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Lattice models in biophysics represent a class of statistical-mechanical models which consider a biological macromacromolecule (such as DNA, protein, actin, etc.) as a lattice of units, each unit being in different states or conformations.
In computational and mathematical biology, a biological lattice-gas cellular automaton (BIO-LGCA) is a discrete model for moving and interacting biological agents, [1] a type of cellular automaton. The BIO-LGCA is based on the lattice-gas cellular automaton (LGCA) model used in fluid dynamics.
A primitive cell is a unit cell that contains exactly one lattice point. For unit cells generally, lattice points that are shared by n cells are counted as 1 / n of the lattice points contained in each of those cells; so for example a primitive unit cell in three dimensions which has lattice points only at its eight vertices is considered to contain 1 / 8 of each of them. [3]
In differential geometry, a triply periodic minimal surface (TPMS) is a minimal surface in that is invariant under a rank-3 lattice of translations. These surfaces have the symmetries of a crystallographic group. Numerous examples are known with cubic, tetragonal, rhombohedral, and orthorhombic symmetries.
The conceptual foundation for DNA nanotechnology was first laid out by Nadrian Seeman in the early 1980s. [2] Seeman's original motivation was to create a three-dimensional DNA lattice for orienting other large molecules, which would simplify their crystallographic study by eliminating the difficult process of obtaining pure crystals.
The history of aperiodic crystals can be traced back to the early 20th century, when the science of X-ray crystallography was in its infancy. At that time, it was generally accepted that the ground state of matter was always an ideal crystal with three-dimensional space group symmetry, or lattice periodicity.
The Bogoliubov inequality, shown above, can be used to find the dynamics of a mean field model of the two-dimensional Ising lattice. A magnetisation function can be calculated from the resultant approximate free energy. [9] The first step is choosing a more tractable approximation of the true Hamiltonian.
The more precise mathematical definition is that there is never translational symmetry in more than n – 1 linearly independent directions, where n is the dimension of the space filled, e.g., the three-dimensional tiling displayed in a quasicrystal may have translational symmetry in two directions.