Packing of Helices: Is Chirality the Highest Crystallographic Symmetry?

Abstract

Chiral structures resulting from the packing of helices are common in biological and synthetic materials. Herein, we analyze the noncentrosymmetry (NCS) in such systems using crystallographic considerations. A comparison of the chiral structures built from helices shows that the chirality can be expected for specific building units such as 3₁/3₂ or 6₁/6₅ helices which, in hexagonal arrangement, will more likely lead to a chiral resolution. In these two systems, we show that the highest crystallographic symmetry (i.e., the symmetry which can describe the crystal structure from the smallest assymetric unit) is chiral. As an illustration, we present the synthesis of two materials ([Zn(2,2’-bpy)₃](NbF₆)₂ and [Zn(2,2’-bpy)₃](TaF₆)₂) in which the 3_n helices pack into a chiral structure.

1. Introduction

Acentric basic building units (BBUs) provide interesting blocks to target noncentrosymmetric (NCS) structures exhibiting potential physical properties, for example piezoelectricity, optical activity, or ferroelectricity. However, one must be able to predict the packing of these BBUs into the 3D acentric structure. Two packings are usually described in the literature: (i) the packing of chiral units of one-handedness into chiral materials and (ii) the alignment of polar units into polar materials [1,2,3,4,5,6,7,8]. As a next step towards the engineering of NCS, it is important to explore the general concepts which drive the BBUs to pack in one of these structural arrangements.

In the solid state, the components tend to crystallize in the highest possible crystallographic symmetry. Thus, about 92% of crystal structures contain only one crystallographically independent molecule or formula unit [9,10]. In most cases, the presence of an inversion center is required to describe the highest crystallographic symmetry. However, systems where the highest symmetry is reached only without an inversion center can also be investigated to understand and predict the noncentrosymmetric arrangements. In this context, the helical systems are of interest. Chiral structures built from helices are common in either biological or synthetic materials [11,12,13]. To describe the packing of helical proteins in biology, Chothia et al. proposed the “ridges and grooves” model in which the proteins tend to pack in the way that the ridges and grooves on the surface intercalate [14]. Using a similar model, Boncheva et al. described the self-assembly of millimeter-sized helical drill bits and bolts into 3D aggregates via capillary forces [15]. These authors showed that, in the case of a high degree of shape complementarity, the components tend to aggregate into chiral structures with enantiomerically pure helices. The shape compatibility depends on the periodicity, handedness and arrangement of the helices and is also a general concept which can be extended to analyze the NCS in the solid state. In organic and inorganic solid state, the interactions between helices are also symmetry-dependent. Thus, Meille et al. previously reported that helical polymers with a hexagonal arrangement have a high probability to crystallize into a chiral space group [16]. They reported that it is not possible to pack two helices of opposite chirality in the hexagonal arrangement and keep them symmetrically equivalent. In parallel to this work, we reported a similar result in hybrid materials. We previously showed that 3₁ helices in the P3₁21 space group (i.e., in hexagonal arrangement) tend to pack with helices of the same handedness because they share identical spatial and translational orientation. From the investigation of these compounds, we emphasized that the packing of helices with opposite handedness does not allow equivalent bridging of the helices. Thus, the packing of helices is also dependent on their symmetries but independent of the system (millimeter-sized helical drill bits, proteins, helical polymers, organic or inorganic structure) or the nature of interactions (capillary forces [15], close-packing of proteins [14,17], covalent [18], or entropic [16]).

2. Results and Discussion

In helical systems, one can determine the arrangements in which every helicate (the smallest repetitive unit of a helix) can crystallize with a similar environment and which are also symmetrically equivalent. In such conditions, the helices can share identical spatial and translational orientations. Symmetry patterns can be drawn by considering different configurations for 3_n helices in hexagonal arrangement (Figure 1a), 6_n helices in hexagonal arrangement (Figure 1b), and 4_n helices in tetragonal arrangement (Figure 1c,d).

In a hexagonal arrangement of 3_n or 6_n helices, the helicates can have the same environment only if the helices are of the same-handednesses (i.e., in a chiral structure) (Figure 1a,b). For every other case in which one would consider the packing of helices of both-handednesses, the helicates would have different environments. Conversely, 4_n helices in a tetragonal arrangement can either crystallize in chiral (Figure 1c) or non-chiral structure (Figure 1d) in order for the helicate to have the same environment.

This observation from the symmetry patterns could also be generalized from group theory. To reach the highest possible crystallographic symmetry in the solid state, the smallest repetitive unit of the helices should also be (if possible) the repetitive unit of the overall arrangement. In other words, the 3D arrangement should be described only from the helicates together with symmetry operations.

Thus, one can identify the space groups in which two conditions are satisfied: (i) the presence of screw axes along which the helix could be described in one direction from a unique repetitive unit, and (ii) the presence of symmetry operations to relate the screw axes to each other along the other two directions. If these conditions are satisfied, the helix can pack in this space group and reach the highest symmetry.

Figure 2 lists the different space groups which include the screw axes related by symmetry [19]. For example, 3_n screw axes are only included in chiral space groups (Figure 2a). It results that the 3_n helices in hexagonal arrangement reach the highest symmetry only when crystallizing in a chiral space group. In the case of the hexagonal arrangement of 6_n helices (Figure 2b), the highest symmetry can also be only reached when crystallizing in a chiral space group. However, for the tetragonal arrangement of 4_n helices (Figure 2c), the resulting structure can either be chiral or non-chiral to reach the highest crystallographic symmetry because 4₁ and 4₃ can be related by symmetry in this arrangement.

This analysis shows that the hexagonal arrangement of helices in the solid state differs from the tetragonal one. The highest symmetry of a hexagonal arrangement of either 3_n or 6_n helices is only reached when the structure is chiral (i.e., when a chiral resolution occurs), whereas the tetragonal arrangement of 4_n helices can reach the highest symmetry when the structure is either centrosymmetric or noncentrosymmetric.

Two new compounds, [Zn(2,2’-bpy)₃](MF₆)₂ (M = Nb⁵⁺ (I) and Ta⁵⁺ (II)), isostructural to the previously reported [A(2,2’-bpy)₃](PF₆)₂ (A = Mn, Ru, Ni, Cu, Zn and Cd), allow us to validate this investigation (Experimental Details in Supplementary Information) [20,21,22]. Chiral metal trischelates [A(2,2’-bpy)₃]²⁺ are described as a pair of enantiomers Δ and Λ-[M(ligand)₃]ⁿ⁺ with the symmetry D₃ (Figure 3). In the structures of I–II and previously reported isostructural compounds, these units are the helicates of helices propagating along the c axis. Thus, π-stacking occurs between the 2,2’-bipyridyne ligands of the helicates. Such interactions are common in organometallic or DNA helices [23,24,25]. The handedness of the helices is induced by the intrinsic character of helicates: the Δ enantiomer leads to the 3₁ helices while the Λ enantiomer leads to the 3₂ helices. [26]. Thus, two enantiomorphic crystal forms (P3₁ and P3₂) are induced by the chiral resolution of Δ and Λ-[Zn(2,2’-bipy)₃]²⁺, respectively. Thus, in order to reach the highest symmetry, the 3₁/3₂ helices in hexagonal arrangement crystallize in the chiral space group P3₁/P3₂. The structure determination carried out on several crystals showed the presence of both configurations in equal amount.

3. Conclusions

In the solid state, most of the materials crystallize with the highest possible symmetry with a single chemical unit per asymmetric unit. The identification of systems in which the highest symmetry is polar or chiral could also be of great interest for the engineering of NCS materials. Thus, one could determine which characteristics a basic building unit must exhibit to target the NCS in the solid state. Our investigation of helical systems shows that 3_n or 6_n helices in hexagonal arrangement likely crystallize into a NCS agglomerate because the packing reaches the highest symmetry when the inversion center is excluded. Conversely, 4_n helices can reach the highest symmetry in either a centro- or noncentrosymmetric tetragonal packing. This new insight on the relationships between NCS and highest symmetry could be extended to other helical systems and could improve our understanding and control of the engineering of chirality in the solid state.