key: cord-0498495-9496vfgo
authors: Tortora, Maxime M.C.; Mishra, Garima; Presern, Domen; Doye, Jonathan P.K.
title: Chiral shape fluctuations and the origin of chirality in cholesteric phases of DNA origamis
date: 2018-11-29
journal: nan
DOI: nan
sha: fcf8e77c6982ca2137e31454cde5781d6e2a9dd4
doc_id: 498495
cord_uid: 9496vfgo

Lyotropic cholesteric liquid crystal phases are ubiquitously observed in biological and synthetic polymer solutions, characterised by a complex interplay between thermal fluctuations, entropic and enthalpic forces. The elucidation of the link between microscopic features and macroscopic chiral structure, and of the relative roles of these competing contributions on phase organisation, remains a topical issue. Here we provide theoretical evidence of a novel mechanism of chirality amplification in lyotropic liquid crystals, whereby phase chirality is governed by fluctuation-stabilised helical deformations in the conformations of their constituent molecules. Our results compare favourably to recent experimental studies of DNA origami assemblies and demonstrate the influence of intra-molecular mechanics on chiral supra-molecular order, with potential implications for a broad class of experimentally-relevant colloidal systems.

Lyotropic cholesteric liquid crystal (LCLC) phases are ubiquitously observed in biological and synthetic polymer solutions, characterised by a complex interplay between thermal fluctuations, entropic and enthalpic forces. 1, 2 The elucidation of the link between microscopic features and macroscopic chiral structure, and of the relative roles of these competing contributions on phase organisation, remains a topical issue. 3, 4 Here we provide theoretical evidence of a novel mechanism of chirality amplification in lyotropic liquid crystals, whereby phase chirality is governed by fluctuation-stabilised helical deformations in the conformations of their constituent molecules. Our results compare favourably to recent experimental studies of DNA origami assemblies and demonstrate the influence of intramolecular mechanics on chiral supra-molecular order, with potential implications for a broad class of experimentally-relevant colloidal systems.

The self-organisation of chiral building blocks into helical super-structures is a phenomenon of broad relevance to many physical and biological processes, from the alpha-helical ordering of amino-acids in protein secondary structures to the synthesis of novel chiroptical meta-materials for plasmonic applications. 1, 2, 5 The hierarchical transfer of chirality from individual molecular units to higher-order assemblies provides a fascinating host of opportunities for the bottom-up design of macroscopic materials with unique functional, mechanical and optoelectronic properties. [6] [7] [8] However, the mechanistic understanding of chirality amplification often constitutes a difficult theoretical task, owing to both the diversity of physico-chemical interactions at play and the wide difference in length-scales between elementary building blocks and super-molecular structures. 1, 2, 9 LCLCs represent a particularly notable illustration of the challenges involved in the description of emergent chirality in self-assembled systems. The macroscopic breaking of mirror symmetry in LCLCs arises from the periodic rotation of the direction of local molecular alignment about a fixed normal axis as one passes through the sample. The dependence of the spatial period of this helical arrangement, termed the cholesteric pitch, on particle structure and experimental conditions has been studied in considerable detail in a variety of model FIG. 1. Ground-state origami conformations. The equilibrium axial twist of the conformations is obtained by elastic energy minimisation using a continuum DNA model. 18 The nucleotide-level depiction corresponds to the finer-grained representation of the oxDNA model, 19 as employed in all mechanical calculations throughout the paper. systems, ranging from filamentous virus suspensions 10, 11 to biologically-relevant collagen assemblies. 12 While theoretical studies of simple particle models have uncovered a few general features of cholesteric organisation, such as the non-trivial link between particle and phase chirality, [13] [14] [15] the remarkable complexity of experimental phase behaviours has so far largely thwarted attempts to rationalise their microscopic underpinnings. The establishment of a quantitative relationship between molecular chirality and supra-molecular helicity in LCLCs has remained a major challenge of soft condensed-matter physics, with broad consequences for their rational applications as bio-inspired multifunctional materials 3, 16 and for our fundamental understanding of the ubiquity of LCLC order in living matter. 17 Significant advances in this direction were recently achieved by exploiting the synergy between colloidal science and DNA origami technology, through which the LCLC organisation of self-assembled origami filaments demonstrated the possibility to tune the micron-scale pitch of the bulk phase via the direct control of single- FIG. 2 . Cholesteric behaviour of ground-state and thermalised origamis. a) Inverse equilibrium cholesteric pitch (P) as a function of particle concentration (c) for ground-state filament conformations. Dashed lines denote values obtained by assuming pure steric interactions, and solid lines by accounting for both steric and Debye-Hückel repulsion. Positive (negative) values of P respectively correspond to LCLC phases bearing right (left) handedness, as illustrated in the right-hand panel. b) Close-approach configuration of idealised, weakly-twisted right-handed filaments, displaying a left-handed arrangement. c) Same as b) in the case of strongly-twisted right-handed filaments, illustrating their entropic preference for right-handed arrangements. d) Same as a) in the case of thermalised filaments. Markers denote experimental measurements (from Ref. 20) . e) Angular configuration minimising the chiral two-body potential of mean force for thermalised 1x-lh origamis ( Supplementary  Fig. 1 ), illustrating the predominance of long-wavelength backbone fluctuations over local axial twist in their LCLC assembly. particle structure at the nanometer level. 20 Through the conjunction of a well-established coarse-grained model of DNA with a classical molecular field theory of LCLCs, we provide a rigorous theoretical analysis of these experimental developments by assessing the detailed influence of particle mechanical properties and thermodynamic state on their ordering behaviour, without the use of any adjustable parameters.

We consider monodisperse B-DNA bundles comprised of 6 double helices crossed-linked in a tight hexagonal arrangement. Such self-assembled filaments may be folded into shapes of programmable twist and curvature through targeted deletions and insertions of base pairs (bp) along each bundle. 21 Following Ref. 20, we here focus on four variants of the filaments comprising 15 224 to 15 240 nucleotides, with experimentally-determined contour lengths of 420 nm and bundle diameters of 6 nm. A continuum finite-element model based on an elastic rod description of DNA 18 predicts the respective ground states of the different designs to bear negligible (s), 360°r ight-handed (1x-rh), 360°left-handed (1x-lh) and 720°l eft-handed (2x-lh) twist about the filament long axis, with negligible net curvature (Fig. 1 ). 20 As a first approximation, we neglect the conformational fluctuations of DNA origamis in solution, and assess the cholesteric arrangement of their respective ground states. To that end, we make use of an efficient and accurate numerical implementation of the Onsager theory extended to the treatment of cholesteric order, 22 which has been extensively discussed elsewhere 23, 24 (Methods). In this framework, the reliable investigation of their LCLC assembly requires the input of a mechanical model capable of resolving the local double-helical arrangement of nucleotides within each duplex. 25 We thus employ the oxDNA nucleotide-level coarse-grained model 19 to represent the origami microscopic structure and interaction potential (Fig. 1 ).

In the absence of electrostatic interactions, the entropy-induced ordering of ground-state filaments is governed by their axial twist, which is found to stabilise anti-chiral LCLC phases -possessing opposite handedness with respect to the origami twist (Fig. 2a) . This seemingly counterintuitive observation is explained by the fact that the pair excluded volume of weakly-twisted, rod-like filaments is generally minimised by oppositehanded arrangements (Fig. 2b) . 22 Conversely, this en-tropic preference is reversed in the case of stronglytwisted filaments (Fig. 2c) , which accounts for the weak right-handed phase predicted for the untwisted (s) origamis in terms of the intrinsic right-handed helicity of DNA. 25 These findings mirror recent results on the LCLC assembly of continuously-threaded particles, for which the quantitative validity of these simple geometric arguments has been investigated in detail. 24 However, these predictions are at odds with the experimental measurements of Ref. 20, which instead revealed a general tendency of origami filaments to stabilise isochiral LCLC phases -bearing the same handedness as their axial twist. Previous theoretical studies have attempted to attribute similar discrepancies to a potential antagonistic influence of electrostatic interactions, 25 although the validity of this argument has been disputed by subsequent numerical investigations. 26 Here, we instead report that the main effect of the inclusion of long-ranged Debye-Hückel repulsion is to simply unwind the predicted cholesteric pitches by penalising close-pair configurations in which the local surface chirality of the origamis is most relevant (see Supplementary Section 2). These results suggest that simple steric and electrostatic repulsion between ground-state filament conformations cannot account for either the handedness or the magnitude of their experimental LCLC pitches, and mirror the conclusions of recent studies on single B-DNA duplexes. 26, 27 To assess the role of conformational statistics on their cholesteric ordering, we make further use of the oxDNA model 19 to probe the detailed thermal fluctuations of the origami filaments. As in Ref. 27 , we extend our theoretical framework to flexible particles through its combination with the numerical sampling of the filament conformational space by single-origami molecular dynamics (MD) simulations (Methods). This hybrid approach, based on the Fynewever-Yethiraj density functional theory, 28 has been shown to be quantitatively accurate in dilute assemblies of long and stiff persistent chains, for which the effects of many-particle interactions on conformational statistics are limited 27 (see Supplementary Section 1). This description is therefore well-suited for our purposes, given the large persistent length (l p ) of the origami structures (l p /l c 5 20,29 ) and the low packing fractions of their stable LCLC phases. 20 Its results display a surprising phase-handedness inversion compared to the LCLC behaviour of the origami ground-states, as well as a considerable tightening of the corresponding equilibrium pitches (Fig. 2d ). The conjunction of these two factors allows for a convincing overall agreement with the experimental measurements of Ref. 20, albeit with a slight offset in the crossover value of the origami twist at which the phase handedness inversion occurs. These effects stem from the emergence of long-wavelength helical deformation modes along the backbone of thermalised origamis, which dominate the chiral component of their potential of mean force over the local surface chirality arising from axial twist (Fig. 2e , see Supplementary Section 2). This long-ranged, super-helical (or solenoidal ) writhe may be quantified by Fourier analysis of the filament backbone conformations (Fig. 3a, Methods) . The fluctuation spectra obtained using the oxDNA model in the limit of long-wavelength deformations are found to be consistent with the asymptotic scaling behaviour of persistent chains for typical experimental values of the filament bending rigidity 20 (Fig. 3b , see Supplementary Section 5). In this regime, the net backbone helicity of each origami variant is found to bear the opposite handedness to the axial twist of its ground state, with left-handed (right-handed) filaments predominantly favouring righthanded (left-handed) helical conformations, respectively (Figs. 3c-d, Methods).

The geometric argument of Fig. 2b , applied to systems of weakly-curled helices, predicts such conformations to display an entropic preference for oppositehanded arrangements. 24 In this case, the stabilisation of iso-chiral phases of twisted origami filaments therefore arises from their propensity for long-ranged, anti-chiral deformations under the effects of thermal fluctuations. This original chirality amplification mechanism is further evidenced by the relative insensitivity of our results to the inclusion of electrostatic interactions (Fig. 2d) , as the typical length-scales of the resulting backbone helicities are considerably larger than the experimental Debye screening length (λ D 0.6 nm) 20 

The origin of this fluctuation-induced solenoidal writhe, and of its dependence on filament twist, lies in the geometric constraints imposed by inter-helical crossovers in the origami design. For instance, the induction of a left-handed axial twist is achieved by reducing the number of base pairs separating adjacent inter-helix junctions along the bundle axis via targeted deletions, resulting in a coherent over-winding strain for each DNA helix. 21 The corresponding torsional stress thus leads to the propagation of a global left-handed twist throughout the filament, as the duplex twist density relaxes towards its equilibrium value (Tw 0 ). When left-handed origamis fluctuate to bear a right-handed helical writhe, one may show that the elastic cost of bending is partially offset by a reduction in the residual over-twist of the DNA heliceswhile left-handed backbone conformations are energetically penalised by a further over-winding of the duplexes (see Supplementary Section 6). Conversely, in the case of right-twisted origamis, the required base-pair insertions lead to an under-winding of the individual DNA helices, which in turn favours a left-handed solenoidal writhe.

In this framework, the observed offset in the filament phase-handedness inversion behaviour could be partially explained in terms of a small misestimate of Tw 0 , as the equilibrium helical pitch of B-DNA within constrained origami structures may slightly differ from the unconfined value 1/Tw 0 10.5 bp assumed in both the computation of the origami ground states 18 and the parametrisation of the oxDNA model. 19 Additional possible sources of error include other potential shortcomings of the oxDNA model, such as our use of sequence-averaged mechanics for DNA or the limitation of soft non-bonded interactions to simple Debye-Hückel electrostatics. 19 The overestimations in the magnitude of our cholesteric pitch predictions, also apparent in Fig. 2d , are further consistent with the symmetry limitations of the theory, in which long-ranged biaxial correlations arising from broken local cylindrical invariance are neglected. 23 The limited extent of these discrepancies, relative to the vast gap between molecular and cholesteric length-scales, combined with the satisfactory experimental agreement achieved in terms of isotropic/cholesteric binodal concentrations (Table I , Methods) and in the magnitude of the underlying macroscopic curvature elasticities (see Supplementary Section 3), nonetheless evidence the ability of the theory to correctly capture the basic physics of LCLC assembly in our case.

To conclude, we have presented the successful application of an extended Onsager theory to the quantita- tive description of LCLC order in systems of long DNA origami filaments. Its combination with an accurate conformational sampling scheme demonstrates that the origin of phase chirality in this case lies in the weak, fluctuation-stabilised solenoidal writhing of the filament backbones. This result represents a marked shift from the prevailing theoretical models, in which the macroscopic breaking of mirror symmetry has generally been attributed to the local chiral structure of the molecular ground state. 25, 30, 31 The link between ground-state and fluctuation-induced chirality is further shown to be nontrivial, as illustrated by the stabilisation of anti-chiral deformation modes through twist-writhe conversion of the filament elastic energy.

This chirality amplification process is grounded in the basic statistical mechanics of the constrained duplexes within each folded origami, and should therefore be quite generally applicable to other supra-molecular assemblies of chiral filament bundles, whose ground-state morphologies have been shown to be widely governed by similar geometric frustration mechanisms. 32 Our findings could thus provide a theoretical basis for the socalled "corkscrew model", previously postulated to explain the puzzling experimental behaviour of filamentous virus suspensions, 11 and more broadly suggest a novel self-assembly paradigm for LCLCs in which subtle, longwavelength conformational features -rather than local chemical structure -dictate macroscopic chiral organisation.

JPKD and MMCT gratefully acknowledgeÉ. Grelet 

MMCT developed the theory, conducted its implementation, carried out the numerical calculations and wrote the manuscript. GM and DP performed the origami simulations. JPKD and MMCT devised the study, analysed the results and proofread the completed manuscript.

The numerical code employed for all densityfunctional and related calculations may be found at https://github.com/mtortora/chiralDFT. The oxDNA simulation package is also available online (https://dna.physics.ox.ac.uk).

Input files will be provided upon request to the authors.

Single-origami simulations were run using the oxDNA coarse-grained model, which represents DNA as a collection of rigid nucleotides interacting through excluded volume, Debye-Hückel, stacking, hydrogen-and covalentbonding potentials. 19 Calculations were performed on GPUs in the canonical ensemble using an Andersen-like thermostat and sequence-averaged DNA thermodynamics, assuming room-temperature conditions (T = 293 K) and fixed monovalent salt concentration c Na + = 0.5 M. This value was chosen in slight excess of the experimental salt concentration c Na + = 0.26 M, employed throughout the rest of the paper, in order to limit computational costs. The effects of this approximation on origami conformational statistics are expected to be minimal in the context of the simplified oxDNA treatment of electrostatics. 19 Relaxation was achieved through equilibration runs of O(10 6 ) MD steps starting from the origami ground state, and production runs of O(10 9 ) steps were conducted to generate O(10 3 ) uncorrelated conformations for each origami variant. The statistical independence of the resulting conformations was assessed by ensuring the vanishing autocorrelation of their end-toend separation distance.

The discretised origami backbones are obtained by averaging the centre-of-mass locations of their bonded nucleotides over the 6 constituent duplexes within each transverse plane along the origami contour. 21 We define the molecular frame R = u v w of each conformation as the principal frame of its backbone gyration tensor, such that u and v correspond to the respective direction of maximum and minimum dispersion of the origami backbone. 27 Shape fluctuations are described by the contour variations of the transverse position vector,

with r(s) the position of the discretised backbone segment with curvilinear abscissa s, assuming the backbone centre of mass to be set to the origin of the frame. Denoting by ∆s the curvilinear length of each segment, the Fourier components of r ⊥ read as

Using the convolution theorem, the spectral coherence between the two transverse components of an arbitrary backbone deformation mode may be quantified by their Fourier-transformed cross-correlation function c vw ,

where r ⊥x = r ⊥ ·x for x ∈ {v, w} and r * ⊥w is the complex conjugate of r ⊥w . It is shown in Supplementary Section 4 that the degree of helicity H(k) of a deformation mode with arbitrary wavenumber k about the long molecular axis u is related to c vw through

with { c vw the imaginary part of c vw . One may check that −1 ≤ H(k) ≤ 1, with H(k) = ±1 if and only if the two transverse Fourier components bear equal amplitudes and lie in perfect phase quadrature. In this case, r ⊥ (k) describes an ideal circular helical deformation mode with pitch 1/k and handedness determined by the sign of H.

We consider a cholesteric phase of director field n and helical axis e z in the laboratory frame R lab ≡ e x e y e z , whose continuum Helmholtz free energy density is expressed by the Oseen-Frank functional, 33

Given the high stiffness of the origami structures and the low packing fractions marking the onset of their LCLC organisation, 20 the mean-field free energy f 0 of their reference nematic state with uniform director n ≡ e x may be written in a generalised Onsager form, 34 based on the second-virial kernel κ 28 (see Supplementary Section 1),

× δ(cos θ 1 − cos θ)δ(cos θ 2 − cos θ ), (6) with δ the Dirac distribution and f the Mayer f -function averaged over all pairs of accessible molecular conformations,

In Eq. (7), U ext (r 12 , R 1 , R 2 ) denotes the extra-molecular interaction energy of two arbitrary origami conformations with centre-of-mass separation r 12 and respective molecular-frame orientations R 1,2 , and · is the ensemble average over the single-origami conformations generated by MD simulations. 27 Local uniaxial order is described by the equilibrium orientation distribution function ψ(cos θ) ≡ ψ(e x · u), quantifying the dispersion of the origami long axes u = R · e x about e x . ψ is obtained by functional minimisation of f 0 at fixed number density ρ and inverse temperature β = 1/k b T , 23

with Z a Lagrange multiplier ensuring the normalisation of ψ. The Oseen-Frank twist elastic modulus K 2 and chiral strength k t read as (see Supplementary Section 1)

×ψ(cos θ 1 )ψ(cos θ 2 )r 2 z u 1y u 2y ,

× ψ(cos θ 1 )ψ(cos θ 2 )r z u 2y , with r z = r 12 · e z , u iy = u i · e y andψ the first derivative of ψ. The equilibrium cholesteric pitch is determined by the competition between chiral torque and curvature elasticity, and is obtained by minimisation of the elastic contribution to the free energy density f (Eq. (5)), 22

Eqs. (6), (9) and (10) are evaluated through optimised virial integration techniques 24 over 16 independent runs of 10 13 Monte-Carlo (MC) steps, using oxDNAparametrised Debye-Hückel and steric inter-nucleotide repulsion for the inter-molecular potential U ext . 19 The conformational average in Eq. (7) is performed by stochastic sampling over the simulated origami conformations in Eqs. (6), (9) and (10). 27 Eq. (8) is solved through standard numerical means. 35 Convergence was ensured by verifying the numerical dispersion of the computed pitches (Eq. (11)) to be less than 10 % across the results of the 16 MC runs. Binodal points were calculated by equating chemical potentials and osmotic pressures in the isotropic and cholesteric phase, and solving the resulting coupled coexistence equations numerically. 23 Mass concentrations were obtained assuming a molar weight of 650 Da per base pair. In the context of classical density functional theory, the Helmholtz free energy of a system of polyatomic molecules may be written in the general form 1,2

where the microscopic density ρ m generally depends on the discrete set of atom positions {r i } i≥1 and bond orientations {R j } j≥1 characterising the full microscopic state of each individual constituent particle. The centre-ofmass position r and molecular orientation R of a given particle in any conformation are uniquely determined by the specification of all internal degrees of freedom {r i } and {R j }, so that one may write, without loss of generality,

with {X } ≡ {r i } i≥2 , {R j } j≥2 . Let r i and R i be the respective expressions of r i and R i in the molecular frame R centred on r,

with R T the matrix transpose of R. The Fynewever-Yethiraj (FY) approximation postulates that ρ m may be cast in the decoupled form 3-5

where {X } ≡ {r i }, {R j } . In Eq. (5), ρ corresponds to the molecular density describing the global distribution of particle centres of mass r and orientations R throughout the sample, while P quantifies the distribution of the conformational degrees of freedom r i and R i in the local molecular frame, subject to the respective normalisation constraints 4ˆd

V dr˛dR ρ(r, R) = N.

In the FY theory, P is assumed to be entirely determined by the intra-molecular interaction potential U int = U int {X } , so as to be independent of the overall position r and orientation R of the molecule. In the absence of external fields, this approximation amounts to neglecting the effects of many-particle interactions on conformational statistics, and is therefore only rigorously justifiable in the case of highly-stiff molecules, for which the accessible conformational space is largely independent of density in the regime of low-to-moderate particle packing fractions. 5 Discarding the effects of extra-molecular interactions, the ideal component F id of the Helmholtz free energy functional reads as 1,6

with λ dB the de Broglie thermal lengthscale. Using Eqs. (5)- (7),

where the characteristic lengthscale λ now reads as

in which we used the change of variables of Eqs. (3) and (4), with unit Jacobian determinant. Note that λ generally depends on intra-molecular properties as well as temperature, but is independent of ρ. In the case of a prolate nematic phase with arbitrary director field n(r), the molecular density function ρ takes the form

where ρ ≡ N/V is the molecular number density, and the orientation distribution function (ODF) ψ describes the ordering of the long molecular axes u ≡ R · e x about the local director n(r). Note that Eq. (9) is only valid in the limit where the spatial fluctuations of n are negligible at the molecular lengthscale, as is typical in experimental cholesterics, and in the absence of long-ranged biaxial correlations, as is commonly assumed in theoretical studies. 7-10 Let us define the unit-Jacobian transformation with T (r) a rotation matrix such that n(r) = T (r) · n(0) ≡ T (r) · n 0 .

Eqs. (8) and (9) immediately yield

with u x ≡ n T 0 · R · e x , which indicates that F id is independent of the configuration of the director field n.

In the case of highly stiff and elongated molecules, the excess component F ex of the Helmholtz free energy may be related to the extra-molecular interaction potential U ext at the second virial level through the Onsager meanfield functional, 6

where the shorthand i ≡ r i , R i , {X i } refers to the full microscopic degrees of freedom of particle i, and f is the so-called Mayer function,

Using Eqs. (5) and (9), Eq. (13) may be recast as 4,5,7 (14) with f the conformational average of the f -function, 2) . (15) Note that the integrand in Eq. (14) is non-zero if and only if f (r 1 , r 2 , R 1 , R 2 ) = 0, i.e., if there exists two molecular conformations with respective centre-of-mass positions r 1,2 and overall orientations R 1,2 such that U ext = 0. It follows that in the case of short-range interaction potentials, a pair of molecules 1 and 2 may physically contribute to the integral in Eq. (14) only if their centre-of-mass separation distance r 12 ≡ r 2 − r 1 is of the order of the typical molecular dimensions. Let us introduce the particle barycentre R = (r 1 + r 2 )/2,

Under the assumptions of Eq. (9), we may thus write 8 n(r i ) n(R) ∓ ∇n(R) · r 12 2 + ∇ 2 n(R) : (r 12 ⊗ r 12 ) 8 .

with ⊗ and : the respective tensor and double dot products. In the case of a cholesteric phase of axis e z and inverse pitch q ≡ 2π/P, the helical modulation of the director field takes the form

where R z ≡ R · e z and we have chosen the laboratory frame such that e x ≡ n 0 . Let T (R) ≡ e x e y e z be a local rotating frame satisfying Eq. (11),

It is straightforward to show that ∇n(R) = q e y ⊗ e z ,

which directly lead to

where primed quantities are expressed in the rotating frame T (R), with u ij ≡ u i ·e j and r z ≡ r 12 ·e z . Plugging Eq. (17) into Eq. (14) , and using the changes of variables of Eqs. (10) and (16), we obtain

in which f 0 is the free energy density of the uniform nematic state with director n 0 ≡ e x ,

where we used Eqs. (1) and (12), and dropped the prime notation from the dummy integration variables. The integration by part of the second-order terms in Eq. (17) with respect to R yields the elastic free energy density f el in the form, to quadratic order in q, 11

which by term-to-term comparison with the Oseen-Frank free energy (Eq. (5) in the main text) leads to the microscopic expressions of the chiral strength k t and twist elastic modulus K 2 as direct generalisations of the Poniewierski-Stecki formulae, 12

which yield Eqs. (9) and (10) of the main text, with cos θ i ≡ u ix .

In the limit of long-wavelength director distortions, it may be assumed that the degree of local orientational order is unaffected by the spatial variations of n, so that the equilibrium ODF ψ eq of the cholesteric phase may be assimilated to that ψ 0 eq of the uniform nematic state. This approximation has been previously shown to be valid for cholesteric pitches as short as a few dozen particle diameters, 7 and is expected to hold without restrictions in our case. ψ eq may then be obtained through the functional minimisation of f 0 (Eq. (18)) in the self-consistent form

with Z a Lagrange multiplier such that

and κ a generalised excluded-volume kernel, 3

As in Refs. 3-5, we sample the conformational distribution P by single-molecule simulations, following the numerical protocol described in the main text (Methods). In this context, the Mayer function f (Eq. (15)) is averaged over all pairs of simulated origami conformations in the computation of Eqs. (20) , (21) and (22) , and the inverse equilibrium cholesteric pitch q eq is finally obtained by minimisation of f el at fixed T and ρ (Eq. (19) ), 11

In the following, let us denote the properties relative to right-and left-handed pair configurations by + and − subscripts, respectively. The angular two-body potential of mean force (PMF) U ± associated with two-particle arrangements of fixed handedness is given by 8

where the configurational average · (θ) ± is defined as

using the same notations as in Sec. 1. In Eq. (24), the Heaviside function Θ mirrors the fact that the handedness of an arrangement of two particles with centre-ofmass separation vector r 12 ≡ r 2 − r 1 and respective long axes u i ≡ R i ·e x is determined by the sign of r 12 ·(u 1 ×u 2 ), and V int represents the total volume spanned by the spatial and angular integrals,

where the factor 1/2 accounts for the equal division of the two-particle configurational space between left-and right-handed arrangements. Note that in the case of flexible particles, Eq. (24) may be further averaged over a representative ensemble of molecular conformations using the numerical procedure outlined in the main text (Methods). 5 In this study, we use for the volume V the smallest cubic box containing all possible interacting configurations of any two origami conformations.

In the context of Eqs. (23) and (24), a system of two particles with fixed inter-axis angle θ 12 will adopt a thermodynamically-stable right-handed configuration if their net repulsion is minimised in a right-handed arrangement -i.e., if U + (θ 12 ) < U − (θ 12 ). Conversely, U + (θ 12 ) > U − (θ 12 ) indicates a thermodynamic preference for left-handed arrangements. The relative stability of chiral two-particle assemblies is thus quantified by the chiral component of the PMF,

In the case of particles with high aspect ratios interacting through short-ranged repulsive potentials, it is easy to verify that only a small statistical fraction of the configurations sampled in Eq. (24) may display a significant interaction energy U ext > 0, so that

The Taylor expansion of Eq. (25) then reads, to leading order in 1 − e −βUext ± ,

and one recovers the definition of the chiral pair excluded volume employed in Refs. 10 and 13 for systems of hard particles, up to a constant multiplicative prefactor. It is apparent from Fig. 1 that the chiral PMFs of thermalised origamis are significantly larger in magnitude than those of their respective ground states, and are also relatively insensitive to the inclusion of electrostatic repulsion. These two observations evidence the ascendency of long-wavelength backbone deformations over local axial twist in their LCLC ordering, as the larger lengthscales associated with solenoidal writhe render the chiral assembly of thermalised filaments fairly independent of the detailed nature of their much shorter-ranged repulsive interactions. The PMFs of thermalised origamis are further found to bear a unique minimum θ m such that θ m < 0 for left-twisted filaments and θ m > 0 for their right-twisted counterparts (Fig. 1b) , thus ensuring their stabilisation of iso-chiral LCLC arrangements; a thorough discussion of the quantitative link between phase handedness and chiral PMFs may be found in Ref. 13 .

Conversely, the PMFs of ground-state filaments interacting purely through steric repulsion display a shallower minimum at large inter-axis angles (θ m +70°, Fig. 1a ), corresponding to the close-approach configuration of ground-state duplexes, as the helical threads of B-DNA form a fixed angle of roughly 35°with respect to the normal to the double-helix axis. 14 This large value is obviously incompatible with the local orientational order of LCLCs, but is nonetheless associated with a regime of weakly-negative values of ∆ c U at smaller angles θ 12 > 0 in the case of the s, 1x-lh and 2x-lh origami variantsand thus leads to their formation of stable right-handed phases. However, the chiral PMF of 1x-rh filaments bears a local secondary minimum θ l at small inter-axis angles of about −20° (Fig. 1a) , arising from their weak righthanded axial twist, which instead stabilises their lefthanded LCLC assembly. Finally, we report that electrostatic interactions greatly reduce the magnitude of the chiral PMFs for all ground-state filaments, indicating that the effects of longer-ranged repulsion mainly "smear out" the local details of their chiral molecular surfacesand therefore unwind their equilibrium pitches.

We reproduce in Fig. 2 the density dependence of the Oseen-Frank twist elastic modulus K 2 and chiral strength k t in the case of thermalised origami filaments, computed following the procedure outlined in the main text (Methods). The orders of magnitudes of the obtained values are in very good agreement with experimental measurements performed in filamentous virus soutions, 15 whose molecular dimensions, relative flexibility and absolute cholesteric pitches are comparable to those of the origamis. 16 The general tendencies apparent in Fig. 2 are also consistent with experimental results on virus assemblies, with both K 2 and k t displaying a marked increase in magnitude with increasing particle concentration. 15 The observed stiffening of twist curvature elasticities upon the inclusion of electrostatic repulsion (Fig. 2a) further mirrors the experimental variations of K 2 with decreasing salt concentration in such systems. 15 The precise experimental determination of these quantities in LCLC phases of origami filaments would be desirable for the thorough investigation of these effects, and for further quantitative comparisons with the theoretical predictions of Fig. 2. 

Let us parametrise an arbitrary backbone conformation of an origami with contour length l c by a continuous curve r(s), where s ∈ [0, l c ] is the curvilinear abscissa. The local unit tangent to the curve reads as

with u the long axis of the conformation as defined in the main text (Methods) and t ⊥ · u = 0. Due to the large bending rigidity of the filaments, we assume the transverse fluctuations of r to be small,

with φ h ∈ [0, 2π]. In the convention of Eq. (45), the handedness of the helix is quantified by the sign of q, with q > 0 (q < 0) corresponding to a right-handed (lefthanded) helicity, respectively. Using the previous notations, the Fourier components of the transverse vector r h q,⊥ associated with Eq. (45) read as

In this case, for any wavenumber k > 0, Eq. (44) reduces to

and it is easy to check that Eqs. (42), (33) and (35) yield

with δ the Kronecker delta and sgn the sign function. Therefore, the handedness of a deformation mode with arbitrary wavenumber k > 0 may be determined by the sign of H(k), with H(k) > 0 and H(k) < 0 respectively describing a right-and left-handed helicity.

Using the notations of Supplementary Section 4, the enthalpic penalty associated with the bending response of a single origami to thermal fluctuations reads as, in the case of weak curvature deformations,

where the bending modulus K is related to the origami persistence length l p through

Substituting Eqs. (28) and (29) for r ⊥ in Eq. (46) yields

(48) Assimilating the different transverse deformation modes in Eq. (48) to decoupled degrees of freedom, the equipartition theorem imposes for r ⊥v and r ⊥w | r ⊥v (k)| 2 = | r ⊥w (k)

Thus, using Eqs. (33) and (47),

valid in the limit of long-wavelength fluctuations (k → 0).

Let us consider a long origami filament whose extremities are firmly clamped to impose the parallel alignment of its backbone end tangents, dr ds s=0 = dr ds s=lc .

The origami backbone curve r is defined as

where the continuous centreline r i (s i ) of the i-th constituent DNA duplex is obtained by contour interpolation of the centre-of-mass positions of its bonded nucleotides. For simplicity, we neglect the effects of duplex splaying at the origami ends, and thus assume Eq. (50) to hold at each of the centre curve extremities, dr i ds i si=0 = dr i ds i si=li .

We further restrict our study to the regime of weak bending deformations of the duplex centrelines about the straight backbone conformation of the origami ground state, and neglect potential fluctuations in their respective contour lengths l i . Under these assumptions, the formulation of the Cȃlugȃreanu-Fuller-White theorem 19 extended to the treatment of open curves 20 states that the linking number Lk i of each individual duplex may be decomposed into twist and writhe contributions,

In this context, Lk i represents the (signed) number of net right-handed turns per unit contour length by which each strand of the duplex winds around the other. These turns may result in both a local twist of the strands about their common centreline r i , as quantified by the twist density Tw i , and/or in a global supercoiling of the centreline itself, as measured by the writhe integral Wr i . It should be noted that the linking number Lk i is generally not a topological invariant in the case of non-circular DNA fragments. Within the origami filament architecture, Lk i is initially constrained by the designed locations of the inter-helical crossovers, but may partially relax towards its preferred unhindered value Lk 0 -thus inducing global axial twist in the origami ground state. 21 Within ground-state B-DNA, the relaxed linking number Lk 0 is entirely absorbed in the form of twist strain, Lk 0 = Tw 0 1 10.5 bp −1 .

The axial twist handedness of an origami filament comprised of duplexes with linking number Lk i is therefore determined by the sign of ∆Lk i ≡ Lk i − Lk 0 , with ∆Lk i > 0 (∆Lk i < 0) respectively denoting a residual over-winding (under-winding) of the duplexes, associated with a global left-handed (right-handed) compensatory twist of the origami. The total elastic energy of a constituent duplex, as defined by an arbitrary centreline curve r i and uniform twist density Tw i , may be obtained as a straightforward generalisation of Eq. (46), 22

with K i and C i the respective effective bending and twisting moduli of B-DNA within the origami structure. Eqs. (51) and (52) immediately yield

(53) It is apparent that the twist elastic contribution in Eq. (53) is minimised by conformations in which ∆Lk i and Wr i bear equal sign and magnitude, leading to a favoured positive (right-handed) supercoiling in the case of left-twisted origamis (∆Lk i > 0, Wr i > 0), and negative (left-handed) supercoiling for their right-handed counterparts (∆Lk i < 0, Wr i < 0). However, this twist relaxation mechanism is hindered by the high penalty in bending energy arising from the curvature of the resulting solenoidal centreline deformations. The competition of these two effects, acting constructively on each duplex within the origami structures, leads to the weak anti-chiral backbone fluctuations underpinning their LCLC assembly.

The Physics of Liquid Crystals

A Treatise on Conic Sections

Proc. Natl. Acad. Sci. USA

where we used the notation of Fig. 3a in the main text, with · the Euclidean norm. Thus,and integrating Eq. (26) yields, to leading order in t ⊥ , r(s) = r 0 + su + r ⊥ (s),where r 0 = r(0) − r ⊥ (0). Consistently with the previous approximations, we further assimilate the filament long axis u with the normalised end-to-end separation vector,so that Eq. (27) imposes simple periodic boundary conditions for r ⊥ ,r ⊥ may then be expressed in the form of an inverse Fourier transform,with discrete wavenumbers k = n/l c for any non-zero integer n and coefficientsLet |·| be the complex modulus, andUsing Eqs. (27)- (29) , the backbone conformation r k associated with a transverse deformation mode of arbitrary wavenumber k is given by the parametric equationIn the most general case, Eq. (32) describes an elliptical helix of axis u and pitch p = 1/k. The shape chirality associated with a deformation mode r k is thus quantified by the anisotropy of its elliptical cross-section, which we now proceed to analyse. In the following, we omit some of the explicit k dependences in order to alleviate the notations when no confusion can arise. Let us denote bythe Euclidean modulus of r ⊥ , and defineUsing Eqs. (32)- (35) , the transverse components of r k may be rewritten aswith ω ≡ 2πk. Eq. (37) then yieldsThus, using Eq. (36),and Eq. (36) immediately yields the further relationPlugging Eq. (40) into Eq. (39) leads to a quadratic equation for r kv and r kw , The respective lengths r ± of the semi-major and semiminor elliptical axes are then related to the respective largest and smallest eigenvalues λ ± of Q through 17 r ± = 1/ λ ∓ , which yields, after rearrangements,Interestingly, Eq. (42) bears a strong resemblance to the Jones vector parametrisation of the polarisation ellipse in classical electrodynamics, 18 which stems from the similarity between Eq. (32) and the field equation of a polarised electromagnetic wave propagating along the direction u.

An explicit expression for H in terms of the Fourier components r ⊥ (k) may be obtained by substituting Eqs. (33) and (34) for θ,and substituting Eqs. (38), (30) and (31) for φ,Eq. (43) may thus be rewritten in the formand one recovers the definition of Eq. 4 in the main text, with c vw (k) the Fourier components of the crosscorrelation function of r ⊥v and r ⊥w as given by the convolution theorem,Using Eqs. (42) and (43), the transverse eccentricity of the elliptical cross-section reads asA necessary and sufficient condition for the deformation mode r k to describe an ideal circular helix is given byConversely, e(k) = 1 ⇐⇒ H(k) = 0 describes the degenerate case in which the elliptical crosssection collapses to a flat line segment, leading to an achiral deformation mode. The magnitude of H(k) may thus be understood as a measure of the degree of circular helicity of the deformation mode r k . The link between the sign of H(k) and the corresponding helical handedness may be elucidated by considering the case of an ideal circular helical conformation of axis u, radius r h > 0 and inverse pitch q = 1/p h . The general parametric equation of such a conformation reads as, in the limit of weak helical curvature (qr h 1), r h q (s) = su + r h cos(2πqs + φ h )v + r h sin(2πqs + φ h )w,