendupwith
amodelthatisessentiallynon-self-avoiding.Thesimplifiednatureofthisproblem,however,
makesitverytractablealsointhepresenceofabendingrigiditypenalty,κ .Weshalltherefore
b
considertheHamiltonian
~b ·~b
H= V (r )−f~·(~r −~r )−κ i i+1 , (6)
3 i,i+1,i+2 N 0 b a2
i i
X X
againwestressthatthethree-bodypotentialisrestrictedonlytoconsecutive(local)triplets.
In this form the stretching response of the model can be characterized exactly both at very
low and very high forces using standard statistical-mechanical procedures [55, 56]. These
twolimitingregimescanbejoinedtogethertoyieldthefollowingapproximateexpressionfor
thestretchingresponseofalocally-thickchainwithbendingrigidity(LTC+BR,forbrevity):
1 2 1 1 2
βaF =2K 1+ − 1+ +
s 2K (1−x)2 s 2K 
(cid:18) (cid:19) (cid:18) (cid:19)
 
1−y(K,∆/a) 1
3 − x (7)
 1+y(K,∆/a) 2K 1+(1/2K)2
 q 
whereK =βκ ,β =1/K T istheinverseBoltzmannfactorand
b B
1− a2 1 + 1 ∆ >0.5
y(K,∆/a)= 2∆2 1−ez z a (8)
( coth(K)(cid:16)− 1 (cid:17) ∆ ≤0.5
K a
with z = a2 K. The two cases in the above equation, reflect the fact that, for ∆ < a/2
2∆2
norestrictionappliestotheangleformedbytwoconsecutivevirtualbondssinceeqn. (3)is
alwayssatisfied.
Inferringtheeffectivethicknessofpolyelectrolytesfromstretchingmeasurements 5
Expression7possessessomenoteworthylimits. First,intheabsenceofthicknessandin
thecontinuumlimit(a→0,K →ξ /a,ξ beingthepersistencelength)[55,56],themodel
p p
reducestothewell-knownMarkoandSiggiaresultfortheWLC:
k T 1 1
f(x)= B − +x (9)
ξ 4(1−x)2 4
p (cid:20) (cid:21)
Secondly, in the absence of both thickness and bending rigidity one recoversthe low-
andhigh-forceresponseofthefreely-jointedchainwithKuhnlengthequaltoa:
3kBTx, x→0;
f(x)≈ a (10)
1 , x→1.
(cid:26) 1−x
It is of interestalso the case offinite thickness∆/a > 0.5 butno bendingrigidity. In
thiscaseoneobtainsthefollowingexpressionforthepersistencelength:
a
ξ =− . (11)
p ln 1− a2
4∆2
Though this expression(cid:0)does not(cid:1)include the non-local self-avoiding effects it will be
shownlatertoprovideagoodapproximationofthepersistencelengthobtainednumerically
forthefullmodelofeqn.1.
Local triplet
i
∆
j
k
∆ k
Non−local
i
triplet j
Figure1. Thefinitethicknessintroduces stericconstraints oflocalandnon-localcharacter
thatforbidconfigurationswherethechainself-intersects. Theseconstraintsareconveniently
treatedwithinthethree-bodyprescriptionofthethick-chainmodel. Withinthisapproachthe
centerlineofaviableconfigurationissuchthattheradii,Rijkofthecirclesgoingthroughany
tripletofpointsonthecurvei, j, karenotsmallerthan∆.
Weconcludethissectionbymentioningthatforthemodelsofeqn.1and6thespacing
ofconsecutivepointsisconstantalongthecenterline,sothatthecontourlengthisunaffected
bytheapplicationofforcesofarbitrarystrength. Theinextensibilitypropertyisobviouslya
simplification of the behaviour found in naturally-occurring polymers which, at very high
forces can undergo isomerization or structural transitions resulting in an “overstretching”
beyondtheirnominalcontourlength.Severalapproximatetreatmentshavebeendevelopedto
correctthestretchingresponseofinextensiblemodelssotoaccountforoverstretching[57,5]
byadoptingadditionalparametersinthetheory.Inthepresentstudyweshallkeepthenumber
ofmodelparametersto a minimumandhencepostponeto futureworkthethe investigation
ofthemostsuitablewaytoincludeoverstretchingintheTCmodel.
Inferringtheeffectivethicknessofpolyelectrolytesfromstretchingmeasurements 6

80100 1.0
Fit with exp(-n/p)
60 10 0.8 p/ a= = 1 62.71 – 0.03
1 0.6
a/p 40 0.10 1 2 3 4 ttii+ni0.4
20
0.2
MC data
Local approx.
0 0.0
0 1 2 3 4 0 25 50 75 100
/a n
(a) (b)
Figure2. (a)ComparisonbetweenthepersistencelengthofathickchainobtainedbyMonte
Carlosampling(dotted-square)andthatobtainedfromthelocalthicknessapproximation(solid
curve)ofeqn.(11). Theinsetillustrates thelimitations ofthelocal approximation forlow
valuesof∆/a. (b)Tangent-tangentcorrelation(dotted-square)obtainedfromtheanalysisof
104(uncorrelated)Monte-Carlo-generatedchainsof1000segmentsandwith∆/a=2.The
errorbarswhichshowtheuncertaintyoftheplottedvaluesaresmallerthanthesquaresymbols.
Thesolidcurveisasingleexponentialfittothenumericaldatayieldingξp=16.71 0.03.
±
Numericalresults
The characterization of the stretching response of the thick chain of eqn. 4 was carried
out using a Monte Carlo scheme: starting from an arbitrary initial chain configuration
satisfyingthethicknessconstraints,theexplorationoftheavailablestructurespacewasdone
by distorting conformations by means of pivot and crankshaft moves. Newly-generated
structures are accepted/rejected according to the standard metropolis criterion (the infinite
strengthofthethree-bodypenaltiesofeqn.2wasenforcedbyalwaysrejectingconfigurations
violatingthecircumradiiconstraints).
The discretization length, a, was taken as the unit length in the problem and several
valuesof∆/awereconsidered,rangingfromtheminimumallowedvalueof0.5tothevalue
of 4.0. This upper limit appears adequate in the present context since the largest nominal
ratiofor∆/aamongthebiopolymersconsideredhereisachievedfordsDNAforwhichone
has ∆/a ≈ 3.7 [47]. For each explored value of ∆/a considered, we considered chains
of length at least ten times bigger than the persistence length estimated through eqn. (11).
The relative elongation of the chain, was calculated for increasing values of the applied
stretching force (typically about 100 distinct force values were considered). For each run,
after equilibration, we measured the autocorrelation time and sampled a sufficient number
of independent conformations to achieve a relative error of, at most, 10−3 on the average
chain elongation. For moderate or high forces this typically entailed the collection of 104
independentstructureswhileatenfoldincreaseofsamplingwasrequiredatsmallforcesdue
tothebroaddistributionoftheend-to-endseparationalongtheforcedirection.
We first discuss the results for the persistence length obtained from the decay of the
tangent-tangentcorrelationsmeasuredat zeroforceoveran ensemble ofsampled structures
pickedattimesgreaterthanthesystemautocorrelationtime. Theresultingdataareshownin
Fig. 2(a),alongwiththecurvecorrespondingtotheapproximateexpressionofeqn. (11). It
Inferringtheeffectivethicknessofpolyelectrolytesfromstretchingmeasurements 7
canbeseenthatthelocalapproximationforthepersistencelengthisverygoodintherange
1.0 ≤ ∆ ≤ 4, where the relative difference from the value found numerically is typically
inferior to 10%. Significant relative discrepancies are, instead found as ∆ approaches the
limiting value of 0.5 (though it should be noted, the single exponentialfit suffers from the
veryrapiddecayofthetangentautocorrelation). Inthiscase, onlyanarrowrangeofvalues
for the angle formedby two consecutivebondsis forbidden. Consequently, the persistence
lengthisverymuchaffectedbythe(non-local)self-avoidanceconditionthatisunaccounted
for by the simple expression of eqn. (11). As intuitively expected the value of ξ found
p
numericallyislargerthantheonebasedonthelocal-thicknessapproximation.

5
100
4 /a = 0.75 /a = 0.75
/a = 1.00 10 /a = 1.00
3 /a = 2.00 /a = 2.00
/a = 3.00 /a = 3.00
f a/kTB 2 f a/kTB 0.11
1
0 0.01
-1 1E-3
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
x x
(a) (b)
Figure 3. Linear (a) and semi-log (b) plots of the force versus relative extension curves
obtainedfromMonteCarlosamplingofchainsofrelativethickness∆/a=0.75,1,2and3.
Theanalysisofthenumericalresultsrevealedseveraldifferentstretchingregimesinthe
elastic response of a thick chain. These are best discussed in connection with analogous
regimes found in the two standard reference models, the freely-jointed chain (FJC) and
the worm-like chain (WLC). We first point out that for both these models, as well as the
generalisation of eqn. 7 the relative elongation, x, depends linearly on the applied force,
f. This result holds also for the thick chain model. However, due to the inclusion of the
self-avoidanceeffectintheTC(absentinboththeFJCandWLC),theHookeanrelationship
betweenf andxdisappearsinthelimitoflongpolymerchainsinfavourofthePincusregime,
f ∼ x1/(1/ν−1), ν ∼ 3/5 being the critical exponent for self-avoiding polymers in three
dimensions[25,16,58],seeFig. 4.
ForintermediateforcesthePincusbehaviourisfoundtobefollowedbyasecondregime
characterisedbythesamescalingrelationfoundintheWLCathighforces,f ∼(1−x)−2.As
showninFig. 4,atstillhigherforcesthesamescalingrelationfoundintheFJCisobserved,
f ∼(1−x)−1. Physically,thefirsttworegimesaredeterminedbyself-avoidanceandchain
stiffnessorpersistencelengthwhilethelastregimeisascribabletothediscretenatureofthe
chain[59,60,55,61].
Inordertoapplythethick-chainmodeltocontextswhereexperimentaldataareavailable
we have analysed the data of the numerical simulations with the purpose of extracting an
analytical expression capturing the observed functional dependence of f on ∆ and a. For
any value of a and ∆ the sought expression should reproduce the succession of the three
Inferringtheeffectivethicknessofpolyelectrolytesfromstretchingmeasurements 8

4
3 slope -2.04 0.3
2
a/kTB 1 slope -1.01 0.2 x f 2/3
fg10 0 x
Lo -1 0.1
-2
MC data for /a = 1.0) MC data for /a = 1.0)
-3 0.0
-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.00 0.04 0.08 0.12
Log10(1-x) fa/kBT
(a) (b)
Figure4.(a)Extensionversusreducedforceforachainofrelativethickness∆/a=1.Data
pointsarepresentedinthe(log(1 x),log(f))planetohighlight theWLC-andFJC-like
−
regimesfoundatmoderateandhighforces. (b)Illustrationofthelow-forcecrossoverfrom
theHookeanregime,x f,tothePincusone,x f2/3forachainof1200segmentsand
∝ ∝
∆/a=1.
regimesdiscussedabove. Amongseveraltrialformulaewefound,aposteriori,thatthebest
interpolationwasprovidedbythefollowingexpression,
k T k x3/2+k x2+k x3
f(x)= B tanh 1 2 3 , (12)
a(1−x) 1−x
(cid:18) (cid:19)
wheretheparametricdependenceon∆andaiscarriedbythefollowingexpressionsforthe
k’s,
k−1 = −0.28394+0.76441∆/a+0.31858∆2/a2, (13)
1
k−1 = +0.15989−0.50503∆/a−0.20636∆2/a2, (14)
2
k−1 = −0.34984+1.23330∆/a+0.58697∆2/a2. (15)
3
Within the explored ranges of ∆ and a, the relative extension obtained from eq. (12)
differsonaverageby1%(andatmostby5%)fromthetruevaluesatanyvalueoftheapplied
force,asshowninFig. 5.
Applicationstoexperimentaldataanddiscussions
We shall now discuss the application of the TC model to data sets obtained in DNA and
RNA stretchingexperimentscarriedoutforvarious[Na+] concentrations. In particular,the
data for dsDNA are taken from ref. [2] (solutions of 10, 1.0 and 0.1 mM [Na+]) while for
single-stranded RNA (poly-U) we considered the recent results of ref. [62] (solutions of
500, 300, 100, 50, 10 and 5 mM [Na+]). The fits of the experimental data is carried out
through the standard procedure of minimizing the χ2 over the TC parameters: the contour
length, L , chain granularity, a and chain thickness, ∆. For the calculation of χ2 we have
c
estimated the effectiveuncertaintyon the force measurements, takingthe relative extension
Inferringtheeffectivethicknessofpolyelectrolytesfromstretchingmeasurements 9

3
10
/a = 0.75 102 /a = 4.00
MC data MC data
f a/kTB 101 Refit f a/kTB 100 Refit
-1
10 -2
10
10-3 10-4
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
x x
(a) (b)
Figure 5. Illustration of the performance of the parametric expression of eqn. (12) in
reproducing the stretching response obtained numerically for chains of relative thickness
∆/a=0.75and∆/a=4.0(figures(a)and(b),respectively).Inbothcasesthediscrepancy
betweenthecomputedandparametrisedvaluesisabout1%onaverage(andalwayssmaller
than5%).
astheindependentvariable,directlyfromthelargedatasetsofpoly-U(morethan400points
for each set). For dsDNA, owing to the more limited number of points (about 20-25) we
propagatedthedeclaredexperimentaluncertaintyonextension.
Theresultsofthefitproceduresareprovidedingraphs6and7andtables1and2.

+
10 mM [Na]
+
10 1.0 mM [Na]
+
N) 0T.C1 mmoMd e[Nl afits]
p
e ( 1
c
or
F 0.1
0.01
5 10 15 20 25 30 35
Extension ( m)
Figure6. Thick-chainfitoftheexperimentaldata(squares,circlesandtriangles)ondsDNA
insolutionsofvariousionicstrengths.Thebest-fitparametersareprovidedinTable1.
Itisparticularlyappealingthat,overtheabout400datapointsavailableforpoly-U,the
χ2 associatedtothethickchainisverycloseto1forthesetofmeasurementscarriedoutfor
[Na+] in the 50 to 500 mM range. In the case of the two smallest concentrations, [Na+]=
5 and 10 mM, a significant increase of the χ2 is observed. The same is true for the fit of
DNA measurements carried out in 0.1 mM [Na+]. The worsening of the TC performance
consequent to the increase of the screening length is reflected, among other effects, by the
progressiveimportanceofaccountingforoverstretching[63,6,62]. Accordingly,thefitting
Inferringtheeffectivethicknessofpolyelectrolytesfromstretchingmeasurements 10

+ +
300 mM [Na] 10 mM [Na]
10
N)10 N)
p p
e ( e (
orc orc 1
F F
1
Experimental data
2 Experimental data
TC fit ( = 1.29) 2
0.1 TC fit ( = 4.21)
0 300 600 900 1200 1500 1800 2100 0 300 600 900 1200 1500 1800 2100
Extension (nm) Extension (nm)
(a) (b)
Figure7. Application ofthe thick-chain modeltoPoly-U stretching data for(a)300mM
[Na+]and(b)10mM[Na+].Experimentaldataareshownasopencircles,thefitwiththeTC
isdenotedwithasolidline.Thebest-fitparametersfortheTCareprovidedinTable2.
dsDNA
[Na+] a ∆ ξ L χ2
p c
(mM) (nm) (nm) (nm) (nm)
10 9±2 12±2 55±2 32.9±0.2 0.6
1.0 25±7 26±4 94±4 32.3±0.2 0.3
0.1 36±16 48±13 242±14 32.3±0.2 1.9
Table1.Best-fitparametersobtainedbyapplyingtheparametricforce-extensionexpressions
oftheTCmodeltotheexperimentaldataonDNAinsolutionsofvariousionicstrengths.
poly-U
[Na+] a ∆ ξ L χ2
p c
(mM) (nm) (nm) (nm) (nm)
500 1.10±0.01 0.64±0.01 0.80±0.01 2126±4 1.49
300 1.09±0.02 0.65±0.01 0.91±0.01 2123±5 1.29
100 1.08±0.03 0.68±0.02 1.11±0.01 2117±8 1.05
50 1.11±0.03 0.73±0.02 1.30±0.01 2134±8 1.33
10 1.33±0.08 0.94±0.04 1.94±0.03 2138±12 4.21
5 1.41±0.10 1.03±0.05 2.26±0.05 2142±12 4.34
Table2.Best-fitparametersobtainedbyapplyingtheparametricforce-extensionexpressions
oftheTCmodeltotheexperimentaldataonpoly-Uinsolutionsofvariousionicstrengths.
parameters obtained at the lowest salt concentrations can be expected to change upon the
introductionofastretchingmodulus.
ForbothDNAandRNAtheviabilityofthestructuralandelasticparametersoftheTC
fitcanbecomparedagainstthoseobtainedbydifferentmodelsandphysicalapproaches.The
mostnaturaltermofcomparisonfortheelasticresponseisprovidedbythewidely-employed
WLC, which is the common reference model for determining the persistence length and