Packing of Helices: Is Chirality the Highest Crystallographic Symmetry?

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 3 1 /3 2 or 6 1 /6 5 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 ](NbF 6) 2 and [Zn(2,2'-bpy) 3 ](TaF 6) 2) in which the 3 n helices pack into a chiral structure.


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 3 1 helices in the P3 1 21 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]).

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).
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]).

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 3n helices in hexagonal arrangement (Figure 1a), 6n helices in hexagonal arrangement (Figure 1b), and 4n helices in tetragonal arrangement (Figure 1c,d).The red arrows and black triangles represent the handedness of helices and the helicates, respectively.In order to be symmetrically equivalent, the helicates must have the same environment.The red arrows and black triangles represent the handedness of helices and the helicates, respectively.In order to be symmetrically equivalent, the helicates must have the same environment.
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 4 1 and 4 3 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.
Crystals 2016, 6, 106 3 of 6 In a hexagonal arrangement of 3n or 6n 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, 4n 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, 3n screw axes are only included in chiral space groups (Figure 2a).It results that the 3n helices in hexagonal arrangement reach the highest symmetry only when crystallizing in a chiral space group.In the case of the hexagonal arrangement of 6n helices (Figure 2b), the highest symmetry can also be only reached when crystallizing in a chiral space group.However, for the tetragonal arrangement of 4n 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.

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 centroor 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.

Figure 1 .
Figure 1.Representation of examples of packings for (a) a chiral assembly of 3n helices in hexagonal arrangement, (b) a chiral assembly of 6n helices in hexagonal arrangement, (c) a chiral assembly of 4n helices in tetragonal arrangement, (d) a non-chiral assembly of 4n helices in tetragonal arrangement.The red arrows and black triangles represent the handedness of helices and the helicates, respectively.In order to be symmetrically equivalent, the helicates must have the same environment.

Figure 1 .
Figure 1.Representation of examples of packings for (a) a chiral assembly of 3 n helices in hexagonal arrangement, (b) a chiral assembly of 6 n helices in hexagonal arrangement, (c) a chiral assembly of 4 n helices in tetragonal arrangement, (d) a non-chiral assembly of 4 n helices in tetragonal arrangement.The red arrows and black triangles represent the handedness of helices and the helicates, respectively.In order to be symmetrically equivalent, the helicates must have the same environment.

Figure 2 .
Figure 2. List of space groups exhibiting (a) 3n screw axes in hexagonal arrangement, (b) 6n screw axes in hexagonal arrangement and, (c) 4n screw axes in tetragonal arrangement.Space groups in red color are chiral.Space groups in bold are polar.No space groups combining 31 and 32 screw axes or 61 and 65 screw axes exist.

Figure 2 .
Figure 2. List of space groups exhibiting (a) 3 n screw axes in hexagonal arrangement, (b) 6 n screw axes in hexagonal arrangement and, (c) 4 n screw axes in tetragonal arrangement.Space groups in red color are chiral.Space groups in bold are polar.No space groups combining 3 1 and 3 2 screw axes or 6 1 and 6 5 screw axes exist.
Figure S1: View along c and b of the structure with the relative orientation of the distortion represented by the shortest Nb-F bond.Along a and b axes, the individual dipole moments are canceled by the 3 n screw axis along c.The alignment of the individual polarizations is collinear to the polar axis of the structure.