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Asymmetry born out of symmetry

Chiral lattices

Two snub hexagonal tilings that were generated in a 2D simulation of point particles interacting via an isotropic potential. The patterns are chiral and are mirror images (plus a rotation) of each other. (Courtesy: APS)

By Hamish Johnston

The chirality – or handedness – of many biological molecules plays an important role in their function. The amino acids that make up proteins only exist in the left-handed form, for example, while the sugars found in DNA are exclusively right-handed.

Why nature seems to favour one handedness over another has long puzzled physicists – particularly because the relevant physical laws that govern the synthesis of such molecules are symmetric and should not be biased towards right- or left-handedness.

The emergence of molecules with a specific chirality in a chemical process is usually understood in terms of chiral-specific catalysis, which accelerates the production of molecules of one handedness over the other. However, it’s also possible that chirality can emerge in much simpler systems that don’t involve complicated chemical reactions.

In order to understand how chirality emerges from symmetrical interactions, Martin Nilsson Jacobi and colleagues at Chalmers University in Sweden have done computer simulations that reveal how point particles acting under a spherically symmetric force can form chiral patterns in 2D. According to the team, the system begins with “maximal a priori symmetry” and therefore the emergence of asymmetric chiral patterns is surprising.

The team began with what it describes as the simplest form of chiral lattice in 2D. This is made from identical scalene triangles – a triangle with no sides of equal length. Such a lattice can be made in two ways, each being a mirror image of the other. However, one lattice cannot be transformed into the other by rotation or translation.

Nilsson Jacobi and colleagues first calculated the Fourier transform of the lattice, which gives its reciprocal lattice. Then, using a technique introduced by the team last year, they were able calculate a potential energy between pairs of lattice points that would result in the creation of the desired chiral lattice. The amazing thing about this potential is that it is spherically symmetric – looking a bit like a 1/r potential with a number of wiggles in it.

To confirm that the potential would indeed result in a chiral structure, the team then used a Monte Carlo simulation to determine what lattice would form if point particles were subject to such a potential. The resulting lattice was indeed a chiral pattern of scalene triangles.

The team then set its sights on a more complicated – and visually appealing – 2D chiral lattice called “snub hexagonal tiling” (see images above). Again, the chiral pattern emerged from the simulation.

While the team has shown that in principle chiral patterns can emerge from simple symmetric systems, this could prove to be very difficult to achieve in a real system. The problem is that the required potentials would be very difficult to recreate in a real-life system and “are not likely to appear in the near future,” according to the physicists.

The simulations are described in this paper in Physical Review Letters.

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