Nanoscale correlated disorder in out-of-equilibrium myelin ultrastructure

Ultrastructural fluctuations at nanoscale are fundamental to assess properties and functionalities of advanced out-of-equilibrium materials. We have taken myelin as a model of supramolecular assembly in out-of-equilibrium living matter. Myelin sheath is a simple stable multi-lamellar structure of high relevance and impact in biomedicine. Although it is known that myelin has a quasi-crystalline ultrastructure there is no information on its fluctuations at nanoscale in different states due to limitations of the available standard techniques. To overcome these limitations, we have used Scanning micro X-ray Diffraction, which is a non-invasive probe of both reciprocal and real space to visualize statistical fluctuations of myelin order of the sciatic nerve of Xenopus Laevis. The results show that the ultrastructure period of the myelin is stabilized by large anti-correlated fluctuations at nanoscale, between hydrophobic and hydrophilic layers. The ratio between the total thickness of hydrophilic and hydrophobic layers defines the conformational parameter, which describes the different states of myelin. Our key result is that myelin in its out-of-equilibrium functional state fluctuates point-to-point between different conformations showing a correlated disorder described by a Levy distribution. As the system approaches the thermodynamic equilibrium in an aged state the disorder loses its correlation degree and the structural fluctuation distribution changes to Gaussian. In a denatured state at low pH, it changes to a completely disordered stage. Our results clarify also the degradation mechanism in biological systems by associating these states with variation of the ultrastructural dynamic fluctuations at nanoscale.


Introduction
A hot topic for material scientists today is the design of chemical systems in quasi stationary nonequilibrium states. 1 In fact it is known that chemical systems can acquire novel functionalities such as for example chiral symmetry breaking, bistability and ultimately life, 2 when kept away from equilibrium. In this context, supramolecular self-assembly is of high interest. It involves multiple weak intermolecular interactions between structural units leading to multiple conformational states with similar energy separated by small potential energy barriers. The interchange among the different structural conformations moves the systems along the complex potential energy landscape, which determines their functionality. Thus, the fluctuations between the nearly isoenergetic conformations constitute a key point to be investigated for a deeper understanding of out-ofequilibrium matter, that occurs in biomaterials, 3 plants, 4 nanotechnology 5 and proteins. 6 In particular, the spatial correlation degree of these fluctuations defines and characterizes the system thermodynamic state related to its function.
It has been proposed in recent theories that the out-of-equilibrium state of living biological matter is close to a critical point. [7][8][9][10] The most significant indicator for proximity to a critical point is critical opalescence where the size of the regions of different phases begin to fluctuate, in a correlated way, over increasingly large length scales. 11 The spatial correlations are described by universal power laws and scaling functions in soft, biological as well as in hard matter. [12][13][14][15] Recently Kaufman group has proposed quantum criticality at origin of life 10 predicting biological matter fluctuations as in a system in proximity to a quantum critical point for metal-insulator transition (MIT). Power law fluctuations in MIT quantum criticality have been well characterized in high temperature superconductors. [16][17][18][19] Myelin 20,21 can be considered the simplest example of a biological ultrastructure since it shows a periodic multilayer structure rich in phospholipid membranes with few active proteins [22][23][24][25][26][27][28] where the competition between fundamental biological interactions 29 determines its quasi steady state out-ofequilibrium. The compact myelin sheath is an elaborated multi-layered membrane wrapping selected axons. Its main role is the control of the propagation speed of action potentials in saltatory conduction, facilitating nerve signal transmission. 20,21 The myelin ultrastructure is usually described as a highly-ordered liquid crystal 22 made of a structural unit constituted by the stacking of the four layers: i) lipidic membrane (lipid polar group, lpg) ii) cytoplasmatic apposition, cyt, iii) a second lipidic membrane, lpg, iv) extracellular apposition, ext. 27,30 The structural unit appears to be stable, as probed by using a wide variety of standard experimental approaches 25,[30][31][32][33][34][35][36][37][38][39][40] giving information only on the average structure, and thus inadequate for the visualization of out-of-equilibrium structural fluctuations, which require highly spatially resolved probes. Besides, on account of the advanced features of the latest generation synchrotron sources and fast acquisition methods, nowadays powerful non-invasive high spatial resolution techniques are available for this purpose. 41,42 In our experimental approach, we have used Scanning micro X-ray Diffraction (SµXRD) to probe the k-space (reciprocal space) order with high resolution in the real space. In this way we have mapped spatial distribution of myelin ultrastructural fluctuations e.g. the fluctuations of myelin sublayers thickness. In a second step, we have applied statistical physics tools to the collected diffraction data to unveil the "statistical distributions of the fluctuating structural order". This approach has recently been used to carry out heterogeneity investigation of complex matter in several systems of interest in different fields from biomedicine to material science. 26,27,[41][42][43] We have applied this approach to investigate the myelin in the sciatic nerves of Xenopus Leavis frogs, which is a Peripheral Nervous System (PNS) representative. We have measured the spatial fluctuations of the four nanoscale layers thickness at thousands of discrete locations in functional, out-of-equilibrium freshly extracted nerves associated with the in vivo state, 33 named fresh samples.
Afterwards, in order to investigate the system degradation towards the thermodynamic equilibrium we quantified the structural statistical fluctuations of myelin in aged nerves after extraction, named unfresh and denatured nerves in acidic pH solution, named denatured. We have used the ratio between the thickness of the hydrophilic and hydrophobic layers called hydrophilic-hydrophobic conformational parameter (HhCF) to characterize the myelin local state. We discovered that the quasi-crystalline periodicity of the myelin lattice in the functional state is maintained thanks to large anti-correlated intrinsic fluctuations between the hydrophobic layers and hydrophilic layers. In fresh out-of-equilibrium samples, the probability distribution of HhCF shows a fat tail, here described by a Levy distribution, [44][45][46][47][48][49] as expected for biological matter near a quantum critical point. [10][11][12][13][14][15][16][17][18][19]50,51 Leaving the functional out-of-equilibrium state, towards the thermodynamic equilibrium the myelin degrades. In the aged state, we have observed the freezing ultrastructure fluctuations leading to a narrow Gaussian distribution of HhCF, while in denatured state, 35 at low pH, we observed a disordered state indicated by the larger uncorrelated fluctuations. These results support the hypothesis that the functional fresh nerve is a system in a non-equilibrium steady state tuned close to a critical point. [7][8][9][10]

Results and discussion
The ultrastructure of myelin is shown schematically in Figure 1a. The repetition of the structural unit, made by the stacking of i) cytoplasmatic (cyt), ii) lipidic (lpg), iii) extracellular (ext) and iv) another lipidic (lpg) 27,32 layers, gives rise to the Bragg peaks well known in literature. 20,26,30,32 The XRD profiles with the indicated Bragg peaks of order 2, 3, 4 and 5, measured in 1 µm 2 area, of    Table 1).
Then, we calculated the Probability Density Function (PDF) of the same quantities (see Let us consider the thickness of the periodic structural unit, d l , (Fig. 2a, 2b). Its average value is found to be µ(d λ ) = 17.350 nm, in agreement with previous works. 26,30 Its spatial fluctuation, , results to be about 0.14%, quite smaller than the fluctuations for the other individual sublayers, having RSD between 1.1-1.7% as reported in Table 1. Here we meet the "paradox" of myelin quasi-crystallinity due to the smaller spatial fluctuation of period in comparison with the larger sublayers' fluctuations.  Table 2: Correlation coefficients c i-j between maps of the spatial fluctuations between the different layers i, j = cyt, ext and lpg in the fresh, unfresh and pH=5 samples. Negatively correlated coefficients are indicated by grey cells. These correlations are well explained by Figure 6 for each sample.
To shed light on this paradox, we have studied the correlations between the spatial fluctuations RSD lpg , RSD cyt and RSD ext . In order to do this, we first calculated the Pearson's correlation coefficients, c i-j , between maps of cytosolic apposition (Fig. 2c), extracellular apposition ( Fig. 2g), lipidic membrane ( Fig. 2e) (see Table 2). We observed that the spatial fluctuations RSD cyt are smaller than RSD ext but d cyt and d ext wave together in the same direction, being c cyt-ext positive. On the other hand, fluctuations RSD lpg go in the opposite direction since c lpg-ext and c cyt-lpg are negative.
This means that the lipid membrane fluctuations are strongly anti-correlated with both cytosolic and extracellular layer fluctuations. It is natural to associate this anti-correlation with the anti-correlated dynamics of the myelin proteins. 25,52,53,54,55 In order to describe the point-to-point spatial structural fluctuations in myelin we have introduced the hydrophilic-hydrophobic conformational parameter (HhCF) described by the ratio x between hydrophilic and hydrophobic layers: A typical map of x measured on the fresh samples is shown in Figure 3a. The red spots represent areas where hydrophilic layers are larger, while in the blue spots they become smaller and the thickness of hydrophobic layers increase. The fact that the x is always less than 0.9 demonstrates that myelin has a high lipid content. 20 The PDF of x shows a skewed line shape modelled by using Levy distributions 44 (see Figure 3b).
The Levy distributions provide a statistical description of complex signals deviating from normal behaviour and in recent years have found increasing interest in several applications in diverse fields. [45][46][47][48][49] The characteristic function of Levy probability distribution is defined by four parameters: stability index α, skewness parameter β, scale parameter γ taking into account the width of the distribution, and location parameter δ with varying ranges of 0<α≤2, -1≤β≤1, γ>0 and δ real. We  2) a state with acid pH (pH=5), named denatured, where the sciatic nerve has been left in an acid buffer solution where the system is expected to go toward degeneration. 35 The ultrastructural fluctuations of the unfresh and denatured samples are shown in Figure 4.
Typical maps and the PDF of the d l , d cyt , d lpg , d ext thicknesses of the unfresh nerve are shown respectively in Figure 4a, 4b, 4c and 4d.  Table 1). These narrower fluctuations are associated with the loss of correlated disorder, as can be seen by the decreasing of the Pearson's correlation coefficient for the unfresh nerve in Table 2. In the case of denaturation with pH shows bigger increasing of fluctuations (see RSD values in Table 1) Table 2. Thus, although the average period in the aged unfresh nerve is unchanged we find i) decreasing amplitude of fluctuations and ii) decreasing spatial correlations between fluctuations. The myelin frozen fluctuations support the hypothesis that the aged system has reached a more static state, approaching the thermodynamic equilibrium.
Let us now consider the denatured sample at pH=5. The maps and the relative PDFs of the d l , d cyt , d lpg , d ext thicknesses are shown in Figure 4e, 4f, 4g and 4h, respectively. The period PDF (Fig. 4e) shows a clear increase in both its mean value and in its standard deviation compared to the fresh sample. In this case µ(d l ) = 17.960 and σ(d l ) = 0.121 nm give spatial fluctuations RSD(d l ) = 0.68%, much larger than fluctuations in both fresh and unfresh states. This is an expected degradation sign of the morphology, indeed, as is commonly known, the degeneration leads to an increase in the period and later breaking of the biological membrane. 30,40 Alongside the morphologically changes, we observed also dramatic changes in the dynamics of the system. Upon inspection of Figure 4f, 4g and 4h we notice a huge increase of the range of spatial fluctuations for all sublayers, compared to the fresh sample. Indeed, now the RSD assume values including between 5.2-8.4%, (see Table 1). The PDFs assume now a non-analytic shape, sign of sample denaturation process. In particular, focusing our attention on cytosolic apposition PDF (Fig.   4f) we can see that there is a long PDF tail at low values of d cyt , which reaches the range of the minimum possible cyt thickness; 25 on the contrary a sharp drop of PDF is visible at high values of d cyt . In order to check the correlation between each sub-layer in the denatured sample, we direct our attention to Pearson's correlation coefficients, c i-j , between maps of cytosolic apposition (Fig. 4f), extracellular apposition (Fig. 4h), lipidic membrane (Fig. 4g). We notice a great enhancement of the coefficient c cyt-ext and a great decreasing of the coefficient c cyt-lpg , that shows the correlation and the anti-correlation between the cytosolic apposition and the other two sub-layers, in comparison with fresh state (see Table 2). At any rate, the bigger coefficient is the anti-correlation c ext-lpg , which is however just a little lower than the fresh state. Therefore, in general, the sign of coefficient c i,j between sublayers in each sample are maintained, suggesting that these correlations are a structural property more than functional. The structural anti-correlation occurs between d lpg and d ext . This is important because it means that the degradation leads to loss of fluctuations compensation between the myelin forming layers and to the initial dismantling of the layers, but not to the loss of anticorrelation between layers. Regarding the pH=5 sample, as for the individual sub-layers, the PDF of x does not follow any clear specific trends, showing the sign of disordering in the denaturation process. There is a substantial expansion of the distribution, associated with an increase in mean value and fluctuation around it. It results to be ξ = 0.9 ± 0.3, that shows a great denaturation because it allows values of ξ larger than 1, changing the lipid rich myelin structure in standard physiological form to the lipid poor myelin with a greater amount of hydrophobic part in degenerated state. Finally, the ultrastructural fluctuations in these two non-functional states can be summarized by visualizing the relative variations of the layers, Δd ext , Δd lpg , Δd cyt and the period, d λ , as a function of ξ. The changes in the aged sample are visualized in Figure 6a, where we have compared the relative variations in unfresh and in fresh sample. Despite the maintained stability of the period, the aged nerve is characterized by: i) decreasing fluctuations and ii) decreasing spatial correlations between fluctuations. This means that the aged system acquires rigidity and order, losing correlated disorder. This is quite intriguing since it tells us that in the living system the functionality is associated to the correlated disorder while the aged state shows the tendency towards a frozen-like state expected at equilibrium with major rigidity and order.
Upon inspection of Figure 6b, As a final point, we propose a brief discussion of our achievements. First of all, it is known that, the myelin sublayers are kept together by Van der Waals (induced dipole -induced dipole) forces which ensure the integrity and stability of myelin structure. 24 In particular, the fluctuations of each sub-layer are due to its protein composition. Indeed, the active dynamics of these proteins cause the various contractions and expansions of each sub-layers.
With aging and the denaturation with pH, the myelin loses this dynamic functional ability.
The results show a loss of protein dynamic fluctuations 25,53,54,55 in the unfresh sample, as expected, owing to the decreasing ATP content. In fact ATP provides the energy to the myelin, controlling its functionality. 57,58 For the denatured sample, the acid pH (pH=5) rebalances the Van der Waals interactions, which manifests in the variation of the sub-layers mean values (d cyt , d lpg , d ext ). We remark that the mean value of the hydrophobic layers remains almost constant, in comparison with the fresh state. On the contrary, the hydrophilic layers show large changes of their mean values: the cytosolic apposition narrows and the extracellular apposition increase, in agreement with its behaviour in the swollen nerve state, 34 as reported in Table 1.

Conclusions
We have obtained information on spatial nanoscale fluctuations of out-of-equilibrium PNS myelin ultrastructure of the sciatic nerve of frog Xenopus leavis using non-invasive SµXRD imaging. This The present results exploit the SµXRD technique to characterize supramolecular chemical structures in quasi stationary state out of equilibrium. Supramolecular assembly is a robust, rapid and spontaneous process that it is poorly understood, although it occurs widely in nature, since it takes place in multiple scales and involves very weak intermolecular interactions. The interests in modelling autonomous supramolecular assembly of materials is a vast challenging area in modern material science as well as in basic physical and chemical sciences. Indeed, wide varieties of supramolecular structures are possible depending on the nature of weak forces involved in complex materials made of components ranging from micro-to nano-structures. It is interesting to remark that particular spatial distributions of the supramolecular structure fluctuations have been predicted theoretically to promote quantum coherence at high temperatures. [16][17][18][19] The SµXRD jointly with the spatial statistics allows the investigation of Levy distribution in quasi stationary states out of equilibrium, 59

Sample preparation
The experimental methods were carried out in accordance with the approved guidelines. Adult female frogs (Xenopus Laevis; 12 cm length, 180-200 g weight, Xenopus express, France) were housed and euthanized at the Grenoble Institute of Neurosciences with kind cooperation of Dr Andre Popov. The local committee of Grenoble Institute of Neurosciences approved the animal experimental protocol. The frogs were individually transferred in water to a separate room for euthanasia that was carried out using a terminal dose of tricaine (MS222) by immersion, terminal anaesthesia was confirmed by the absence of reflexes. Death was ensured by decapitation. Two sciatic nerves were ligated with sterile silk sutures and extracted from both thighs of freshly sacrificed Xenopus frog at approximately the same proximal-distal level through a careful dissection of the thigh. After dissection, the sciatic nerves were equilibrated in culture medium at pH 7.3 for at least 3 hours at room temperature. The culture medium was a normal Ringer's solution, containing 115 mM NaCl, 2.9 mM KCl, 1.8 mM CaCl 2 , 5 mM HEPES (4-2-hydroxyethyl-1-piperazinyl-ethanesulfonic). Following equilibration, one of the freshly extracted nerves was immediately placed in a thin-walled quartz capillary, 1 mm diameter, sealed with wax and mounted perpendicular to the sample holder, for the SµXRD imaging measurements. In total we acquired 20301 X-ray diffraction patterns, for the fresh sample.
Another sciatic nerve, after dissection, was left in Petri dish and equilibrated in culture medium at pH 7.3 for 18 hours at room temperature. Following equilibration in the same described conditions, the nerve was prepared for a further SµXRD imaging session in the same day. In total we acquired 8989 X-ray diffraction patterns, for the unfresh sample.
The third sciatic nerves, after equilibration, were placed in a solution at pH 5 for 3 hours at room temperature. The culture acidic solution was an acetate buffer solution, made up by mixing 847 ml of 0.1 M acetic acid (CH 3 COOH) and 153 ml of 0.1 M sodium acetate tri-hydrate. Following equilibration, the nerve was placed in a thin-walled quartz capillary, sealed with wax and mounted on the sample holder for a third SµXRD imaging session. In total we acquired 10201 X-ray diffraction patterns, for the denatured with pH sample.

Experimental and data analysis
The Scanning micro X ray Diffraction measurements of myelin of frog's sciatic nerve were performed on the ID13 beamline 26,27 of the European Synchrotron Radiation Facility, ESRF, France. The source of the synchrotron radiation beam is a 18 mm period in vacuum undulator. The beam is first monochromatized by a liquid nitrogen cooled Si-111 double monochromator (DMC) and then is focused by a Kirkpatrick-Baez (KB) mirror system. This optics produces an energy Xray beam of l=12.6 KeV on a 1x1 µm 2 spot. The sample holder hosts the capillary-mounted nerve with the horizontal (y) and vertical (z) translation stages with 0.1 µm repeatability. The sample was scanned by using a step size of 5 µm in both y and z direction, in order to avoid a sample damaging and autocorrelation between measured points. A Fast Readout Low Noise (FReLoN) camera (1024x1024 pixels of 100x100µm 2 ) is placed at a distance of 565.0 mm from the sample to collect the 2-D diffraction pattern in transmission. Diffraction images were calibrated using silver behenate powder (AgC 22 H 43 O 2 ), which has a fundamental spacing of d 001 =58.38Å. We choose an exposure time of 300 ms for minimizing the radiation damage and keeping good photon counting statistics at the same time. 26,27 The crossed bundle is of approximately 50 myelinated axons. Therefore, the diffraction frames are an average of these axons. Considering the scale of our problem, this is an acceptable average.
We measured different regions of interest (ROIs) in the central part of the nerves around their axis to minimize the capillary geometry effect on the X-ray absorption. A typical 2-D diffraction pattern with the expected arc-rings corresponding to the Bragg diffraction orders h = 2, 3, 4, 5 is shown in where the phases were taken from literature: 30,32 ℎ = ℎ ?@ ℎ = ±| (ℎ)| since the nerve is a centrosymmetric structure, so we consider just the real terms of Fourier series.
The EDD obtained from the diffraction patterns measured at different sample positions in Fig. 1b, are shown in Figure 1c. From the differences between two adjacent maxima in the EDD profile the widths of the inter-membrane spaces at the cytoplasmic (d cyt ) and extracellular (d ext ) appositions and the thickness of the lipid bilayer (d lpg ) were obtained. From these we got the period of the structural unit 26,32 d l =2d lpg +d ext +d cyt . A scheme of the myelin ultrastructure is shown in Figure 1a. The extrapolation of EDD at each pixel of the ROI, was performed using a customized in-house developed code written in MatLab (Mathworks, Natick, MA).