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Correlated and Decorrelated Positional and Orientational Order in the Nucleosomal Core Particle Mesophases

We investigate the orientational order of transverse polarization vectors of long columns of nucleosomal core particles and their coupling to positional order in high density mesophases discovered recently. Inhomogeneous polar ordering of these columns precipitates crystalization of the 2D sections with different orientations of the transverse polarization vector on each column in the unit cell. We propose possible scenarios for going from the 2D hexagonal phase into distorted lamellar and related phases observed experimentally.

-histone protein complexes, that make a fundamental unit of chromatin: the nucleosome [1]. A nucleosome is composed of a 147 bp long DNA fragment, wrapped 1.75 times around the histone octamer core composed of four different histone proteins and a DNA linker that connects two consecutive nucleosomes [2]. Digestion of the linker DNA between different nucleosomes along the eucaryotic genome gives rise to the nucleosome core particle (NCP). NCP has an approximately cylindrical shape of height and radius ∼ 55Å, and a structural charge of ∼ -165. This complex is stable in aqueous solutions from mM to 750 mM monovalent salt ionic strength [3].
At high enough concentration the solution of NCPs ceases to be isotropic. Competing, as yet poorly understood [4,5], microscopic interactions between NCPs in concentrated solutions give rise to a slew of liquid crystalline phases of fascinating complexity [6]. At low salt concentrations and sufficiently high osmotic pressure NCPs aggregate into columns of almost ideal cylindrical shape which contain dozens of individual particles. These columns then aggregate to form a lamellar phase [7]. At higher salt concentrations this phase gives way to a 2D hexagonal columnar liquid crystalline phase leading at still higher salt concentrations to a 3D hexagonal crystal [8].
The high asymmetry of the NCP particle, with a pronounced dyadic axis marking the entry-exit point of the wrapped nucleosomal DNA, allows one to actually observe the local orientational order of the NCPs, described by its in-plane dyadic axis vector, conventionally called transverse polarization [9]. In the lamellar phase the dyadic axes of individual NCPs are strongly correlated along the column leading to a non-zero average transverse polarization vector P of a column. Cryomicroscopy studies [7] give strong evidence that the dyadic axes, and consequently the vectors P, of columns in the two layers of a lamella are in an antiparallel direction. In addition, the columns in the two layers are shifted to a small but constant distance in such a way that the antiparallel dyadic axes form a periodic polar vector field (Fig. 1). SAXS refine this picture [8] by showing that there are strong correlations between the NCP columns within a single lamella but different lamellae appear to be completely decorrelated (Fig. 1). In this sense the lamellar phase is decorrelated at a macroscopic scale and corresponds to simple smectic ordering. 2D hexagonal phase is characterized, in turn, by a positional order in the plane perpendicular to the columns, but dyadic axes of NCPs are orientationally disordered along the columns (Fig. 1). The corresponding average polar vector of the column is zero.
In what follows we will investigate the nature of positional and orienta-tional order in phases that are induced by a condensation of the periodic antiparallel transverse polarization field modes. Basing our assumptions on experiments we postulate that the transverse polarization order in a lamella described with P(ρ), where ρ = (x, y), is periodic and highly anisotropic. We will show that the transition between the 2D colloidal crystal with disordered dyadic axes and the smectic-like lamellar phase with the in-plane periodic polar order can be understood by introducing a common subphase substructure for both phases. The common subphase is a 2D colloidal crystal with lamellar structure and with the in-plane polar order of individual lamellae similar to that in the actual lamellar phase. However, the correlations between the positions of columns and the orientations of their dyadic axes from one lamella to another make this phase crystalline and different with respect to the smectic-like structure. The transformation from the hexagonal colloidal crystal to the smectic lamellar phase can take place as a sequence of two second-order transitions through the intermediate crystal lamellar phase, or as a direct first-order transition. Order parameters of both transformations can be easily expressed in terms of the periodic transverse polarization fields.
If the transverse polarization vector field condensation starts from a 2D hexagonal phase then in general one would have several different choices for the direction of the wave vector k with respect to the symmetry planes of the 2D hexagonal phase. In the hexagonal coordinate system the wave vector of the condensation into the crystal lamellar phase has to be described in Kovalev's notation [10] by k 12 = 1 2 b 1 , where b 1 is one of the reciprocal basis vectors. The irreducible star of the k 12 vector has three arms given by k 1 = k (1) . The basis functions of a polar vector field with this periodicity are given by ψ 1 = (2ê x +ê y ) e ik 1 ·r , ψ 2 = (ê x + 2ê y ) e ik 2 ·r and ψ 3 = (−ê x +ê y ) e ik 3 ·r . The physical realisation of the order parameter is given by the symmetry coordinates composed of the column's polar vectors P i . Denoting by i the index of the column, in such a way that its (x, y) hexagonal unit cell and the 2D polar vector components by (P x , P y ), one gets the following components of the order parameter The Landau free energy of this system is now given via the orientation probability density function ρ(x, y, z) For the sake of clarity, we have omitted in Eq. 2 the terms stemming from the large-scale elastic deformations of this order. Consequently, the free energy of the system can be considered as a simple function of the order parameter components. Let us also stress an evident duality between the polar vector carried by a NCP column and the polar vector of its displacement in the solvent which has important consequencies for the structure of its ordered phases. Since the displacements u i of NCP columns and the polar vectors P i of the dyadic axes span the same irreducible representation of the symmetry group of the 2D hexagonal phase the components of the order parameter can be expressed in the same form in terms of the columns' displacements (u xi , u yi ). Minimization of the free energy dependent on both P i and u i shows that they are proportional. This fact permits one to obtain also the positional order and thus also the final structures of the phases.
To establish the exaustive list of the phases induced by the considered condensation mechanism we use the fact that F (η i ) has to be an invariant function of the irreducible representation spanned by the periodic transverse polarization vector field. Since the basis invariants of this represenatation are I 1 = η 2 1 + η 2 2 + η 2 3 , I 2 = η 2 1 η 2 2 + η 2 2 η 2 3 + η 2 3 η 2 1 and I 3 = η 2 1 η 2 2 η 2 3 , the free energy Eq. 2 has to have the form F = F (I 1 , I 2 , I 3 ). The minimization of this free energy w.r.t η i gives the equation of state. There are six different solutions of this equation of state which we list below in terms of η i , the unit cell multiplication in the ordered state with respect to the 2D hexagonal phase i.e. V ord /V hex and their space groups. Let us show that the first solution, i.e. η 1 = 0, η 2,3 = 0 describes the crystal lamellar state. From Eq. 1 it follows that in this case P 1 = P 3 and P 2 = P 4 , or in other words, that the orientations of dyadic axes carried by columns 1 and 3 or columns 2 and 4 are parellel and have the same degree of correlation along the column. Thus, due to the duality, the displacements of the columns 1 and 3 as well as 2 and 4 are also identical. This means that the periodic transverse polarization field induces a lamellar structure, where a single lamella is defined by the columns 1, 3, 1, 3... for the top part and 2, 4, 2, 4... for the bottom part of a single lamella. The order parameter in this crystalline lamellar phase induced by the periodic transverse polarization vector ordering is given by The simplest phenomenological Landau free energy expansion of an isolated transition from a hexagonal 2D crystal η i = (0, 0, 0) to a lamellar 2D crystal with η i = (η, 0, 0), with basis invariants I 1 = η 2 , I 2 = η 4 and I 3 = η 6 can be written as This free energy describes a second order transition between a hexagonal and a lamellar crystalline phases, driven trivially by the parameter a 1 , that can depend on various solution conditions, e.g. the salt concentration. The same periodic transverse polarization vector field can induce also more complicated transitions. Assuming now a general form of the order parameter we can start with the following free energy ansatz The phase diagram in this case is presented in Fig. 2. One can observe that in addition to the lamellar phase, stable for b 1 > 0, the diagram shows the (η, η, 0) phase, to be discussed below, stable for b 1 < 0. The growth of the complexity of the phase diagram as more non-linear terms are included into the Landau free energy is clearly apparent. E.g. if one starts from the free energy this leads to the form of the phase diagram presented on Fig. 2, where an additional phase, this time (η 1 , η 2 , 0), intercalates between the (η, η, 0) and the (η, 0, 0) phases. The structure of other phases listed in Table 1, induced by the periodic ordering of the transverse polarization vector field obtained in highly nonlinear free energy models can be quite complicated. We have already analysed the lamellar phase, beside this we also have: 1. (η, η, η). The corresponding vector field is given in this phase by the vectors of the different columns in the form : The vectors of orientation (and, consequently, the displacements) of the columns are expressed in terms of one polar vector (P x , P y ). Note that, because of the specific interaction between the columns in this phase, column 2 is frustrated : its average polar vector is zero. Consequently, its displacement with respect to the position in the 2D hexagonal phase is zero and the dyadic axes of individual NCPs are disordered along this column. All three components of the order parameter are equal to the following combination of P i vectors : η = P x1 + 2P y1 + P x3 − P y3 − 2P x4 − P y4 . Analyzing the displacement field in this phase, see Fig. 2, it is obvious that the columns of a hexagonal phase are now displaced to make an inverted hexagonal phase, observed also in experiments [6], whose vertices are given by the columns 3 ′ , 2, 1 ′ , 2, 4 ′ and 2.
2. (η, η, 0). The corresponding vector displacement fields in this phase are given by Both non-zero components of the order parameter in this phase are equal to η = 3 2 (P x1 + P y1 ) + 1 2 (−P x2 + P y2 ) + 1 2 (P x3 − P y3 ) − 3 2 (P x4 + P y4 ). The columns of a hexagonal phase undergoing the orientational and displacement fields, see Fig. 2, form now an orthorhombic phase with a large unit cell. It can be understood as a succession of two series of columns undergoing 1D displacement waves : in the first series the wave is transeverse and in the second one it is longitudinal. Note that the point symmetry of this structure shows only slight deviation from the hexagonal symmetry. 3. Along the same line the vector fields and the structures of the lowsymmetry phases (η 1 , η 2 , 0), (η 1 , η 1 , η 2 ) and (η 1 , η 2 , η 3 ) can be obtained.
We have shown how a 2D ordering of a transverse vector polarization field can induce a transition into a lamellar crystalline phase. The experimentally observed lamellar phase is however not of this type and is basically equivalent to a smectic phase. The difference is subtle and referrs to the fact that in SAXS in addition to the smectic peaks only reflections with two non-zero indices from the NCPs correlated along the column and within a single lamella are present. No reflection with three non-zero indices which can evidence the correlation between different lamellae is observed. The lamellae are thus decorrelated and the 3D long range order of the transverse polarization vector is destroyed. This means that the macroscopic symmetry of this phase is higher than that of the crystal lamellar phase with η i = (η, 0, 0) and equivalent to the symmetry of a usual lamellar smectic phase (G = D ∞h ⊗ T z d ). One can speculate that this decorrelation is related to the difference in correlation lengths of microscopic interactions which insure the stability of NCP mesophases on the one hand and their colloidal character on the other hand. Indeed, the displacements of columns in a 2D colloidal crystal are much greater than typical values of atomic shifts in a close-packed hexagonal crystal. Resulting distance d L between the lamellae in our case can become greater than the correlation length of the interaction between the NCPs but still smaller then the correlation length of the smectic density wave ξ ρ(z) . Since the symmetry of the crystal lamellar phase is lower Figure 2: Displacement fields for the transition from the hexagonal to lamellar (η, 0, 0) (columns forming a single lamella are enclosed within a gray rectangle), orthorhombic (η, η, 0) and inverted hexagonal (η, η, η) phases. Below, the phase digram stemming from Eqs. 5, 6.
than that of the actual lamellar smectic phase, this decorrelation transition must be accompanied by vanishing of a physical quantity responsible for this structural change. In other words the order parameter of the lamellar correlation transition can be defined. Group-subgroup relationship between these two lamellar phases permits one to show that the physical quantity vanishing through decorrelation transition is the macroscopic transverse polarization vector wave with the wave vector parallel to the smectic layer and the polarization directed perpendicular to the single lamella (1D transverse vector wave). Condensation of this wave leads directly from the smectic-like decorrelated structure to a correlated crystal lamellar phase of the η i = (η, 0, 0) type, with periodic distribution of transverse polarization in columns in each layer. The order parameter of the transition is given by the first harmonic of the wave i.e. u z (k)ê z exp(iky) + u z (−k)ê z exp(−iky), where u z (k) is the amplitude of this harmonic and k is the wave vector perpendicular to the direction of the columns within a lamella, with the magnitude k = 2π a , where a is the distance betweem neighboring columns within a single lamella.
These polarization waves exist localy for each lamella, but the distribution of the origins of these waves in each lamella is random for the whole sample and consequently the average macroscopic value of the wave is zero in the decorrelated phase (Fig. 1). Similar scenario but in a different context was first proposed by P. Bak [11]. The order-disorder character of the transition and the wave nature of the order parameter were clarified. It was also shown that the wave vector of the macroscopic order parameter for the transitions of this type is equal to the wave vector of the local waves (wave vector of an atomic chain in [11] or of the polar vector wave in an individual lamella in our case). The transition from the decorrelated smectic phase into the correlated η i = (η, 0, 0) type phase is described trivially via the Lifshitz free energy [12] F = α 1 |u z (k)| 2 + α 2 |u z (k)| 4 + . . . and is thus of second order.
In this correlated-decorrelated transition scenario the crystalline lamellar phase is a common subphase of both the hexagonal crystalline phase as well as the smectic lamellar phase. The hexagonal crystal −→ crystal lamellar phase transition is driven by condensation of the 2D periodic transverse polarization field, whereas the smectic lamellar −→ crystal lamellar phase transition is driven by the condensation of the 1D polar vector wave and is characterised by the correlation between the transverse polarization vector distributions in each individual lamella.
Let us finally discuss the influence of chirality on the state of NCP columns. Each NCP particle being chiral their aggregation results in a chiral column with the rotational symmetry of a cylinder without neither mirror planes nor inversion, and the periodicity approximately equal to the indi-vidual NCP height. According to the Curie principle ordering of the polar vector P i along a chiral column leads to a simultaneous ordering of the axial vector t i , since both P i and t i span the same representation of the column symmetry group. In a chiral periodic aggregate the axial vector has an evident physical interpretation. Non-zero average value of t i corresponds to the correlated tilt of individual NCPs with respect to the column axis (Fig. 1). In the 2D hexagonal phase the NCP columns carry zero average polar vector, then in a column the tilt of the particles is also zero : the axes of individual particles (usually called S-axes) are parallel to the column axis. On the contrary, in the lamellar phase the ordering of the polar dyadic axes induces a tilt of the NCPs S-axes. This simple consequence of the NCP chirality can explain peculiarities of the SAXS spectrum observed in [8].
One of the authors (R.P.) would like to thank F. Livolant and her collaborators for their hospitality while being a Chercheur Associé fellow of C.N.R.S at Orsay in 2002.