Chiral shape fluctuations and the origin of chirality in cholesteric phases of DNA origamis
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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.Wang13 (); Yash16 () 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.Wang18 (); Baec18 () 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.
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.Wang13 (); Yash16 (); Hent17 () 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.Iams14 (); Feng17 (); Lee18 () 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.Broo99 (); Wang13 (); Yash16 ()
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 systems, ranging from filamentous virus suspensionsDogi00 (); Grel03 () to biologically-relevant collagen assemblies.DeSa11 () 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,Wens11 (); Frez14 (); Duss16 () 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 materialsLage14 (); Wang18 () and for our fundamental understanding of the ubiquity of LCLC order in living matter.Mito17 ()
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-particle structure at the nanometer level.Siav17 () 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.Diet09 () Following Ref. Siav17, , we here focus on four variants of the filaments comprising 15224 to 15240 nucleotides, with experimentally-determined contour lengths of and bundle diameters of . A continuum finite-element model based on an elastic rod description of DNAKim17 () predicts the respective ground states of the different designs to bear negligible (s), right-handed (1x-rh), left-handed (1x-lh) and left-handed (2x-lh) twist about the filament long axis, with negligible net curvature (Fig. 1).Siav17 ()
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,Stra76 () which has been extensively discussed elsewhereTort17-1 (); Tort17-2 () (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. Tomb05 () We thus employ the oxDNA nucleotide-level coarse-grained modelSnod15 () 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 opposite-handed arrangements (Fig. 2b).Stra76 () Conversely, this entropic preference is reversed in the case of strongly-twisted 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.Tomb05 () 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.Tort17-2 ()
However, these predictions are at odds with the experimental measurements of Ref. Siav17, , which instead revealed a general tendency of origami filaments to stabilise iso-chiral 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,Tomb05 () although the validity of this argument has been disputed by subsequent numerical investigations.Cort17 () 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.Cort17 (); Tort18 ()
To assess the role of conformational statistics on their cholesteric ordering, we make further use of the oxDNA modelSnod15 () to probe the detailed thermal fluctuations of the origami filaments. As in Ref. Tort18, , 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, Fyne98 () 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 limitedTort18 () (see Supplementary Section 1). This description is therefore well-suited for our purposes, given the large persistent length () of the origami structures (Kaue11 (); Siav17 ()) and the low packing fractions of their stable LCLC phases.Siav17 ()
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. Siav17, , 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 rigiditySiav17 () (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 right-handed (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 opposite-handed arrangements.Tort17-2 () 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 ()Siav17 () (Figs. 3c-d, Methods).
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.Diet09 () 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 (). 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 helices — while 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 , as the equilibrium helical pitch of B-DNA within constrained origami structures may slightly differ from the unconfined value assumed in both the computation of the origami ground statesKim17 () and the parametrisation of the oxDNA model.Snod15 () 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.Snod15 () 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. Tort17-1 () 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 1, 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 quantitative 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.Tomb05 (); Tomb06 (); deMi16 () The link between ground-state and fluctuation-induced chirality is further shown to be non-trivial, 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.Hall16 () Our findings could thus provide a theoretical basis for the so-called “corkscrew model”, previously postulated to explain the puzzling experimental behaviour of filamentous virus suspensions,Grel03 () and more broadly suggest a novel self-assembly paradigm for LCLCs in which subtle, long-wavelength conformational features — rather than local chemical structure — dictate macroscopic chiral organisation.
Acknowledgements.JPKD and MMCT gratefully acknowledge É. Grelet and Z. Dogic for helpful discussions. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 641839. The authors would like to acknowledge the use of the University of Oxford Advanced Research Computing (ARC) facility in carrying out this work (http://dx.doi.org/10.5281/zenodo.22558). MMCT made use of time on the ARCHER UK National Supercomputing Service (http://www.archer.ac.uk) granted via the UK High-End Computing Consortium for Biomolecular Simulation, HECBioSim (http://www.hecbiosim.ac.uk). MMCT is grateful to the UK Materials and Molecular Modelling Hub for computational resources, which is partially funded by EPSRC (EP/P020194/1). GM is grateful for financial support from the Department of Science and Technology India in the form of an INSPIRE faculty grant (DST/INSPIRE/04/2014/002085), and acknowledges the computational facilities provided by Calcul Québec (www.calculquebec.ca) and Compute Canada (www.computecanada.ca). DP is supported by the EPRSC Centre for Doctoral Training in Theory and Modelling in the Chemical Sciences (EP/L015722/1).
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.
MD simulations setup
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 covalent-bonding potentials.Snod15 () Calculations were performed on GPUs in the canonical ensemble using an Andersen-like thermostat and sequence-averaged DNA thermodynamics, assuming room-temperature conditions () and fixed monovalent salt concentration . This value was chosen in slight excess of the experimental salt concentration , 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.Snod15 () Relaxation was achieved through equilibration runs of MD steps starting from the origami ground state, and production runs of steps were conducted to generate uncorrelated conformations for each origami variant. The statistical independence of the resulting conformations was assessed by ensuring the vanishing autocorrelation of their end-to-end 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.Diet09 () We define the molecular frame of each conformation as the principal frame of its backbone gyration tensor, such that and correspond to the respective direction of maximum and minimum dispersion of the origami backbone.Tort18 () Shape fluctuations are described by the contour variations of the transverse position vector,
with the position of the discretised backbone segment with curvilinear abscissa , assuming the backbone centre of mass to be set to the origin of the frame. Denoting by the curvilinear length of each segment, the Fourier components of 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 ,
where for and is the complex conjugate of . It is shown in Supplementary Section 4 that the degree of helicity of a deformation mode with arbitrary wavenumber about the long molecular axis is related to through
with the imaginary part of . One may check that , with if and only if the two transverse Fourier components bear equal amplitudes and lie in perfect phase quadrature. In this case, describes an ideal circular helical deformation mode with pitch and handedness determined by the sign of .
Molecular theory of cholesteric order
We consider a cholesteric phase of director field and helical axis in the laboratory frame , whose continuum Helmholtz free energy density is expressed by the Oseen-Frank functional,deGe93 ()
Given the high stiffness of the origami structures and the low packing fractions marking the onset of their LCLC organisation,Siav17 () the mean-field free energy of their reference nematic state with uniform director may be written in a generalised Onsager form,Onsa49 () based on the second-virial kernel Fyne98 () (see Supplementary Section 1),
with the Dirac distribution and the Mayer -function averaged over all pairs of accessible molecular conformations,
In Eq. (7), denotes the extra-molecular interaction energy of two arbitrary origami conformations with centre-of-mass separation and respective molecular-frame orientations , and is the ensemble average over the single-origami conformations generated by MD simulations.Tort18 () Local uniaxial order is described by the equilibrium orientation distribution function , quantifying the dispersion of the origami long axes about . is obtained by functional minimisation of at fixed number density and inverse temperature ,Tort17-1 ()
with a Lagrange multiplier ensuring the normalisation of . The Oseen-Frank twist elastic modulus and chiral strength read as (see Supplementary Section 1)
with , and the first derivative of . 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 (Eq. (5)),Stra76 ()
Eqs. (6), (9) and (10) are evaluated through optimised virial integration techniquesTort17-2 () over 16 independent runs of Monte-Carlo (MC) steps, using oxDNA-parametrised Debye-Hückel and steric inter-nucleotide repulsion for the inter-molecular potential .Snod15 () The conformational average in Eq. (7) is performed by stochastic sampling over the simulated origami conformations in Eqs. (6), (9) and (10).Tort18 () Eq. (8) is solved through standard numerical means.Herz84 () Convergence was ensured by verifying the numerical dispersion of the computed pitches (Eq. (11)) to be less than 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.Tort17-1 () Mass concentrations were obtained assuming a molar weight of per base pair.
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