Orientation Kinetics of Chiral-Nematic N* Domains in Suspensions of Charged DNA Rods

Many biological systems exhibit natural regulation mechanisms as necessary functions. Proteins, peptides, charged receptors, RNAs, and DNAs clearly demonstrate such varieties of structural dynamics in their biocomplexity. To understand the proper structural functionality of biomolecules, this requires integrated information from initiation, regulation, transfer and transport to desir-able sustainable processes. Here, we demonstrate the orientation kinetics of stable chiral-nematic (N*) domains that are consistently formed in the suspensions of charged DNA rods at low ionic strengths. This is enhanced by large access of released dissociated condensed ions and the mobile diﬀusive ions of DNA rods, acting onto the perpendicular motions of the DNA rods in bulk. The quantiﬁcation of collective orientations for these interacting charged DNA rods is extracted by image-time correlation (ITC) performed in Fourier transforms (FTs) for overall spectral density distributions. Three distinguishable length scales are clearly shown corresponding to the relevant motion of the N* domain in parallel, perpendicular and the optical pitch within the domains as well as the kinetics of local distributions in FTs. Particularly strong concentration transitions are conﬁrmed by replica symmetry breaking of elastic deformations in N* domains in terms of the average twist angle and the order parameter. This work can be interesting for suﬃcient cooling of the given concentration of charged DNA rods, at low ionic strengths (below the critical value), mimicking super-cooled liquid and orientation glass in other biomacromolecules.


Introduction
The structures of crystals as disordered and amorphous morphologies can be characterized by diffraction and scattering methods, as introduced by A. Guinier, including the fibrous structures as para-crystals consisting of domains (with aligned fibers) or irregular anisotropic particles [1]. Protein mixtures are natural systems mimicking colloidal behaviors [2] and the structural fluctuation of orientational glass is also found for a small protein (levoglucose) away from the structural glass that has initial caging, with non-exponential relaxations in broader length scales [3]. Typically, spectral density starts to broaden when observable structures are formed via fluctuations of domains. It is therefore interesting to see how the elasticity of inner structures are arranged within the domains. Anomalies are essential in elasticity so as to be enhanced near the orientation phase transition in such a way that the kinetics of orientational motions slow down in terms of order parameter and microscopic stresses [4]. In the case of charged rod glasses, the invariant of rotor dynamics is often referred to as the Wigner distribution function (for a finite system), where the time evolution can be deformed conveniently in the polar presentation of orientations or in the Fourier space [5]. Dynamical arrests can exist in glasses, where a slow time variable evolves in non-ergodic relaxations at different length scales either as the frozen or the intermediate states between fast and slow processes [6]. Thus, tracing the system with proper time scales is essential for the hierarchical classes, preserving unknown degrees of freedoms. Such commonly known examples are the orientation orders for the block copolymers: the grain size of the microdomains of cylindrical block copolymers are analyzed by Fourier filtering, depicting the local order parameter for the relevant correlation length in the nematic state [7]. Furthermore, the orientational orders of the microdomains of 2D striped patterns consisting of PS-PI diblock polymers are investigated using the kinetic scaling law for the smectic layers of topological constraints that originated from the annihilation of disclinations, annealing temperatures and time. The exponent power of the annealing time is found to be lower than 0.25, as was observed in the 2D nematic system (0.5), due to the dynamic growth of domains [8]. Furthermore, the self-assembly of nanocolloidal rods is considered to be useful for employing the orientational dynamics and performing efficient diffraction of the planar devices for the spacing of layers with controlled orientations [9]. The kinetics of longranged ordering are also shown for thin metallic films (of anodic aluminum), forming porous structures during anodization, which leads to domains of different growth rates (for a time exponent of 0.2) in plane orientations [10]. Irrespective of the above findings, the "cluster reversion" is interestingly found to be the metastable ordering in the orientational glass of cyanoadamantane with the characteristic feature of relevant kinetics (see temperature-time diagram in Fig. 14 in Ref. [11]). The observation is interpreted as the occurrence of orientational glass, accompanied by the freezing of random strain field of a cluster coupled to the orientational order parameter. The reorientation kinetics of the nematic director are then measured by a molecular level of deuterium nuclear magnetic resonance (NMR) in aramid solution, showing a small noise of the intrinsic twist viscosity for frictional forces among the neighboring domains [13].
The mechanism of a glass-forming system is described theoretically by a general network theory for various systems of metallic alloys, silicates, and proteins [16]. In particular, the protein HIV protease is simulated by a "pebble game" or a bead-spring model [17,18]. The model simplifies the rigid region of a mechanical network away from the soft one as a dynamic mode of proteins approached by the classical Lagrangian methods.The question arises regarding how many particles are engaged for active sites as well as the shape consisting in the concentration or the volume fractions. To date, there have been a lack of experiments on biological systems demonstrating orientational glasses in bulk. In addition, most orientation order glasses are studied in the thermotropic system or multi-component alloys near the transition point. Here, the suspensions of charged DNA rods in equilibrium phase behaviors at low ionic strengths are introduced as a good lyotropic system, where the orientation kinetics are analyzed and characterized by means of an advanced method of image-time correlations (ITCs) in the Fourier transforms (FTs). The equilibrium phase behaviors are explored systematically by varying the concentration of DNA rods at a given low ionic strength: the chiral-nematic N* phase and the hierarchical chiral mesophases, Xpattern and helical domain (HD) phase are observed above the isotropic-nematic coexistence concentrations [14]. In particular, the intermediate concentration of DNA rods, the Xpattern is formed as a new type of glass (forming the cavity loops, a chiral glass) where replica symmetry breaking (RSB) occurs, in contrast to the structural glass [19]. The results of experimental findings clearly demonstrate differences in the averaged domains and orientation angles in both time and concentrations.
The key questions addressed in this paper are as follows: (I) To what extent the twist angles of collective orientations in chiral-nematic N* domains occur by varying the concentration and waiting time? (II) What are the orientation kinetics and order parameters approaching the equilibrium time? (III) Are there any differences in the results of ITCs in real space and FTs in terms of determining the critical concentration (of RSB) and hierarchical chiral mesophases? (IV) What are the characteristics of N* domains, in the lyotropic orientational system? (V) What are the "effective" reduced parameters in the orientation kinetics of N* domains? (VI) Are there any possible relations to the thermotropic system resembling the thermodynamic system? If so, then how and why is the case?

Results
The quantification of collective orientations for charged DNA rods provides the timedependent averaged orientations of chiral nematic N* domains, effectively performed by image-time correlations (ITCs) in FTs. The characteristic features are then obtained from the results of data analyses as more direct evidence of replica symmetry breaking of N* domains for the ensemble averaging of orientations. The particular aim here is to perform ITCs in FTs by varying the concentrations and waiting times developed by the in-house automated program that is rigorously used for the conversion of real images in morphology to FTs, followed by the calculations of ITCs for spectral intensity distributions in FTs.
Further details of data analyses are described in the later section on methods.

Concentration-dependent orientations of twisted chiral nematic N*-domains
The polarized optical morphologies are collected from the birefringence of alignments to investigate orientations of charged DNA rods under the crossed polarizers, resulting in the degree of optical anisotropy. Then, the order parameter can typically be estimated or calculated for proper average orientations with statistical tools for long waiting times. It is therefore a rather demanding task to acquire the data to capture the whole slow dy- the visible differences of intensity distributions in FT lobes appearing between the q D, and q D, ⊥ for all the concentrations above the N-N* transition ( Fig. 2 and Fig. 3). In particular, the most drastic changes in FTs are seen in the middle concentrations of the X-pattern occurring as a critical concentration at low ionic strengths [19]. Also, the local orientational intensity distribution profiles show that both axes contribute to unique behaviors between these two orthogonal directions of N* domains by varying the concentration of DNA rods (see supplementary data movies, Movies F-I in Fig. 2). Temporal changes of the orientation distributions of the N* domains are shown in the FTs in Fig. 3, where the N* domains with optical pitch vary very slowly over time (135-240 hours). By further increasing the concentration of DNA rods in the X-pattern approaching the X-HD transition (see comparison on 10.5 mg/ml (Movie K) and 14 mg/ml (Movie L)), half-sized domains appear, resulting in an FT spacing that is twice as large (Fig. 4). Further comparison of the critical concentrations (near the X-pattern) are later shown at even longer waiting times, t W , between the N*-X transition (at 4.7 mg/ml) and the X-pattern (5.4 mg/ml) (see Fig. 6).

Kinetics of orientation distributions of N*-domains, image-time correlations in FTs
Image-time correlation (ITC) spectroscopy is used to extract more characteristic features in the FT images. Compared to the ITCs in real space [19], the ITCs in FTs turn out to be a rather direct way of determining the average orientational motions of N* domains from the changes of spectral intensity distributions. The main features of equilibrium phases at a low ionic strength below the critical value (at 1.2 mM Tris/HCl buffer) are as follows: N* phase is stabilized by oriented N* domains that appear orthogonally together with the N* optical pitch (stripes in the N* domains) inside. The corresponding FT of the N* phase (2.6 mg/ml) is shown in Fig. 3(a) for the averages of distributions in the orientations of overall N* domains in the center FTs, compared to a more pronounced stable N* phase (3.8 mg/ml) in Fig. 3(b). However, above the N* phase (5.4 mg/ml), a unique phase of X-pattern occurs such that the N*-X pattern transition has noticeably different intensity distributions; the reflection symmetry is broken in the axes of the N* domain perpendicular. q D, ⊥ is also independently shown as the diverging intensity profiles in FTs (see Fig. 3(c)), while the parallel component q D, appears to localized spherical distributions.
The average size of the spectral domains, < q D >, also increases with an increase of the concentration of DNA rods due to the smaller sizes of the domains that are seen in real space. Furthermore, in Fig. 3(d), for the X-pattern at a higher concentration, twice as large spacing is observed in FTs compared to Fig. 3(a) as well as more visible differences between q D, and q D, ⊥ . The system is even more clearly presented in the orientations, such that  Intriguingly, the overall sum of the intensity distributions interacts well between the N* domains and the chiral pitch within the domains. This therefore supports the notion that the orientations of N* domains are affected by the elastic deformations, leading to rather sharp transitions near the N*-X pattern and X-HD concentration, found as the RSB for symmetry breaking [19]. A more detailed description of the elastic properties of N* domains near the X pattern is given in the following subsection.
To quantify the characteristic features of morphological changes in the time-lapsed images, the two-dimensional image matrix is considered to be converted as the numbers of all 2D array intensity values for each pixel [25]. The image-time correlation function is defined as follows: I(t) is the instantaneous transmitted intensity detected by a given pixel of the CCD camera. For the time traces recorded for all these pixels, the image-time correlation function CV (t) is defined as: where V indicates the "video", or time-lapsed images, and the brackets < · · · > denote the averaging of the whole field of views in the CCD camera pixels at 2D (i, j) matrix indices.
Each individual image in a time trace is used to construct an image correlation function that is variable for the region of interest in the square (e.g, 512 × 512) pixels and int time, depending on the application. The image-time correlation is also applied to various other systems to extract different features of dynamical images [25]- [28]. Here, the application of ITCs in FTs is focused on obtaining the information of collective orientation degrees of freedom for charged DNA rods observed in slow times and the effective order parameter.
Elastic kink of N* domains near the X-pattern (a chiral glass) The mechanical behavior of the (network) glasses can be physically demonstrated by the existing rigidity of the molecules in variations of the soft phonon mode and the discrete glassy percolation [22], which often occur through the optical contrast by he mechanical anisotropy of crystallinity and molecular orientations. In practice, the optical birefringence and elastic moduli are then realized by the anisotropic shape of rod-like molecules due to their rigidity in the core, embedded in an elastic medium [23,24]. In addition, a computational algorithm of generic rigidity in 2D percolation (called a "pebble game") is used to predict the order parameter for first-and second-order transitions with few critical exponent powers.
Calculations of central force rigidity percolations are then mapped to the heat capacity, distinguishing the free energy distributions of bond and site percolation as the result of the "floppy" model in the glassy system. However, they are still restricted to 2D and not yet available in 3D [23,24]. Therefore, it is worthwhile to validate the RSBs in the current lyotropic system, which is uniquely observed in the X-pattern, as a chiral glass resembling 3D orientation glass [19].
The results of systemic quantification for overall changes are shown in Fig. 5, with the average coherence for orientations analyzed by an image-time correlation (ITC) function,  (t < t eq ), large variations of correlations can still be seen in FTs, as can be seen in the data analysis (see also Fig. 7(a)).
More direct evidence of physical observations for the replica symmetry breaking (RSB) appears in the middle concentrations (of the X-pattern), occasionally in the fast time scale with mechanical kinks [14]. Without any loss of information, maintaining reliable continu- balance breaking in the X-pattern through RSB. However, when the density increases at a higher concentration of the X-HD transition, the replicas of smaller helical domains appear again, similar to the optical morphology of N* domains in the equilibrium phase [19].
The RSB may then be relevant with a sudden reverse of cluster (or domains) in the development of orientation orders, forming a microscopic lattice at the (thermotropic) glasslike transitions (Fig. 13 of Ref. [11]). Then, the intensity fluctuations of orientations are a precursor to such transitions that are steady until the actual development of the cluster occurs in a limited space (e.g. a sudden jump or mechanical kink) and is followed by a weak time dependence for a long period of time (see effective temperature-time diagram in Fig. 14 of Ref. [11]). An unusual isotope effect is also observed in a high-temperature superconductor [12], which compromises an open question as to whether or not the X-pattern is a coexistent of the partially molten state of N* domains against more disordered (in isotropic) or ordered out of the plane (homeotropic nematic) states in bulk.  [19]. When the duration time is similar or larger than the equilibrium time, as in Fig. 7(a), rather more significant changes are seen in the critical concentration regimes (see example of t ∼ t eq or t > t eq ). In contrast, for the shorter duration time, t < t eq , in ( Fig. 8(b)). Due to the finite Fourier component analysis, the background value here is seen as required for the stationary value corresponding to the N* domains. In addition, the average twist angle, < θ tw > is converted from the decay constant of the ITC in FTs as the multiplication of a complete 2 π turn (see Fig. 8(a)) so that the sum of possible spherical intensities can be considered. The values of the concentration-dependent averaged order parameter, < S >, is included in Fig. 8(c) and 8(d), estimated from the effective Debye screening length and dissociation constants for the release of the condensed ions at a given ionic strength to clearly highlight the middle-concentration gap presented as the RSB (see pink range in Fig. 8).

Summary
The equilibrium phase diagram of charged DNA rods is provided in Fig. 9  in the equilibrium diagram in Fig. 9 (see X-pattern below and LTKA3 on the right) with dominant peaks in the center zone in FTs. Further details on the equilibrium phase diagram of charged DNA rods at low ionic strengths can be found in Ref. [14,19].
In summary, the current system demonstrated well the orientations of charged DNA rods, resembling the slow dynamics of orientation glasses that are often found in metallic alloys (see Fig. 5 in Ref. [16]) or the rigid rod-shape molecules. Furthermore, the intermediate concentration of the X-pattern between the N*-X and X-HD phase boundaries at a low ionic strength is found to be a unique RSB (as a chiral glass exhibiting cavity loops) [19], which has now also been confirmed by a For lower ionic strengths, the same procedures are followed to measure the concentration of DNA suspensions by repeating the sampling process 5 times. For the optical measurements, a commercially available Quartz transparent cylinder cuvette with a thickness of 1 mm and a diameter of 20 mm (120 QS 1mm, Hellma Precision in Spectro-Optics) is used to contain an approximately 380µl sample volume. The sample holder is placed between two crossed polarizer sheets to capture polarized images of the birefringent orientation texture.
In addition, the large field of view is captured by a long-distance telescopic (InfiniProbe, Infinity, Boulder, CO, USA) lens, placed in front of the CCD camera (AxioCam Color A12-312). The entire measurement is performed by an automatic save setting for the slow-motion sequences of images for a time interval of every 30 minutes recorded for 10-30 (90) days by the image software (Axivert, Carl Zeiss).

Orientation distributions and image-time correlation spectroscopy in FTs
The conversion of FFT transforms of the real images is performed by writing a short script to read the movie data (avi) files as stacks. The FFT is then applied to each image and saved as a new image stack using the image software available online (via online ImageJ/Fiji version  [25]. Here, in this work, the images of FTs are additionally obtained from the corresponding images, captured in the real space for the long-time traces (120-240 hours) to extract the averaged overall orientational fluctuations. The in-house-developed ITC program is very robust and effective. For instance, a single image-time correlation only takes a few seconds for the whole sequence of time-lapsed images (or the video data, typically for 512x512 pixels at a total of 500 time frames) via the user-friendly window setting (developed by Dr. H. Kriegs, IBI-4, FZJ). The quantification of orientation changes in the time-lapsed FT images here is performed by 2D array intensity values prepared from the reconstruction of image data to the (i, j) index intensity values for black/white 2D ASCII formatted data to calculate the image-time correlation function in the pixel-pixel intensity auto-correlation, which was first introduced in Ref. [25]. The reconstruction of image data is taken from the collected time-lapsed images in such a way that each image is subtracted from the overall averaged intensity value at a given time. Then, the image-time correlation function in the pixel-pixel intensity auto-correlation, the black/white 2D ASCII formats, is kept for reading the instantaneous transmitted intensity taken by a given pixel of the ROI in the 22 CCD camera. The image-time correlation function C V (t) is calculated for the time traces recorded for all these pixels, with the averaging of CCD camera pixels used to subtract from the individual pixel-pixel intensities. Each single image in a time trace is used to construct an image correlation function that has a variable region of interest as the square pixels (e.g. 300 × 300 or 512 × 512). The initial time can then be a good estimate for depicting the average of the dynamical images in time sequences, followed by a normalization with the total sum of the pixel intensity correlation at an initial time frame. Here, the results of ITC in FTs, C θ (q D , θ tw , τ ), are fitted by a single decay function in time to characterize the 3 representative parameters, which are interpreted as A ∼ S, B ∼< q D >, and Γ ∼< θ tw >, for the order parameter, average size of N* domains, and twist angle, respectively.