Self-assembly of short DNA duplexes: from a coarse-grained model to experimentsthrough a theoretical link

Cristiano De Michele 1,∗, Lorenzo Rovigatti 1, Tommaso Bellini 2 and Francesco Sciortino 31

1 1 Dipartimento di Fisica, “Sapienza” Universita di Roma,P.le A. Moro 2, 00185 Roma, Italy 2 Dipartimento di Chimica,

Biochimica e Biotecnologie per la Medicina, Universita di Milano,I-20122 Milano, Italy and 3 Dipartimento di Fisica and CNR-ISC,

“Sapienza” Universita di Roma, P.le A. Moro 2, 00185 Roma, Italy(Dated: April 5, 2012)

Short blunt-ended DNA duplexes comprising 6 to 20 base pairs self-assemble into polydispersesemi-flexible chains due to hydrophobic stacking interactions between terminal base pairs. Abovea critical concentration, which depends on temperature and duplex length, such chains order intoliquid crystal phases. Here, we investigate the self-assembly of such double-helical duplexes with acombined numerical and theoretical approach. We simulate the bulk system employing the coarse-grained DNA model recently proposed by Ouldridge et al. [ J. Chem. Phys. 134, 08501 (2011) ].Then we evaluate the input quantities for the theoretical framework directly from the DNA model.The resulting parameter-free theoretical predictions provide an accurate description of the simulationresults in the isotropic phase. In addition, the theoretical isotropic-nematic phase boundaries are inline with experimental findings, providing a route to estimate the stacking free energy.

PACS numbers:

I. INTRODUCTION

Self-assembly is the spontaneous formation throughfree energy minimization of reversible aggregates of basicbuilding blocks. The size of the aggregating units, e.g.simple molecules, macromolecules or colloidal particles,can vary from a few angstroms to microns, thus mak-ing self-assembly ubiquitous in nature and of interest inseveral fields, including material science, soft matter andbiophysics [1–5]. Through self-assembly it is possible todesign new materials whose physical properties are con-trolled by tuning the interactions of the individual build-ing blocks [6–9].

A relevant self-assembly process is the formation offilamentous aggregates (i.e. linear chains) induced bythe anisotropy of attractive interactions. Examples areprovided by micellar systems [10–12], formation of fibersand fibrils [13–16], solutions of long duplex B-form DNAcomposed of 102 to 106 base pairs [17–20], filamentousviruses [21–25], chromonic liquid crystals [26] as well asinorganic nanoparticles [27].

If linear aggregates possess sufficient rigidity, the sys-tem may exhibit liquid crystal (LC) phases (e.g. ne-matic or columnar) above a critical concentration. Inthe present study we focus on the self-assembly of short(i.e. 6 to 20 base pairs) DNA duplexes (DNADs) [28–30] in which coaxial stacking interactions between theblunt ends of the DNADs favor their aggregation intoweakly bonded chains. Such a reversible physical poly-

∗To whom correspondence should be addressed. Tel:+390649913524; Fax: +39064463158; Email: cris-tiano.demichele@roma1.infn.it

merization is enough to promote the mutual alignment ofthese chains and the formation of macroscopically orien-tationally ordered nematic LC phases. At present, stack-ing is understood in terms of hydrophobic forces actingbetween the flat hydrocarbon surfaces provided by thepaired nucleobases at the duplex terminals, although thedebate on the physical origin of such interaction is still ac-tive and lively [31, 32]. In this respect, the self-assemblyof DNA duplexes provides a suitable way to access andquantify hydrophobic coaxial stacking interactions.

In order to extract quantitative informations fromDNA-DNA coaxial stacking experiments, reliable compu-tational models and theoretical frameworks are needed.Recent theoretical approaches have focused on theisotropic-nematic (I-N) transition in self-assembling sys-tems [33, 34], building on previous work on rigid andsemi-flexible polymers [35–43]. In a recent publica-tion [44] we investigated the reversible physical poly-merization and collective ordering of DNA duplexes bymodeling them as super-quadrics with quasi-cylindricalshape [45] with two reactive sites [46, 47] on their bases.Then we presented a theoretical framework, built onWertheim [48–50] and Onsager [51] theories, which is ableto properly account for the association process.

Here, we employ this theoretical framework to studythe physical properties of a realistic coarse-grained modelof DNA recently proposed by Ouldridge et al. [52], wherenucleotides are modeled as rigid bodies interacting withsite-site potentials. Following Ref. [44], we compute theinputs required by the theory, i.e. the stacking free en-ergy and the DNAD excluded volume, for the Ouldridgeet al. model [52]. Subsequently we predict the poly-merization extent in the isotropic phase as well as theisotropic-nematic phase boundaries.

To validate the theoretical predictions, we perform

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large-scale molecular dynamics (MD) simulations in theNVT ensemble of a bulk system comprising 9600 nu-cleotides, a study made possible by the computationalpower of modern Graphical Processing Units (GPUs).The parameter-free theoretical predictions provide anaccurate description of the simulation results in theisotropic phase, supporting the theoretical approach andits application in the comparison with experimental re-sults.

The article is organized as follows. Section II providesdetails of the coarse-grained model of DNADs, of theMD computer simulations and presents a summary ofthe theory. Section IV describes the protocols imple-mented to evaluate the input parameters required by thetheory via MC integrations for two DNADs. We alsodiscuss some geometrical properties of the bonded dimerconfigurations. We then compare the theoretical predic-tions with simulation and experimental results. Finally,in Section V we discuss estimates for the stacking freeenergy and present our conclusions.

II. METHODS

A. Model

We implement a coarse-grained model for DNA re-cently developed by Ouldridge et al. [52, 53]. In suchmodel, designed via a top-down approach, each nu-cleotide is described as a rigid body (see Figure 1). Theinteraction forms and parameters are chosen so as toreproduce structural and thermodynamic properties ofboth single- (ssDNA) and double- (dsDNA) strandedmolecules of DNA in B-form. All interactions betweennucleotides are pairwise and, in the last version of themodel [52], continuous and differentiable. Such featuremakes the model convenient foe MD simulations.

The interactions between nucleotides account for ex-cluded volume, backbone connectivity, Watson-Crick hy-drogen bonding, stacking, cross-stacking and coaxial-stacking. The interaction parameters have been adjustedin order to be consistent with experimental data [52, 54,55]. In particular, the stacking interaction strength andstiffness have been chosen to be consistent with the ex-perimental results reported for 14-base oligomers by Hol-brook et al. [55]. Hydrogen-bonding and cross-stackingpotentials were adjusted to give duplex and hairpin for-mation thermodynamics consistent with the SantaLuciaparameterization of the nearest-neighbor model [54]. In-teraction stiffnesses were also further adjusted in order toprovide correct mechanical properties of the model, suchas the persistence length. The model does not have anysequence dependence apart from the Watson-Crick pair-ing, meaning that the strength of the interactions actingbetween nucleotides is to be considered as an averagevalue. In addition, the model assumes conditions of highsalt molarity (0.5 M). In this model, the coaxial-stackinginteraction acts between any two non-bonded nucleotides

and is responsible for the duplex-duplex bonding. It hasbeen parametrized [56] to reproduce experimental datawhich quantify the stacking interactions by observingthe difference in the relative mobility of a double strandwhere one of the two strands has been nicked with respectto intact DNA [57, 58] and by analyzing the melting tem-peratures of short duplexes adjacent to hairpins [59].

To cope with the complexity of the model and the largenumber of nucleotides involved in bulk simulations, weemploy a modified version of the CPU-GPU code used ina previous work [60], and extend it to support the force-fields [61]. Harvesting the power of modern GraphicalProcessing Units (GPUs) results in a 30-fold speed-up.The CPU version of the code, as well as the Python li-brary written to simplify generation of initial configura-tions and post-processing analysis, will soon be releasedas free software [62].

B. Bulk simulations

To compare numerical results with theory, we performBrownian dynamics simulations of 400 dsDNA moleculesmade up by 24 nucleotides each, i.e. 400 cylinder-likeobjects with an aspect ratio of ≈ 2 (see Figure 1 (d)).The integration time step has been chosen to be 0.003 insimulation units which corresponds, if rescaled with theunits of length, mass and energy used in the model, toapproximately 1× 10−14 seconds.

We study systems at three different temperatures,namely T = 270 K, 285 K and 300 K, and for differentconcentrations, ranging from 2 mg/ml to 241 mg/ml. TheT = 270 K state point, despite being far from the ex-perimentally accessed T , is here investigated to test thetheory in a region of the phase diagram where the degreeof association is significant. To quantify the aggregationprocess we define two DNADs as bonded if their pair in-teraction energy is negative. Depending on temperatureand concentration, we use 106 − 107 MD steps for equi-libration and 108 − 109 MD steps for data generation onNVIDIA Tesla C2050 GPUs, equivalent to 1− 10 µs.

III. THEORY

We build on the theoretical framework previously de-veloped to account for the linear aggregation and col-lective ordering of quasi-cylindrical particles [44]. Here,we provide a discussion of how such a theory can beused to describe the reversible chaining and ordering ofoligomeric DNADs at the level of detail adopted by thepresent model. According to Ref. [44], the free energy of

3

FIG. 1: (a) Schematic representation of the coarse graining of the model for a single nucleotide. (b) Model interaction sites.For the sake of clarity, stacking and hydrogen-bonding sites are highlighted on one nucleotide and the base repulsion site on theother. For visualization reasons, in the following strands will be shown as ribbons and bases as extended plates as depicted in(c). (d) A 12 base-pairs DNAD in the minimum energy configuration. (e) An equilibrium configuration of the same object atT = 300 K. The nucleotides at the bottom are not bonded, the so-called fraying effect. (f) A chain of length Nb = 48 extractedfrom a simulation at c = 200 mg/ml.

a system of equilibrium polymers can be written as

βF

V=

∞∑l=1

ν(l) ln [vdν(l)]− 1+

+η(φ)

2

∞∑l=1l′=1

ν(l)ν(l′)vexcl(l, l′)

− β∆Fb

∞∑l=1

(l − 1)ν(l) +

∞∑l=1

ν(l)σo(l) (1)

where V is the volume of the system, φ ≡ vdρ (ρ =N/V is the number density of monomers) is the packingfraction, ν(l) is the number density of chains of length l,normalized such that

∑∞l=1 l ν(l) = ρ, vd is the volume of

a monomer, ∆Fb, as discussed in the subsection IV B, isa parameter which depends on the free energy associatedto a single bond and vexcl(l, l

′) is the excluded volumeof two chains of length l and l′. η(φ) is the Parsons-Leefactor [63]

η(φ) =1

4

4− 3φ

(1− φ)2(2)

and σo(l)[43] accounts for the orientational entropy thata chain of length l loses in the nematic phase (includingpossible contribution due to its flexibility). The explicitform for σo(l) can be found in Ref. [44].

The free energy functional (Eq. 1) explicitly accountsfor the polydispersity inherent in the equilibrium poly-merization using a discrete chain length distribution andfor the entropic and energetic contributions of each singlebond through the parameter ∆Fb.

1. Isotropic phase

In the isotropic phase σ0 = 0 and the excluded volumecan be written as follows (see Appendix A):

vexcl(l, l′, X0) = 2BIX

20 l l′ + 2vdkI

l + l′

2(3)

where the parameters BI and kI can be estimated via MCintegrals of a system composed by only two monomers(see Appendix A) and X0 is the aspect ratio of themonomers. We assume that the chain length distribu-tion ν(l) is exponential [44] with an average chain lengthM

ν(l) = ρM−(l+1)(M − 1)l−1 (4)

where

M =

∑∞l=1 l ν(l)∑∞l=1 ν(l)

. (5)

With this choice for ν(l) the free energy in Eq. (1) be-comes:

βFIV

= −ρβ∆Fb(1−M−1) +

+ η(φ)

[BIX

20 +

vdkIM

]ρ2 +

+ρ

M

[ln(vdρM

)− 1]

+

+ ρM − 1

Mln(M − 1)− ρ lnM. (6)

4

FIG. 2: Snapshots taken from simulations at T = 300 K.At low concentrations (c = 2 mg/ml, top) chain formation isnegligible and the average chain length is approximately 1.As the concentration is increased (c = 80 mg/ml, bottom),DNADs start to self-assemble into chains and the averagechain length increases.

Minimization of the free energy in Eq. (6) with respectto M provides the following expression for the averagechain length M(φ):

M =1

2

(1 +

√1 + 4φekIφη(φ)+β∆Fb

). (7)

2. Nematic phase

In the nematic phase the monomer orientational dis-tribution function f(θ) depends explicitly on the angle θbetween the particle and the nematic axis, i.e. the di-rection of average orientation of the DNAD, since thesystem is supposed to have azimuthal symmetry around

such axis. We assume the form proposed by Onsager [51],i.e.:

fα(θ) =α

4π sinhαcosh(α cos θ) (8)

where α controls the width of the angular distribution.Its equilibrium value is obtained by minimizing the freeenergy with respect to α. As discussed in Appendix A,we assume the following form for the excluded volume inthe nematic phase:

vexcl(l, l′, X0, α) = 2BN (α)X2

0 l l′+2vdk

HCN (α)

l + l′

2(9)

where the term 2vdkHCN (α) is the end-midsection contri-

bution to the excluded volume of two hard cylinders (seeAppendix B) and

BN (α) =π

4D3(η1 +

η2

α1/2+η3

α

). (10)

In Eq. (10), D is the diameter of the monomer and ηkwith k = 1, 2, 3 are three parameters that we chose inorder to reproduce the excluded volume calculated fromMC calculations as discussed in Appendix A.

Inserting Eqs. (9) and (4) into Eq. (1) and assum-ing once more an exponential distribution for ν(l) oneobtains after some algebra:

βFNV

= σo − ρβ∆Fb(1−M−1) +

+ η(φ)

[BN (α)X2

0 +vdk

HCN (α)

M

]ρ2 +

+ρ

M

(ln[vdρM

]− 1)− ρ lnM +

+ ρ ln(M − 1)M − 1

M(11)

where σo ≡∑l σo(l)ν(l). The explicit calculation of the

parameters BN and kHCN is explained in Appendices Aand B.

Assuming that the orientational entropy σo can be ap-proximated with the expression valid for long chains [43],minimization with respect to M results in

M =1

2

(1 +

√1 + αφekN (α)φη(φ)+β∆Fb

). (12)

while using the approximated expression for shortchains [43], one obtains

M =1

2

(1 +

√1 + 4αφekN (α)φη(φ)+β∆Fb−1

). (13)

The equilibrium value of α is thus determined by fur-ther minimizing the nematic free energy in Eq. (11),which has become only a function of α. The parame-ter α is related to the degree of orientational orderingin the nematic phase as expressed by the nematic orderparameter S as follows:

5

S(α) =

∫(3 cos2 θ − 1)fα(θ)π sin θ dθ ≈ 1− 3/α. (14)

Further refinements of the theory could be obtained byincluding a more accurate description of the orientationaldistribution fα(θ) in the proximity of the I-N phase tran-sition, along the lines of Eqs. (40)-(42) of Ref. [44]. Forthe sake of simplicity we have just presented the basictheoretical treatment. However, in the theoretical cal-culations in Sec. IV we will make use of the refined andmore accurate free energy proposed in Ref. [44].

3. Phase Coexistence

The phase boundaries, at which the aggregates ofDNAD are sufficiently long to induce macroscopic ori-entational ordering, are characterized by coexistingisotropic and nematic phases in which the volume frac-tion of DNADs are, respectively, φN = vdρN and φI =vdρI . The number densities ρI and ρN can be calculatedby minimizing Eq. (6) with respect to MI and by mini-mizing Eq. (11) with respect to MN and α. In addition,the two phases must be at equal pressure, i.e. PI = PN ,and chemical potential, i.e. µI = µN . These conditionsyield the following set of equations:

∂

∂MIFI(ρI ,MI) = 0

∂

∂MNFN (ρN ,MN , α) = 0

∂

∂αFN (ρN ,MN , α) = 0

PI(ρI ,MI) = PN (ρN ,MN , α)

µI(ρI ,MI) = µN (ρN ,MN , α) (15)

IV. RESULTS AND DISCUSSION

A. Properties of the model

To characterize structural and geometrical propertiesof the simulated DNADs monomers and aggregates weanalyze conformations of duplexes extracted from large-scale GPU simulations (see Figure 2 for some snapshots).

In the following, the volume vd occupied by a singleDNAD of length X0D and double helix diameter D (D '2 nm) will be calculated as the volume of a cylinder withthe same length and diameter, i.e. vd = πX0D

3/4. Whencomparing numerical and experimental results with the-oretical predictions we use the number of nucleotides Nbin place of X0 (X0 ' 0.172Nb) and the concentration cinstead of the packing fraction φ, which can be relatedto the former via:

φ =0.172D3π

8mNc (16)

where mN = 330 Da is the average mass of a nucleotide.Hence, in the following cI and cN will be used in placeof φI and φN .

First we calculate the dimensions (height L and widthD) of the DNADs for different c and T . We observe noconcentration dependence on both quantities, while thevariation in T is negligible (of the order of 0.1% betweenDNADs of samples at 270 K and 300 K). The effect of thissmall change does not affect substantially the value ofthe aspect ratio, which we consider constant (X0 = 2.06)throughout this work.

The geometrical properties of end-to-end bonded du-plexes are not well-known since there are no experimentalways to probe such structures. In a very recent work, theinteraction between duplex terminal base–pairs has beenanalyzed by means of large-scale full-atom simulationsby Maffeo et al. [64]. They found that blunt-ended du-plexes (i.e. duplexes without dangling ends) have pref-erential binding conformations with different values ofthe azimuthal angle γ, defined as the angle between theprojections onto the plane orthogonal to the axis of thedouble helix of the vectors connecting the O5’ and O3’terminal base pairs. They report two preferential valuesfor γ, namely γ = −20 and γ = 180.

20 25 30 35 40 45 50 55 60 65 70

θ [ ° ]

0

1

2

3

4

P(θ

)

T = 270 KT = 285 KT = 300 K

(a)

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55r [ nm ]

0

2

4

6

8

10

P(r

)

T = 270 KT = 285 KT = 300 K

(b)

FIG. 3: Probability distributions for (a) the azimuthal angleγ and (b) the end-to-end distance r.

6

In the present model the continuity of the helix underend-to-end interactions is intrinsic in the model and theazimuthal angle probability distribution is peaked arounda single value γ0 ≈ 40 (see Figure 3(a)). This is veryclose to the theoretical value γ ≈ 36 given by the pitchof the B-DNA double helix. The qualitative differencebetween the conformations of bonded DNADs found inthis work and in Ref. [64] should be addressed in futurestudies describing the coaxial end-to-end interaction in amore proper way.

In addition, we calculate the average distance r be-tween the centres of masses of the terminal base pairs.Figure 3(b) shows P (r), the probability distribution ofr. P (r) is peaked at 0.39 nm, whereas Maffeo et al. [64]found an average distance of r ≈ 0.5 nm. This differ-ence can be understood in terms of the effect of the saltconcentration which, being five times higher than theone used in Ref. [64], increases the electrostatic screen-ing, thus effectively lowering the repulsion between DNAstrands.

The effect of the temperature is small, as lowering Tleads only to more peaked distributions for both P (γ)and P (r) (and a very small shift towards smaller anglesfor γ) but does not change the overall behavior.

B. Stacking free energy and excluded volume

In this section we discuss the procedure employed toevaluate the input quantities required by the theory,namely ∆Fb and vexcl(l, l

′). To this aim we perform aMonte Carlo integration over the degrees of freedom oftwo duplexes. ∆Fb is defined as [44]

β∆Fb = ln

[2

∆(T )

vd

](17)

where [65]

∆(T ) =1

4

⟨∫Vb

[e−βV (r12,Ω1,Ω2) − 1] dr12

⟩. (18)

Here r12 is the vector joining the center of masses ofparticles 1 and 2, Ωi is the orientation of particle i and〈. . .〉 represents an average taken over all the possible ori-entations. Vb is the bonding volume, defined here as theset of points where the interaction energy V (r12,Ω1,Ω2)between duplex 1 and duplex 2 is less than kBT . Tonumerically evaluate ∆(T ) we perform a MC integrationusing the following scheme:

1. Produce an ensemble of 500 equilibrium configura-tions of a single duplex at temperature T .

2. Set the counter Ntries = 0 and the energy factorF = 0.

3. Choose randomly two configurations i and j fromthe generated ensemble.

4. Insert a randomly oriented duplex i in a randomposition in a cubic box of volume V = 1000 nm3.Insert a second duplex j in a random position andwith a random orientation. Compute the interac-tion energy V (i, j) between the two duplexes i andj and, if V (i, j) < kBT , update the energy factor,F = F +

(e−βV (i,j) − 1

). Increment Ntries.

5. Repeat from step 3, until ∆(T ) ∼= 14

VNtries

F con-verges within a few per cent precision.

The employed procedure to compute vexcl(l, l′) is fairly

similar except that it is performed for duplexes with avarious number of bases (i.e. with different X0) and thequantity F counts how many trials originate a pair con-figuration with V (i, j) > kBT (i.e. in step 4, F = F +1).In the nematic case, the orientations of the duplexes areextracted randomly from the Onsager distribution givenby Eq. 8. With such procedure,

vexcl(l = 1, l′ = 1, X0) =V

NtriesF (19)

We calculate vexcl for 8 values of α, ranging from 5 to45 (see Appendix B). Since the X0 and l dependences ofEqs. (3) and (9) are the same and the X0 dependence ofthe numerically calculated vexcl on the shape of DNADsis negligible, the evaluation of the excluded volume asa function of X0 provides the same information as theevaluation of vexcl as a function of l.

Fig. 4 shows ∆(T ) for all investigated T in a ln ∆ vs1/T plot. A linear dependence properly describes thedata at the three T . An alternative way to evaluate ∆(T )is provided by the limit ρ → 0 of Eq. (7). Indeed inthe low density limit M and ∆(T ) are related via thefollowing relation:

∆(T ) =M(1−M)

2ρ. (20)

Therefore it is also possible to estimate ∆(T ) by ex-trapolating the low density data for M at T = 270 K,285 K and 300 K. The results, also shown in Fig. 4, arein line with the ones obtained through MC calculations.The Arrhenius behavior of ∆(T ) suggests that bondingentropy and stacking energy are in first approximationT independent. The coaxial stacking free energy GST isrelated to ∆(T ) as follows

GST = −kBT ln[2ρ∆(T )]. (21)

Substituting the fit expression provided in Fig. 4for ∆(T ) results in a stacking free energy G0

ST =−0.086 kcal/mol at a standard concentration 1 M ofDNADs and T = 293 K comprising a bonding entropy of−30.6 cal/mol K and a bonding energy of−9.06 kcal/mol.

7

3.3 3.4 3.5 3.6 3.7

1/T [ x 10-3

K-1

]

-2.5

-2

-1.5

-1

-0.5

ln[

∆/D

3 ]

fit: -17.6769+4559/TEq. 20two-body MC

FIG. 4: ∆(T ) calculated with the procedures described inSec. IV B for all investigated T .

C. Isotropic phase: comparing simulation resultswith theoretical predictions

Fig. 5 shows the concentration c dependence of M cal-culated from the MD simulation of the Nb = 12 sys-tem. The average chain length increases progressivelyon increasing c. The figure also shows the theoreticalpredictions calculated by minimizing the isotropic freeenergy in Eq. (6) with respect to M using the previouslydiscussed estimates for ∆Fb and vexcl. The theoreticalresults properly describe the MD simulation data up toconcentrations around 200 mg/ml, which corresponds toa volume fraction φ ≈ 0.20. In Ref. [44] similar observa-tions have been made and the discrepancy at moderateand high φ has been attributed to the inaccuracy of theParsons decoupling approximation. The M values cal-culated using the excluded volume of two hard cylinders(HC) are also reported, to quantify the relevance of theactual shape of the DNA duplex. Indeed the HC pre-dictions appreciably deviate from numerical data beyond100 mg/ml.

D. Phase Coexistence: Theoretical predictions

A numerical evaluation of the phase coexistence be-tween the isotropic and the nematic phases for the coarse-grained model adopted in this study is still impossible toobtain given the current computational power. We thuslimit ourselves to the evaluation of the I-N phase coexis-tence via the theoretical approach discussed in Sec. III.Fig. 6 shows the theoretical phase diagram in the c-Nbplane for T = 270 K and 300 K. As expected, both cIand cN decrease on increasing Nb, since the increase ofthe number of basis result in a larger aspect ratio. Ondecreasing T , theory predicts a 10% decrease of cI and a

0 50 100 150 200 250c [ mg/ml ]

1

1.1

1.2

1.3

M

HCT = 270 KT = 285 KT = 300 K

FIG. 5: Average chain length M in the isotropic phase atlow concentration. Symbols are numerical results and dashedlines are theoretical predictions. Dotted lines are theoreti-cal predictions using the excluded volume of HCs vHC

excl (seeAppendix B).

similar decrease of cN , resulting in an overall shift of theI-N coexistence region toward lower c values. This trendis related to the increase of the average chain length Mwith increasing β∆Fb (see Fig. 4).

Fig. 6 also shows the phase boundaries calculated usingthe excluded volume of two hard cylinders. AssimilatingDNADs to hard cylinders results in a 10–15% wideningof the isotropic-nematic coexistence region.

E. Comparison between theory and experiments

The theoretical predictions concerning the isotropic-nematic coexisting concentrations can be compared tothe experimental results reported in Refs. [28] and [30]for blunt-ended DNADs.

Figure 7 compares the experimentally determined ne-matic concentrations cN at coexistence with the valuescalculated from the present model for T = 293 K. De-spite all the simplifying assumptions and despite the ex-perimental uncertainty, the results provide a reasonabledescription of the Nb dependence of cN . The experimen-tal data refer to different base sequences and differentsalt concentrations. According to the authors cN is af-fected by an error of about ±50 mg/ml. In particular forthe case Nb = 12 the critical concentrations cN for dis-tinct sequences shows that blunt-end duplexes of equallength but different sequences can display significantlydifferent transition concentrations. Hence, for each du-plex length, we consider the lowest transition concentra-tion among the ones experimentally determined, sincethis corresponds to the sequence closest to the symmet-ric monomer in the model. Indeed the dependence of cN

8

6 8 10 12 14 16 18 20N

b

350

400

450

500

550

600

650c

[ m

g/m

l ]

I

I+N

N

T=270 K

6 8 10 12 14 16 18 20N

b

350

400

450

500

550

600

650

c [

mg

/ml

]

I

I+N

NT=300 K

FIG. 6: I-N phase diagram in the c vs Nb plane for T = 270 K(top) and 300 K (bottom). Dotted lines are theoretical phaseboundaries calculated using the excluded volume of HCs vHC

excl

(see Appendix B).

on the DNADs sequence is expected to be larger for theshortest sequences, i.e. Nb < 12, for which DNAD bend-ing could be significant [30]. Unfortunately, quantitativeexperimental data on this bending effect are still lack-ing. In general it is possible that cN for Nb < 12 (forwhich a large number of sequences have been studied,see Fig. 7), would be corrected to lower values if a largernumber of sequences were explored. For more details onthis phenomenon, we refer the reader to the discussionsin Refs. [30, 44, 66].

The overestimation of the phase boundaries for Nb ≥12 with respect to experimental results suggests thatthe DNA model of Ouldridge et al. [52] overestimatesthe coaxial stacking free energy. Such discrepancycan perhaps be attributed to the restricted number ofmicrostates allowing for bonding states in the DNAmodel [52, 56], as discussed in Sec. IV. Indeed, allow-ing DNADs to form end-to-end bonds with more thanone preferred azimuthal angle would increase the entropy

of bonding, thus effectively lowering GST . Allowing forboth left- and right-handed binding conformations, a pos-sibility supported by the results of Maffeo et al. [64],would double ∆(T ) in Eq. (21) and hence add an entropiccontribution equal to −kBT ln(2) to GST , which wouldresult in a value G0

ST − 0.403 kcal/mol = −0.49 kcal/molfor T = 293K. Fig. 7 also shows the theoretical predic-tion for such upgraded GST value.

In Fig. 7 theoretical calculations of the I-N transitionlines are shown for GST = −0.4 kcal/mol and GST =−2.4 kcal/mol at T = 293 K as the upper and lowerboundaries of the grey band respectively. To calculatethese critical lines we retain the excluded volume cal-culated in the subsection IV B and, given the value ofGST , we evaluate ∆Fb according to Eqs. (17) and (21)for T = 293 K and ρ corresponding to the standard 1 Mconcentration.

The selected points with Nb ≥ 12 fall within thegrey band shown in Fig. 7 enabling us to provide anindirect estimate of GST between −0.4 kcal/mol and−2.4 kcal/mol. For the points withN < 12, where duplexbending might play a role, it would be valuable to havemore experimental points corresponding to more straightsequences in order to validate the theoretical predictions.

It is worth observing that for all DNAD lengths Nbthe electrostatics interactions are properly screened. ForNb = 20 a concentration 1.2 M of NaCl has been addedto the solution resulting in a Debye screening lengthk−1D ≈ 0.23 nm. For all other lengths (i.e. Nb ≤ 18) we

note that at the lowest DNA concentration of 440 mg/mlcorresponding to Nb = 14 k−1

D ≈ 0.40 nm. Therefore the

experimental k−1D is always smaller than the excluded vol-

ume diameter for the backbone-backbone interaction ofour coarse-grained model [52] (≈ 0.6 nm), thus enablingus to neglect electrostatic interactions.

On the other hand, if electrostatics interactions arenot properly screened the effective aspect ratio for suchDNAD sequences would be smaller than the ones usedin our theoretical treatment and this would result in aunderestimate of cN . To account for this behavior oneshould at least have a reasonable estimate of the effec-tive size of DNADs when electrostatics interactions arenot fully screened. Moreover, the role of electrostaticsinteractions can be subtle and not completely accountedfor by simply introducing an effective size of DNADs. Apossible route to include electrostatics in our treatmentcan be found in Ref. [20] and it will be addressed in futurestudies.

F. Comparison with Onsager Theory

The experimental average aggregation numbers are es-timated in Refs. [13, 28] by mapping the self-assembledsystem onto an “equivalent” mono-disperse system ofhard rods with an aspect ratio equal to MX0. InRef. [44] it has been shown that the theoretically es-timated isotropic-nematic coexistence lines for the case

9

6 8 10 12 14 16 18 20N

b

300

400

500

600

700

c N

[ m

g/m

l ]

Theory

Th. GST

= -0.49 kcal/mol

Experiments

Exp. other seqs.

FIG. 7: Critical nematic concentrations cN as a functionof the number of base pairs per duplex Nb for the presentmodel, calculated theoretically at T = 293 K using the com-puted stacking free energy G0

ST (short dashed lines), GST =−0.49 kcal/mol (long dashed lines), and for experiments [28](circles and squares). Squares are cN for different sequencesat the same Nb = 12. The grey band has been built consid-ering for GST an upper bound of −0.4 kcal/mol and a lowerbound of −2.4 kcal/mol.

of polymerizing superquadric particles in the MX0 − φplane, parametrized by the stacking energy, are signifi-cantly different from the corresponding Onsager originalpredictions (as re-evaluated in Ref. [35]). In light of therelevance for interpreting the experimental data, we showin Figure 8 the same curves for the DNA model investi-gated here. In this model, a clear re-entrant behavior ofthe transition lines in the c − MX0 plane is observed.The re-entrant behavior occurs for values of the stack-ing free energy accessed at temperatures between 270 Kand 330 K. We believe that the re-entrancy of the tran-sition lines in the c −MX0 plane is a peculiar mark ofthe system polydispersity resulting from the reversibleself-assembling into chains, and as such it should be alsoobservable experimentally.

V. CONCLUSIONS

In this article, we have provided the first study of abulk solution of blunt ended DNA duplexes undergoingreversible self-assembly into chains, promoted by stack-ing interactions. The simulation study, carried out at dif-ferent concentrations and temperatures, provides a clearcharacterization of the c and T dependence of the aver-age polymerization length M and an indirect estimateof the stacking free energy. We have provided a theo-retical description of the self-assembly process based ona theoretical framework recently developed in Ref. [44].

250

300

350

400

450

500

550

600

650

c [

mg/m

l ]

3332/(MX0)

4244/(MX0)

Nematic

T=270 K

T=330 K

T=300 K

T=230 K

T=360 K

Isotropic

Nb=12

5 10 15 20 25 30 35 40M X

0

250

300

350

400

450

500

550

600

c [

mg/m

l ]

Nematic

Isotropic

T=270 K

T=330 K

T=300 K

T=230 K

T=360 K

Nb=20

FIG. 8: Isotropic-nematic coexistence lines in the averageaspect ratio MX0 and concentration c plane for two valuesof Nb, namely Nb = 12 (top) and Nb = 20 (bottom). Solidlines indicate theoretical predictions, dashed lines indicate theOnsager original predictions, as re-evaluated in Ref. [35] forcI and cN . Symbols along the isotropic and nematic phaseboundaries at coexistence are joined by dotted lines, to indi-cate the change in concentration and average chain length atthe transition.

The inputs required by the theory (the DNAD excludedvolumes and the stacking free energy) have been nu-merically calculated for the present DNA model, allow-ing a parameter free comparison between the moleculardynamic results and the theoretical predictions. Suchcomparison has been limited to the isotropic phase, dueto the difficulties to simulate the dense nematic phaseunder equilibrium conditions. The description of theisotropic phase is satisfactory: quantitative agreementbetween theory and simulations is achieved for concen-trations up to c ≈ 200 mg/ml. The stacking free en-ergy value that properly accounts for the polymerizationprocess observed in the molecular dynamics simulationsis G0

ST = −0.086 kcal/mol at a standard concentration1 M of DNADs and T = 293 K comprising a bond-

10

ing entropy of −30.6 cal/mol K and a bonding energy of−9.06 kcal/mol.

Theoretical predictions for the I-N transition have beencompared with experimental results for several DNAlengths, ranging from 8 to 20 bases. For Nb ≥ 12 themodel predicts values for cN which are higher than ex-perimental ones. This suggests that the DNA model em-ployed overestimates GST . In view of the recent resultsof Maffeo et al. [64], we speculate that the bonding en-tropy is underestimated, in agreement with the observa-tion that the probability distribution of the azimuthalangle between two bonded DNADs, which is designedto be single-peaked, is too restraining. In this respect,the present study call for an improvement of the coarse-grained potential [52] in regard to the coaxial stackinginteraction.

The value of GST can also be used as a fitting pa-rameter in the theory for matching cN with the experi-mental results, retaining the excluded volume estimatescalculated for the coarse-grained DNA model. Such pro-cedure shows that values of the stacking free energy be-tween −0.4 kcal/mol and −2.4 kcal/mol are compatiblewith the experimental location of the I-N transition line.In the work of Maffeo et al., the authors report a quitesmaller value of GST , namely GMST = −6.3 kcal/mol, avalue which was confirmed by the same authors by per-forming an investigation of the aggregation kinetic in avery lengthy all-atom simulation of DNAD with Nb = 10.

If such GST value is selected as input in our theoreticalapproach (maintaining the same excluded volume term),then one finds cMN ≈ 250 mg/ml, a value significantlysmaller than the experimental result (cN = 650 ± 50mg/ml). This casts some doubts on the effectivenessof the employed all-atom force-field to properly modelcoaxial stacking.

Finally, our work draws attention to the errors af-fecting the estimate of the average chain length Mvia a straightforward comparison of the nematic co-existing concentrations with analytic predictions basedon the original Onsager theory for mono-disperse thinrods [13, 28]. We have found that such approximationsignificantly underestimates M at the I-N transition con-centration cN . In addition, the theoretical approach pre-dicts a re-entrant behavior of the transition lines in thec-MX0 plane, a distinct feature of the polydisperse na-ture of the equilibrium chains.

ACKNOWLEDGMENTS

We thank Thomas Ouldridge, Flavio Romano andTeun Vissers for fruitful discussions. We acknowl-edge support from ERC-226207-PATCHYCOLLOIDSand ITN-234810-COMPLOIDS as well as from NVIDIA.

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Appendix A: Excluded volume contributions

Here we further discuss the calculation of the excludedvolume term vexcl(l, l

′) for the present model. Follow-ing Ref. [44], the excluded volume is assumed to be thefollowing second order polynomial in l and l′:

vexcl[l, l′; f(θ)] = 2

∫f(θ)f(θ′)D3 [Ψ1(γ,X0)+

+l + l′

2Ψ2(γ,X0)X0 + Ψ3(γ,X0)X2

0 l l′]

dΩ dΩ′ (A1)

where the functions Ψα, α = 1, 2, 3, describe the angulardependence of the excluded volume. The orientationalprobability f(θ) is normalized such that∫

f(θ)dΩ = 1. (A2)

The three contributions to the excluded volumein Eq. (A1) come from end-end, end-midsection andmidsection-midsection steric interactions [44] betweentwo chains.

In the isotropic phase the orientational distributiondoes not have any angular dependence, i.e. f(θ) = 1/4π,

and Eq. (A1) reduces to the form

vexcl(l, l′, X0) = BIX

20 l l′ + kIvd

l + l′

2+AI . (A3)

The parameters BI , kI and AI appearing in Eq. (A3)can be calculated via MC integration procedures as dis-cussed in the subsection IV B and in Ref. [44]. We expectthat these parameters do not depend on X0 because eachDNADs comprises Nb stacked base pairs which are allidentical with respect to excluded volume interactions(i.e. they all have the same shape). In particular, thecalculated excluded volume of two DNADs is reportedin Fig. 9 for 5 different aspect ratios, together with theresulting values for the above parameters.

2 3 4 5 6 7X

0

10

12

14

16

18

20

22

24

vex

cl/v

d

Fit: 6.1677 + 2.4412 X0

Two-body MC

HC

FIG. 9: Excluded volume in the isotropic phase together withanalytic approximations. From the linear fit one has BI =0.959D3 and kI = 3.084, while we assume AI = 0

Using the Onsager angular distribution fα(θ) inEq. (A3), the excluded volume in the nematic phase de-pends also on the parameter α, i.e. the general form inEq. (A1) reduces to

vexcl(l, l′, X0, α) = BN (α)X2

0 l l′ + kN (α)vd

l + l′

2+ AN (α). (A4)

Assuming that AN (α) = 0, kN (α) = kHCN (α) andBN (α) is given by Eq. (10), the three parameters ηkwith k = 1, 2, 3 have to be estimated. For l = l′ = 1 andseveral values of α (α = 5 . . . 45 in steps of 5) and X0

we calculated numerically the nematic excluded volumefor two DNADs. The results are shown in Fig. 10, wherewe plot vexcl/vd vs X0 for various α. The dashed linesshown in Fig. 10 are obtained through a two-dimensionalfit to numerical data for vexcl(1, 1, X0, α) using Eq. (9)as fitting function.

13

Appendix B: Excluded volume of hard cylinders

2 3 4 5 6 7

X0

10

12

14

16

18

20

vex

cl/

vd

2D fit

α = 5

α = 15

α = 25

α = 35

FIG. 10: Excluded volume as a function of aspect ratio X0 inthe nematic phase together with analytic approximations forseveral α. The dashed lines are obtained plotting the functionreported in Eq. (9) and setting η1 = 0.386419, η2 = 1.91328and η3 = −0.836354.

For two rigid chains of length l and l′ which are com-posed of hard cylinders (HCs) of diameter D and lengthX0D, vexcl(l, l

′) can be described by

vHCexcl[l, l′; f(θ)] =

∫f(θ)f(θ′)D3

[ π2

sin γ +π

2X0

(1 + | cos γ|+ 4

πE(sin γ)) +

+l + l′

2+ 2X2

0 sin γ l l′]dΩ dΩ′(B1)

where cos γ = u ·u′, u and u′ are the orientations of twoHCs and E(sin γ) is the complete elliptical integral

E(sin γ) =1

4

∫ 2π

0

(1− sin2 γ sin2 ψ)1/2dψ. (B2)

The integrals in Eq. (B1) can be calculated exactly inthe isotropic phase, while in the nematic phase the calcu-lation can be done analytically only for suitable choicesof the angular distribution f(θ). Here we assume thatthe angular distribution is given by the Onsager functionin Eq. (8).

Using the Onsager orientational function the follow-ing approximate expressions for the coefficients kN (α),BN (α) and AN (α) can be derived [43]

BN (α) = D3(π/4)ρa(α)

kN (α) = πD3X0

vd

(1− 1

α

)AN (α) = D3 (π/4)

2ρa(α) (B3)

where

ρa = 4(πα)−1/2

(1− 15

16α+

105

512α2+

+315

8192α3

)(B4)

We evaluate numerically the excluded volume in Eq.(B1) for many values of α and, building on the expres-sions in Eqs. (B3), we perform a fit to this data usingthe following functions:

BHCN (α) ' D3(π/4)(ρa(α) +

c4α9/2

+c5

α11/2

)(B5)

kHCN (α) = 4

(1− 1

α

)+

∞∑i=2

biαi' 4

π

4∑i=0

diαi

(B6)

AHCN (α) ' D3(π/4)2(ρa(α) +

c4α9/2

+c5

α11/2

)(B7)

The coefficient values resulting from the fitting proce-dure are c4 = 1.2563, c5 = −0.95535, d0 = 3.0846, d1 =−4.0872, d2 = 9.0137, d3 = −9.009 and d4 = 3.3461.